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+SciPost Physics
+Submission
+Anomalies, Representations, and Self-Supervision
+Barry M. Dillon, Luigi Favaro, Friedrich Feiden, Tanmoy Modak, Tilman Plehn
+Institut für Theoretische Physik, Universität Heidelberg, Germany
+January 13, 2023
+Abstract
+We develop a self-supervised method for density-based anomaly detection using contrastive
+learning, and test it using event-level anomaly data from CMS ADC2021. The Anomaly-
+CLR technique is data-driven and uses augmentations of the background data to mimic
+non-Standard-Model events in a model-agnostic way. It uses a permutation-invariant Trans-
+former Encoder architecture to map the objects measured in a collider event to the represen-
+tation space, where the data augmentations define a representation space which is sensitive
+to potential anomalous features. An AutoEncoder trained on background representations
+then computes anomaly scores for a variety of signals in the representation space. With
+AnomalyCLR we find significant improvements on performance metrics for all signals when
+compared to the raw data baseline.
+Contents
+1
+Introduction
+2
+2
+Dataset
+4
+3
+AnomalyCLR
+5
+3.1
+Contrastive learning
+5
+3.2
+CLR for anomaly detection
+6
+4
+Application to event-level anomalies
+8
+5
+Anomaly scores
+10
+6
+Results
+11
+6.1
+Comparison of methods
+11
+6.2
+The effect of anomaly-augmentations
+12
+6.3
+The effect of representation dimension
+13
+7
+Summary & conclusions
+14
+References
+15
+1
+arXiv:2301.04660v1  [hep-ph]  11 Jan 2023
+
+SciPost Physics
+Submission
+1
+Introduction
+Model-agnostic new physics searches are one of the most interesting analysis prospects for the
+LHC and other colliders. Over the past decade the LHC has searched for new physics based on
+model-specific hypothesis testing. Despite these efforts there has been no strong evidence of new
+physics found. It is possible that new physics does exist at the scales probed by the LHC, and has
+not been uncovered due to the particular signal not being covered by previous analysis hypotheses.
+The ATLAS and CMS collaborations have both implemented model-agnostic new physics searches
+to deal with this [1, 2], however these methods suffer some drawbacks. For example scanning
+high-dimensional parameter spaces can lead to large look-elsewhere effects, or methods can lack
+the ability to make full use of the high-granularity low-level information collected in the experi-
+ments. Recent progress in machine learning based high-energy physics tools are making significant
+advances in solving many problems of such classical methods [3].
+The main machine learning tools to date for data-driven model-agnostic searches are based
+either on density-related scores, or on classification scores using a background-dominated control
+sample. The latter, typically known as CWoLa methods (Classification Without Labels) [4–6] have
+been shown to be very successful in applications such as bump hunting [7–14] and semi-visible jet
+searches [15], providing both anomaly scores and background estimates. However they run into
+difficulty when the dimension of the input space or number of observables becomes large, and so
+the question of whether or not they can be used on low level data is still uncertain. CWoLa tools
+have already been adopted by the ATLAS collaboration [16].
+Density-based methods use machine learning to estimate the density in the phase space, and
+then identify anomalies as those laying in the low density regions. These tools typically work
+on high-dimensional inputs and so can be used on low-level data. The first density-based meth-
+ods were the AutoEncoder studies [17, 18], where the network is optimised to compress and
+reconstruct the kinematics of a jet or event. While this is not strictly density estimation, the
+optimisation is highly aligned with learning a density, since regions of the phase space which
+are most populated are those which should be reconstructed the best and thus have the low-
+est anomaly score. There has been significant progress with the AutoEncoder tools and other
+density-based anomaly detection methods in recent years [19–33], with studies covering inter-
+pretability of AutoEncoders [34, 35], topic modelling [36, 37], null hypothesis tests for anomaly
+detection [38], ABCD methods [39], the Normalised AutoEncoder (NAE) [40], and normalising
+flow techniques [41–44]. For a comprehensive summary of many different anomaly detection
+methods we refer the reader to the community challenge papers in Refs [45,46].
+One issue with the density-based approaches [44,47] is that the score is not invariant under
+simple transformations in the phase space. This means that a simple re-mapping of the momenta
+or coordinates fundamentally changes what the anomaly score is. This poses the question of how
+to choose a representation of the data for use in density-based anomaly detection tasks. It is
+also worth noting that despite the great progress that more sophisticated neural network archi-
+tectures and the implementation of symmetries in networks has brought to supervised classifi-
+cation [48–51], they have not yet led to the same progress in anomaly detection. In this work
+we develop a new approach to density-based anomaly detection using self-supervision, which de-
+fines the representation of the data in a model-agnostic way using the power of highly expressive
+networks such as transformers or graph networks to boost anomaly detection performance.
+Supervised machine learning methods use the idea of a truth-label to optimise the neural net-
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+works, usually to classify between data with different truth labels. Unsupervised methods are
+those which do not require truth labels, instead optimising a network using a reconstruction loss
+or a negative log likelihood, for example. Self-supervision on the other hand uses ‘pseudo-labels’,
+labels generated from the data without knowledge of a truth label, to optimise the networks. In
+contrastive learning [52], these labels correspond to a link between an original event and an aug-
+mented event. We define the augmentation as some physical modification of the event kinematics.
+Contrastive learning uses the pseudo-labels to devise an auxiliary task for the network optimisation
+through the contrastive loss function. Now the network learns how to process high-dimensional
+correlations in the data, and thus the representations learned by these networks can be very useful
+for downstream tasks. We introduced the self-supervised JetCLR method in [53] and demonstrated
+its ability to construct highly expressive representations for classification tasks. In [54] this same
+technique was used to construct representations for CWoLa-based anomaly detection. In addition
+to these works, other self-supervised / representation learning techniques have been applied in
+particle physics [55,56] and in other scientific disciplines such as astrophysics [57–60]. In [53,54]
+the augmentations corresponded to transformations of the event to which the underlying physics
+should be invariant to rotations or translations, but also soft-collinear parton splittings.
+We introduce AnomalyCLR, a new method based on the idea of ‘anomaly-augmentations’.
+These anomaly-augmentations are modifications of the original event to which the underlying
+physics is not invariant. In fact these augmentations are chosen to mimic very general features
+that anomalous events might have, such as high multiplicity, large MET, or large pT. Despite choos-
+ing explicitly the augmentations, the approach does not target any specific new physics model, and
+we will see from the results that the approach is model agnostic. AnomalyCLR projects the kine-
+matics of each event to a representation vector, which we then use to train an AutoEncoder and
+define the anomaly scores. It enriches the representation space using known invariances in the
+data, such as invariance to azimuthal rotations, and known generic features of anomalies. Self-
+supervised anomaly detection methods have gained prominence in the machine learning literature
+recently [61–64], and while the approaches are necessarily domain specific, we have drawn on
+these methods. The anomaly score can be computed in different ways, and we opt for the Au-
+toEncoder approach. So the workflow is as follows; train AnomalyCLR to obtain a representation
+vector for each event in the dataset, then train an AutoEncoder on these representations to obtain
+the anomaly scores. This is in contrast to the typical approach of training the AutoEncoder directly
+on the raw kinematical data from the events. We test AnomalyCLR on the CMS Anomaly Detection
+Challenge dataset [65], and, compared to the raw data baseline, we find significant improvements
+on all signals.
+In Section 2 we will discuss the dataset and the different backgrounds and signals. In Section 3
+we will then introduce the AnomalyCLR idea, first discussing contrastive learning and then how
+this can be modified for use in anomaly detection. The specifics of the application to event-level
+collider data such as the CMS ADC dataset is given in Section 4. The discussion on how we estimate
+anomaly scores is given in Section 5, where the architecture and optimisation of the AutoEncoder
+we use is discussed. The results are presented in Section 6, along with an analysis of how different
+anomaly-augmentations and different representation dimensions affect the results. We conclude
+in Section 7 with a discussion of the results and future directions.
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+2
+Dataset
+To test the performance of the AnomalyCLR representations compared to raw data in an anomaly
+detection task we use the CMS anomaly detection challenge dataset [65], which contains simu-
+lated proton-proton collisions with a 13 TeV centre-of-mass energy. The events are selected to
+have at least one e or µ with transverse momenta pT >23. The pseudo-rapidity (|η|) is required
+to be <3 and <2.1 respectively for e and µ. Further, the events are allowed to have up to 10 jets
+with pT > 15 GeV and |η| < 4, up to 4 muons pT > 3 GeV and |η| < 2.1, up to 4 electrons pT > 3
+GeV and |η|<3 and missing transverse energy (MET). The dataset is generated with Pythia 8.240
+generator [66] with a fast detector simulation using Delphes 3.3.2 [67] with the Phase-II CMS
+detector card. The jets are reconstructed using anti-kt algorithm [68]. In the provided dataset
+each event is formatted such that the first entry is assigned for MET, next eight are assigned for
+electrons and muons respectively and, the final 10 entries are for jets. For each particle object
+the data set contains information of pT, η, φ and particle id such that the shape of an event in
+the data frame is [N,19,4] where N is the total number of events. Note that if an event has less
+than the maximum allowed of a type of object, the remaining entries in that case are zero padded.
+The background dataset consists of a number of Standard Model processes and to determine the
+performance of the anomaly detection algorithm four light BSM scenarios are considered.
+Backgrounds
+For the SM background a collection of events are generated from production channels with at
+least a single lepton in the final state. The fraction of events to be included in the SM for each
+process is fixed by its trigger efficiency and the LO cross section. Thus, four leading processes are
+considered: W and Z inclusive productions, QCD multijet contributions, and t¯t production. The
+proportions between the four processes are given in [69] as:
+pp → W ± + jets → ℓ±νℓ + jets
+(59.2%)
+pp → Z + jets → ℓ+ℓ− + jets
+(6.7%)
+pp → t¯t + jets
+(0.3%)
+pp → jets
+(33.8%) .
+(1)
+with ℓ = e,µ,τ. The QCD multijet production is by far the largest production process at the
+LHC. Although leptons in QCD multijet backgrounds are rarely present and mainly originate from
+decays of unstable hadrons, the sheer volume of QCD multijet production makes it one of the
+largest processes in the data stream for the challenge.
+New physics signals
+The signal datasets provided by the challenge consist of events simulated from the following signal
+models:
+• Leptoquark (LQ): A 80 GeV LQ decaying in to a b and τ.
+• Neutral scalar boson A: A 50 GeV neutral scalar boson A. The production mechanism
+pp → A+X → Z∗Z∗+X (with X is inclusive activity) followed by both Z∗ decaying into charged
+leptons.
+• Scalar boson h0: A scalar boson 60 GeV h0 with pp → h0 + X → τ+τ− + X production.
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+• A charged scalar h±: Charged scalar with 60 GeV mass and pp → h±+X → τν+X production.
+The most distinguishing high-level features of these signals when compared with the background
+processes are the electron, muon, and jet multiplicities and the pT and MET distributions † .
+3
+AnomalyCLR
+In this section we describe the AnomalyCLR method ‡. Contrastive learning of representations
+(CLR) [52] is a technique used to construct highly-expressive representations of data for use in
+downstream tasks, in our case this task is anomaly detection. It is self-supervised in that the
+technique does not require any ‘truth’ labels for the training data. The advantage of this from the
+collider physics perspective is that the technique could be run directly on experimental data rather
+than on simulation. Due to the ability of deep learning methods to learn non-trivial correlations
+in data that is not expected to be well-modelled by simulation, this is an important aspect of CLR
+for anomaly detection.
+3.1
+Contrastive learning
+The basic idea is that some function f (·) (typically a neural network) is used to map from the data
+space D to a representation space R, with the function being optimised to solve some auxiliary
+task which does not require truth labels. This auxiliary task is framed as an optimisation problem
+using ‘pseudo-labels’. In the anomaly detection scenario addressed in this work, the function that
+performs the mapping from D to R is optimised only on background data. Given that the collider
+events or objects such as jets typically consist of unordered sets of particles reconstructed by the
+experiment, we opt for a permutation-invariant function to perform the mapping from D to R.
+Specifically, we use a transformer encoder neural network, there are more details on this later in
+the section.
+The auxiliary task that our function is optimised to solve uses augmentations of the collider
+data. In the traditional contrastive learning approach these augmentations are used to define two
+types of pseudo-labels:
+1. Positive-pair labels
+These labels match each data point in the sample to an augmented version of itself.
+2. Negative-pair labels
+These labels match each data point in the sample to every other data point which is not itself
+or an augmented/transformed version of itself.
+The function f (·) is then trained to map from the raw data to the representation space such that
+positive-pairs are close together in R and negative-pairs are far apart in R. These two optimi-
+sation goals are referred to as alignment (of positive-pairs) and uniformity (of negative-pairs),
+respectively. The augmentations are chosen to be modifications of the data that should leave the
+underlying physics unchanged, for example a symmetry in the physical system or an augmentation
+that could mimic a detector resolution effect.
+†We note that since the publication of previous papers using this dataset, a bug fix in the simulation has resulted in
+a new dataset, and so it is difficult to make direct comparisons between new and old results.
+‡The code will be made available at https://github.com/bmdillon/AnomalyCLR.
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+Each data point is described by an array of data xi with the subscript labelling the specific data
+point. We denote an augmentation of a data point as x′
+i, with the positive-pairs and negative-pairs
+being defined as the sets {(xi, x′
+i)} and {(xi, x j)}∪{(xi, x′
+j)} for i ̸= j, respectively. The contrastive
+loss function that the network is trained to minimise then is
+LCLR = −log
+es(zi,z′
+i)/τ
+�
+j̸=i∈batch
+�
+es(zi,zj)/τ + es(zi,z′
+j)/τ� ,
+(2)
+where zi = f (xi) and z′
+i = f (x′
+i) are the outputs of the mapping function. The cosine similar-
+ity measure s(·,·) is used to compare events and measure distances between them in the new
+representation space,
+s(zi,zj) =
+zi · zj
+|zi||zj| = cosθi j .
+(3)
+In this way, s(·,·) projects each vector zi to the surface of a unit hypersphere and computes the
+cosine distance between each pair. As it stands, s(·,·) is not a proper distance metric, however we
+could form one by taking di j = θi j/π as the distance between each event in the representation
+space, although we do not explore this here. The numerator of the contrastive loss in Eq. (2)
+accounts for the positive-pair and alignment, where distances between events and their augmented
+counter-parts enter. While the denominator accounts for the negative-pairs and uniformity, where
+distances between completely different events are accounted for. The degree to which we trade
+off between the different tasks is determined by the temperature hyper-parameter τ in the loss
+function.
+3.2
+CLR for anomaly detection
+While contrastive learning has been shown to be very useful in generating representations for
+downstream classification tasks [53], there is a potential issue when using this approach for down-
+stream anomaly detection tasks. For the classification task, for example in [53], the function f (·)
+is optimised on data from both the background and signal classes, despite not using their truth-
+labels explicitly in the optimisation. Through the contrastive learning this allows the function to
+encode non-trivial features of both the background and signal data in the representations. When
+using contrastive learning for a downstream anomaly detection task however, the function f (·) is
+optimised on just the background data (or at least a significantly background-dominated dataset).
+This means that the representation learned by the function f (·) will focus solely on features rele-
+vant for the background data. This could mean that anomalous data is not out-of-distribution and
+so may not lead to competitive performance in downstream anomaly detection tasks. This will
+become evident when we look at the results in Section 6. To remedy this we introduce Anoma-
+lyCLR, a modified approach to contrastive learning for anomaly detection in particle physics. At
+the core of this approach is the introduction of ‘anomaly-augmentations’, such that we now have
+two categories for augmentations:
+1. Physical augmentations
+These are augmentations of the data that we would like the mapping to be invariant to.
+2. Anomaly-augmentations
+These are unphysical augmentations of the data that are supposed to mimic potential anoma-
+lies, we want the representations to be highly discriminative towards these augmentations.
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+We add a third pseudo-label:
+3. Anomaly-pair labels
+These labels match each data point in the sample to an anomaly-augmented version of itself.
+The advantage of anomaly-augmentations is that we can increase the sensitivity of the anomaly
+detection tools to anomalies using just the background data, potentially the data directly measured
+at colliders. This keeps the approach in line with the original data-driven CLR idea. We can then
+define the anomaly-augmented contrastive loss function as
+LAnomCLR = −log
+e[s(zi,z′
+i)−s(zi,z∗
+i )]/τ
+�
+j̸=i∈batch
+�
+es(zi,zj)/τ + es(zi,z′
+j)/τ� ,
+(4)
+where we denote the representations of the anomaly-augmented events by z∗, and so the anomaly-
+pair is defined as {(xi, x∗
+i )}. Note that the anomaly-augmentations only enter in the numerator
+of Eq. (4), and without these the loss function becomes the regular contrastive loss function.
+Introducing the anomaly-pairs we expose the network to data features that are outside of the
+background distribution. The CLR portion of the loss function still optimises for alignment and
+uniformity, however this uniformity is now disrupted by the anomaly-pair term. As a result the
+background data will not be uniformly distributed in the representation space, with some regions
+encoding features of the anomaly-augmented data. This means that anomalous data with features
+similar to those generated by the anomaly-augmentations should be out-of-distribution in this
+representation space.
+We did some minor testing on alternative forms of this loss function, for example including
+the anomaly-augmentations in the denominator of the loss function with the negative-pairs. How-
+ever since the anomaly-augmentations compute distances between a data point and its augmented
+counter-part, and not between other data points (i.e. i ̸= j), it is more intuitive to include this
+term in the numerator. The denominator in Eq. (4) is used to encode features in the represen-
+tation space that discriminate between the different data points used during training, which for
+anomaly detection is the background data. This is not necessary for anomaly detection, and the
+anomaly-pairs should provide the representations with all the discriminative power they need,
+so we experimented with removing the denominator in Eq. (4) altogether, and found that this is
+sufficient. In this case the loss function is written as
+L+
+AnomCLR = −log e[s(zi,z′
+i)−s(zi,z∗
+i )]/τ =
+s(zi,z∗
+i ) − s(zi,z′
+i)
+τ
+,
+(5)
+where the plus sign in L+
+AnomCLR indicates that only positive-pairs are used. This results in a much
+less computationally expensive loss function, since we no longer need to compute pair-wise cor-
+relations between each entry in a batch the complexity scales as Nbatch rather than N 2
+batch. We also
+remove the dependence on τ in L+
+AnomCLR, since there is no longer a trade-off between positive-
+and negative-pairs. We could of course introduce a term to control the trade-off between the
+physical and anomaly-augmentation terms, but we do not explore that here. In our results we will
+compare the performance of both loss functions.
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+4
+Application to event-level anomalies
+The application of AnomalyCLR to different physical scenarios requires an understanding of the
+data and the physics in order to construct the physical and anomaly-augmentations. For the event-
+level dataset discussed in Section 2 we consider three physical augmentations the data:
+1. Azimuthal rotations
+The whole final state is rotated by an angle φ randomly sampled from [0,2π].
+2. η − φ smearing
+The (η,φ) coordinate of every object in the event is resampled according from a Normal
+distribution centred on the original coordinate and with a variance inversely proportional to
+the pT, i.e. η′ ∼ N (η,σ(pT)) and φ′ ∼ N (φ,σ(pT)).
+3. Energy smearing
+The pT of every object in the event is re-sampled according to p′
+T ∼ N (pT, f (pT)) with f (pT)
+determining the strength of the smearing.
+These augmentations reflect both the symmetries in the data and the experimental resolution of
+the detector. Detectors are imperfect, especially in measuring jet energies, and we encode this in
+the representations of the data through the energy-smearing augmentation. Here we re-sample
+the jet pT’s as p′
+T ∼ N (pT, f (pT)), where f (pT) =
+�
+0.052p2
+T + 1.502pT is the energy smearing
+applied by Delphes (the pT’s are normalised by 1GeV). If not explicitly mentioned, we always
+assume units of GeV for energy. For the anomaly-augmentation we consider some very simple
+scenarios:
+1. Multiplicity shift, x′
+i = m(xi)
+For each event m(·) adds a random number of electrons, muons, and jets to the event. The
+number is chosen randomly within the limits (ne,4− ne), (nµ,4− nµ), and (nj,10− nj) for
+electrons, muons, and jet, respectively. The azimuthal angle and pseudo-rapidities are also
+chosen randomly within the limits allowed, and the pT for each object is chosen as a random
+fraction of the maximum pT in the event. Once the objects have been added, the MET of the
+event is recalculated and updated.
+2. Multiplicity shift, keeping MET and the total pT constant, , x′
+i = m(xi)
+This is similar to the above augmentation, but now m(·) generates the extra objects by splitting
+the existing objects and smearing the η−φ coordinates using the function used in the physical
+augmentations above.
+3. pT and MET shifts, x′
+i = spT (xi)
+Here spT (·) shifts the pT’s in the event by the same random factor. We randomly choose whether
+we shift just the MET, just the reconstructed object pT’s, or both. And we ensure that the the
+trigger selection is not spoiled by these shifts.
+With the physical augmentations we apply all of them simultaneously, whereas for the anomaly-
+augmentations we apply just one augmentation to each event. The augmentation that is applied is
+selected randomly and uniformly. We do not apply both a physical augmentation and an anomaly-
+augmentation to the events in s(zi,z∗
+i ), since this would conflict with the optimisation goal of the
+s(zi,z′
+i) term. It would also be possible to have an anomaly-augmentation that removes objects
+from the event, however this effect is already captured by the augmentation that adds objects to
+the event. Many of the events in the background have the minimal multiplicity allowed by the
+applied cuts, so the effect of an anomaly-pair with a low multiplicity background event and the
+same event augmented to have more objects is the exact same as the effect of an anomaly-pair
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+with a high-multiplicity background event augmented to have less objects. This is because of the
+symmetry in the distance function s(zi,z∗
+i ). So the anomaly-augmentations here are as general
+as can be, and do not target any specific new physics scenario, therefore the technique should be
+model-agnostic.
+Architecture and training
+The collider event data being used has a well-defined structure:
+• MET: one entry with (pT,η,φ)
+• Electrons: four entries, each with (pT,η,φ)
+• Muons: four entries, each with (pT,η,φ)
+• Jets: ten entries, each with (pT,η,φ).
+This amounts to a 19 × 3 array, with the electrons, muons, and jets being ordered by pT and hav-
+ing zero-padded entries where there is less than the maximum allowed number of reconstructed
+objects. The multiplicity is typically much less than the maximum allowed, so the data for a
+single collider event can have many zeros. The transformer allows us to avoid this by having a
+permutation-invariant and variable length input format. Because the data is now processed in
+a permutation-invariant way, the information on which entry corresponds to which object (MET,
+electron, muon, or jet) is lost. We reinstate this information by adding a one-hot encoded ID vec-
+tor to (pT,η,φ), with a 1 indicating the correct ID. This means that each reconstructed object is
+now represented by a 7D vector. Before passing the kinematic data to the transformer we do some
+very minor preprocessing to make sure that the numbers the networks see are O(1). Specifically,
+we divide all MET and pT values by the average pT of all objects (electrons, muons, jets) in the
+background dataset, we do not shift the values to be centred on zero because the distribution is
+highly peaked at zero and we want the preprocessed data to have the same sparsity as the original
+data. We then divide all η and φ values by 4 and π, respectively. When training the AutoEncoder
+networks discussed in the next section we use the same preprocessing of the data, this ensures
+that any difference in the results can be attributed to AnomalyCLR.
+The transformer starts by projecting each object to a larger vector whose dimension is deter-
+mined by the embedding dimension. The embeddings for each object are then passed through the
+transformer, with a feed-forward network between each transformer layer. The output from the
+transformer has a dimension of (n× model dimension) with n being the number of objects in the
+event. The last steps are to sum over the n vectors in this output, which enforces the permutation-
+invariance, and to pass this vector through a fully-connected head network. The output of this
+head network is what is passed to the loss function. For more details on the architecture we re-
+fer the reader to [53], here we only list the hyper-parameters used in training the network in
+Table 1. The representation used in the anomaly detection task is taken from the output of the
+transformer network, before being passed through the head network. It is well documented in the
+machine learning literature that these intermediate representations from self-supervised networks
+generally contain more discriminating features, for example in [52].
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+hyper-parameter
+model (embedding) dimension 160
+feed-forward hidden dimension 160
+output dimension
+160
+# self-attention heads
+4
+# transformer layers (N)
+4
+# layers
+2
+dropout rate
+0.1
+hyper-parameter
+optimiser
+Adam(β1=0.9, β2=0.999)
+learning rate
+5 × 10−5
+batch size
+128
+# epochs
+500
+Table 1: Default setup of the transformer-encoder network and the AnomalyCLR train-
+ing, unless noted explicitly.
+5
+Anomaly scores
+The basic flow in an AutoEncoder involves two steps; (i) mapping high-dimensional input data
+to a compressed latent space using a neural network called an encoder, and (ii) mapping the
+compressed latent space representation to a reconstructed version of the input data using a neural
+network called a decoder. We refer to the encoder network as e(·) and the decoder network as d(·).
+With input data of dimension D, and a bottleneck of dimension B, the encoder maps e : �D → �B,
+while the decoder maps d : �B → �D, with the AutoEncoder defined as h = e ◦ d : �D → �D.
+Acting on a single input ⃗x, the AutoEncoder returns ⃗x′ = h(⃗x), and is optimised to minimise the
+mean-squared-error (MSE) loss function between the input and reconstructed input,
+L(⃗x,θ) =
+�
+⃗x − ⃗x′�2 ,
+(6)
+where θ represents the learnable parameters of the AutoEncoder. In the limit where the AutoEn-
+coder is able to reconstruct inputs perfectly, which is guaranteed to be possible when D = B, the
+function hθ(·) is simply the identity. But with B < D the AutoEncoder may not be able to perfectly
+reconstruct all features in the data, and therefore it should learn to reconstruct only the most
+common or prominent features in the data. This means that events containing rare or anomalous
+features should have a larger ‘reconstruction loss’, i.e. L(⃗x,θ), and this can then be used as the
+anomaly score.
+The encoder and decoder networks have 5 feed forward layers each with 256, 128, 64, 32, and
+16 neurons, connected by a 5-dimensional bottleneck. The activation function between layers is
+a LeakyReLU with default slope. The decoder is a mirrored version of the encoder. We don’t apply
+regularization techniques during training. The training is performed using Adam optimiser with
+learning rate 0.001 for 100 epochs, the batch size is 4096, and the number of SM events used is
+106. Note that we have not optimised the AutoEncoder architecture, simply choosing the same
+architecture used in [39]. Instead we have only ensured that they are trained to convergence and
+that the training is stable. The AutoEncoder is trained on both the representations obtained from
+contrastive learning and the raw data. In the case of the raw data we apply the same preprocessing
+to the data as is applied to the data in the contrastive learning network. In this way we ensure
+that any differences in the anomaly detection performance can be attributed to the contrastive
+learning methods.
+10
+
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+6
+Results
+In this section we present some results using the different techniques discussed in the preceding
+sections. The results here are three-fold; we first compare the different methods based on anomaly
+detection performance, we then study the effects of the different anomaly-augmentations on the
+AnomalyCLR performance, and lastly we look at the effect of the representation dimension on the
+performance.
+6.1
+Comparison of methods
+We compare the methods using the ROC (Receiver Operating Characteristic) curves, the SI (Signif-
+icance Improvement) curves, and the AUC (Area Under the ROC Curve). The baseline we compare
+to is the AutoEncoder trained on raw kinematic data. We present results using the CLR method
+without anomaly-augmentations (LCLR), and the CLR method with anomaly-augmentations (both
+LAnomCLR and L+
+AnomCLR). So we have 4 methods in total to compare. For all results on the raw
+data we have trained 5 AutoEncoder networks and taken the central value and the error estimation
+from the mean and standard deviation of the results. For the CLR methods we also aggregate over
+5 different runs, where in each run we train a different transformer network and a different Au-
+toEncoder. The CLR representations have a dimension of 160 and where anomaly-augmentations
+are used we have used them all as outlined in Section 4. In Fig. 1 we present AnomalyCLR results
+using L+
+AnomCLR and see that it leads to significant improvements over the raw data representations,
+not only in the AUC but also at all signal efficiencies. In the Significance Improvement (SI) curves
+we also see large improvements, with the SI being between ∼ 3.5−4 for A → 4l and h+. We can see
+from Table 2 that the L+
+AnomCLR loss function is clearly advantageous over LAnomCLR, beating it on
+all signals with the exception of A → 4l, where LAnomCLR achieves better performance at εs =0.3.
+A point of interest here is that the AutoEncoder on raw data outperforms the AutoEncoder on the
+CLR representations in most cases. This is likely due to the fact that traditional CLR optimises
+for uniformity, and since it is trained on background only, the mapping is not optimised to sepa-
+rate SM-like background events from any event which may look different to that. The benefit of
+Signal
+AE-Raw
+CLR
+AnomCLR
+AnomCLR+
+AUC
+A
+0.885(2)
+0.880(7)
+0.907(6)
+0.909(3)
+h0
+0.755(2)
+0.740(5)
+0.765(4)
+0.776(2)
+h+
+0.900(4)
+0.87(1)
+0.913(2)
+0.930(1)
+LQ
+0.856(2)
+0.841(9)
+0.854(3)
+0.880(1)
+ε−1
+b (εs =0.3)
+A
+47(2)
+80(22)
+156(34)
+139(20)
+h0
+14.9(7)
+11(1)
+18(1)
+23(1)
+h+
+60(10)
+28(6)
+98(9)
+171(7)
+LQ
+24.4(6)
+18(2)
+28(3)
+39(1)
+SI(εs =0.3)
+A
+2.05(5)
+2.7(4)
+3.7(4)
+3.5(2)
+h0
+1.16(3)
+0.99(4)
+1.26(4)
+1.44(3)
+h+
+2.3(2)
+1.6(2)
+3.0(1)
+3.9(1)
+LQ
+1.48(2)
+1.3(1)
+1.6(1)
+1.88(2)
+Table 2: Comparison of the different CLR loss functions, with and without anomaly-
+augmentations, and the AE trained on raw data.
+11
+
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+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵs
+100
+101
+102
+103
+ϵ−1
+b
+AE-Raw
+A, AUC=0.885(2)
+h0, AUC=0.755(2)
+h+, AUC=0.900(4)
+LQ, AUC=0.856(2)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵs
+100
+101
+102
+103
+ϵ−1
+b
+AnomCLR+
+A, AUC:0.909(3)
+h0, AUC:0.776(2)
+h+, AUC:0.930(1)
+LQ, AUC:0.880(1)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵs
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+SI
+AE-Raw
+A
+h0
+h+
+LQ
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ϵs
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+SI
+AnomCLR+
+A
+h0
+h+
+LQ
+Figure 1: Comparison between the AE on raw data and the AE on the CLR representa-
+tions trained with the L+
+AnomCLR loss function.
+anomaly-augmentations here is strikingly clear.
+6.2
+The effect of anomaly-augmentations
+We now want to study how the addition of the individual anomaly-augmentations affects the
+anomaly detection performance. For this we use just L+
+AnomCLR , however we expect the results
+with LAnomCLR to be similar. We use a representation dimension of 160 and obtain the error
+estimate from just 2 runs due to the computational cost of the scan.
+We can see from Fig. 2 that the affect of the augmentations together results in the best over-
+all performance. One thing we noticed is that it can be difficult to determine from the affect of
+individual augmentations, or subgroups of them, what the performance of all of them together
+will be. For example, in most cases if we take just the m(x) augmentation, i.e. the multiplicity
+augmentation that simply adds reconstructed objects, we see that it alone decreases performance
+below baseline for three out of four signals. However when used in combination with the others
+it increases the performance. This is most clear for the leptoquark signal, where all augmenta-
+12
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+0.800
+0.850
+0.900
+A
+AUC
+0.700
+0.750
+h0
+0.800
+0.850
+0.900
+0.950
+h+
+m(x)
+m(x)
+spT(x) m(x)/m(x)
+all
+anomaly augmentations
+0.750
+0.800
+0.850
+LQ
+Figure 2: Results of a scan on the anomaly-augmentations used with the L+
+AnomCLR loss
+function. The augmentations are defined in Section 4. The dashed lines here correspond
+to the AutoEncoder on raw data baseline performance.
+tions taken individually result in a performance which is at or below baseline, but when taken
+together we get a significant boost in the AUC. We also see the interplay between the m(x) and
+m(x) augmentations, since individually these augmentations do not seem to help much, but when
+they are both applied in the same optimisation we see a reduced error and in most cases better
+performance. When drawing conclusions here we should keep in mind that only two runs for each
+combination have been used to compute the mean and error estimation.
+6.3
+The effect of representation dimension
+With CLR we can project our raw data from D to a representation of any dimension we like.
+We would expect that the larger the representation dimension the more information that can be
+encoded in the space. However we also expect that this would plateau or even peak at some point,
+and this what we want to investigate here. For this we use just L+
+AnomCLR , however we expect the
+results with LAnomCLR to be similar. Here we also obtain the error estimate from just 2 runs due
+to the computational cost of the scan.
+In Fig. 3 we see that increasing the representation dimension certainly improves the perfor-
+mance of the anomaly detection, at least up until a certain point. The A, h0, and LQ signals
+all appear to achieve peak performance somewhere between dimensions 120 and 200, while the
+h+ signals performance increases right up until 400. There is no fundamental limitation related
+to the representation size which we would expect to cause a degradation at larger dimensions,
+however there are two points we should keep in mind here. The first is simple, these means and
+variances are calculated with only two runs, so more runs might present a clearer picture. The
+13
+
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+Submission
+0.850
+0.875
+0.900
+0.925
+A
+AUC
+0.740
+0.760
+0.780
+h0
+0.900
+0.920
+0.940
+h+
+4
+8
+12
+20
+40
+80
+120
+160
+200
+300
+400
+500
+representation dimension
+0.840
+0.860
+0.880
+LQ
+Figure 3: Results of a scan on the representation dimension used with the L+
+AnomCLR loss
+function. The dashed lines here correspond to the AutoEncoder on raw data baseline
+performance.
+second point is that we have not optimised the AutoEncoder architecture or hyper-parameters as
+the representation size increases. While it is beyond the scope of this paper, it is possible that
+an independent hyper-parameter optimisation for each representation dimension would improve
+these results, particularly at larger dimensions. What these results show is that there is a clear
+tendancy for the results to improve as we increase from dimensions of ∼ 4 to ∼ 100, as we would
+naturally expect.
+7
+Summary & conclusions
+In this paper we have introduced AnomalyCLR§, a new method for density-based anomaly detec-
+tion in high-energy physics. It makes use of anomalous augmentations of collider data to build a
+representation space from which to construct anomaly scores with a range of methods, for exam-
+ple using AutoEncoders. It is a self-supervised method, based on the contrastive learning idea. We
+tested this method on the CMS ADC dataset, and compared to the raw data baselines we find large
+improvements on all signals. At a fixed signal efficiency of 0.3 and a fixed representation dimen-
+sion of 160 we find significance improvements for the different signals in the range of 14−70%,
+and a decreased relative error on the significance improvement in each case. Allowing for varying
+signal efficiencies and representation dimensions would improve these performance markers even
+§The AnomalyCLR code, along with the event-level anomaly detection application, will be made available at
+https://github.com/bmdillon/AnomalyCLR.
+14
+
+SciPost Physics
+Submission
+further.
+Density-based anomaly detection, using AutoEncoders or normalising flows, suffer from the
+ambiguity that a change in the ‘coordinate system’ or representation of the data results in a fun-
+damental change in how the anomaly score is defined. This makes it difficult to choose a suitable
+representation by hand, for example a simple re-mapping of pT’s along with some re-scaling of
+numerical inputs. These simple choices are difficult to motivate from a physics perspective and
+can drastically change the results of the anomaly detection. This change can be for better or for
+worse, and typically depends on the signal models used to test the algorithm.
+AnomalyCLR addresses this by constructing a representation of the data using self-supervised
+contrastive learning with the addition of anomaly-augmented data. The anomaly-augmented data
+is constructed from the background data through feature augmentation, designed to emulate a
+generic anomaly. We have discussed in detail how we do this for the event-level anomalies in
+the CMS ADC dataset, however this would of course be different in different physics cases. We
+proposed a new loss function which we use to train a deep transformer-based neural network. This
+network projects the events to a new representation, in which the anomaly-augmented events are
+far from their original counterparts, while being close to events which are similar. The transformer
+network then learns a highly discriminative representation of the events which is sensitive to
+the presence of potential anomalies. We have seen that the choice of these augmentations is
+quite model agnostic. This model-agnostic nature of the approach can be seen in how the results
+improve across all four signals considered.
+We have shown the effectiveness of self-supervision and the idea of anomaly-augmentations
+in significantly enhancing anomaly detection performance in a model-agnostic way. This opens
+the door to further studies, such as improving the density-estimation portion of the method with a
+more sophisticated hyper-parameter optimisation of the AutoEncoders, using normalising flows,
+or even using the Normalised AutoEncoder. More generally, the use of anomaly-augmented data
+could be explored further in other anomaly detection approaches.
+Acknowledgements
+We would like to thank Jernej Kamenik and Ben Nachman for their helpful comments on the
+manuscript. BMD acknowledges funding from the Alexander von Humboldt Foundation. LF, TM,
+and TP are funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
+under grant 396021762 – TRR 257: Particle Physics Phenomenology after the Higgs Discovery
+and Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Ex-
+cellence Cluster).
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diff --git a/-dE3T4oBgHgl3EQfrgpm/content/tmp_files/load_file.txt b/-dE3T4oBgHgl3EQfrgpm/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf,len=1056
+page_content='SciPost Physics  Submission  Anomalies, Representations, and Self-Supervision  Barry M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Dillon, Luigi Favaro, Friedrich Feiden, Tanmoy Modak, Tilman Plehn  Institut für Theoretische Physik, Universität Heidelberg, Germany  January 13, 2023  Abstract  We develop a self-supervised method for density-based anomaly detection using contrastive  learning, and test it using event-level anomaly data from CMS ADC2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The Anomaly-  CLR technique is data-driven and uses augmentations of the background data to mimic  non-Standard-Model events in a model-agnostic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' It uses a permutation-invariant Trans-  former Encoder architecture to map the objects measured in a collider event to the represen-  tation space, where the data augmentations define a representation space which is sensitive  to potential anomalous features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' An AutoEncoder trained on background representations  then computes anomaly scores for a variety of signals in the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' With  AnomalyCLR we find significant improvements on performance metrics for all signals when  compared to the raw data baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Contents  1  Introduction  2  2  Dataset  4  3  AnomalyCLR  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='1  Contrastive learning  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='2  CLR for anomaly detection  6  4  Application to event-level anomalies  8  5  Anomaly scores  10  6  Results  11  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='1  Comparison of methods  11  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='2  The effect of anomaly-augmentations  12  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3  The effect of representation dimension  13  7  Summary & conclusions  14  References  15  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='04660v1  [hep-ph]  11 Jan 2023  SciPost Physics  Submission  1  Introduction  Model-agnostic new physics searches are one of the most interesting analysis prospects for the  LHC and other colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Over the past decade the LHC has searched for new physics based on  model-specific hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Despite these efforts there has been no strong evidence of new  physics found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' It is possible that new physics does exist at the scales probed by the LHC, and has  not been uncovered due to the particular signal not being covered by previous analysis hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  The ATLAS and CMS collaborations have both implemented model-agnostic new physics searches  to deal with this [1, 2], however these methods suffer some drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For example scanning  high-dimensional parameter spaces can lead to large look-elsewhere effects, or methods can lack  the ability to make full use of the high-granularity low-level information collected in the experi-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Recent progress in machine learning based high-energy physics tools are making significant  advances in solving many problems of such classical methods [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  The main machine learning tools to date for data-driven model-agnostic searches are based  either on density-related scores, or on classification scores using a background-dominated control  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The latter, typically known as CWoLa methods (Classification Without Labels) [4–6] have  been shown to be very successful in applications such as bump hunting [7–14] and semi-visible jet  searches [15], providing both anomaly scores and background estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' However they run into  difficulty when the dimension of the input space or number of observables becomes large, and so  the question of whether or not they can be used on low level data is still uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' CWoLa tools  have already been adopted by the ATLAS collaboration [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Density-based methods use machine learning to estimate the density in the phase space, and  then identify anomalies as those laying in the low density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' These tools typically work  on high-dimensional inputs and so can be used on low-level data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The first density-based meth-  ods were the AutoEncoder studies [17, 18], where the network is optimised to compress and  reconstruct the kinematics of a jet or event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' While this is not strictly density estimation, the  optimisation is highly aligned with learning a density, since regions of the phase space which  are most populated are those which should be reconstructed the best and thus have the low-  est anomaly score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' There has been significant progress with the AutoEncoder tools and other  density-based anomaly detection methods in recent years [19–33], with studies covering inter-  pretability of AutoEncoders [34, 35], topic modelling [36, 37], null hypothesis tests for anomaly  detection [38], ABCD methods [39], the Normalised AutoEncoder (NAE) [40], and normalising  flow techniques [41–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For a comprehensive summary of many different anomaly detection  methods we refer the reader to the community challenge papers in Refs [45,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  One issue with the density-based approaches [44,47] is that the score is not invariant under  simple transformations in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This means that a simple re-mapping of the momenta  or coordinates fundamentally changes what the anomaly score is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This poses the question of how  to choose a representation of the data for use in density-based anomaly detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' It is  also worth noting that despite the great progress that more sophisticated neural network archi-  tectures and the implementation of symmetries in networks has brought to supervised classifi-  cation [48–51], they have not yet led to the same progress in anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In this work  we develop a new approach to density-based anomaly detection using self-supervision, which de-  fines the representation of the data in a model-agnostic way using the power of highly expressive  networks such as transformers or graph networks to boost anomaly detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Supervised machine learning methods use the idea of a truth-label to optimise the neural net-  2  SciPost Physics  Submission  works, usually to classify between data with different truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Unsupervised methods are  those which do not require truth labels, instead optimising a network using a reconstruction loss  or a negative log likelihood, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Self-supervision on the other hand uses ‘pseudo-labels’,  labels generated from the data without knowledge of a truth label, to optimise the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In  contrastive learning [52], these labels correspond to a link between an original event and an aug-  mented event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We define the augmentation as some physical modification of the event kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Contrastive learning uses the pseudo-labels to devise an auxiliary task for the network optimisation  through the contrastive loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Now the network learns how to process high-dimensional  correlations in the data, and thus the representations learned by these networks can be very useful  for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We introduced the self-supervised JetCLR method in [53] and demonstrated  its ability to construct highly expressive representations for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In [54] this same  technique was used to construct representations for CWoLa-based anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In addition  to these works, other self-supervised / representation learning techniques have been applied in  particle physics [55,56] and in other scientific disciplines such as astrophysics [57–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In [53,54]  the augmentations corresponded to transformations of the event to which the underlying physics  should be invariant to rotations or translations, but also soft-collinear parton splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  We introduce AnomalyCLR, a new method based on the idea of ‘anomaly-augmentations’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  These anomaly-augmentations are modifications of the original event to which the underlying  physics is not invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In fact these augmentations are chosen to mimic very general features  that anomalous events might have, such as high multiplicity, large MET, or large pT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Despite choos-  ing explicitly the augmentations, the approach does not target any specific new physics model, and  we will see from the results that the approach is model agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' AnomalyCLR projects the kine-  matics of each event to a representation vector, which we then use to train an AutoEncoder and  define the anomaly scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' It enriches the representation space using known invariances in the  data, such as invariance to azimuthal rotations, and known generic features of anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Self-  supervised anomaly detection methods have gained prominence in the machine learning literature  recently [61–64], and while the approaches are necessarily domain specific, we have drawn on  these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The anomaly score can be computed in different ways, and we opt for the Au-  toEncoder approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' So the workflow is as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' train AnomalyCLR to obtain a representation  vector for each event in the dataset, then train an AutoEncoder on these representations to obtain  the anomaly scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This is in contrast to the typical approach of training the AutoEncoder directly  on the raw kinematical data from the events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We test AnomalyCLR on the CMS Anomaly Detection  Challenge dataset [65], and, compared to the raw data baseline, we find significant improvements  on all signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  In Section 2 we will discuss the dataset and the different backgrounds and signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In Section 3  we will then introduce the AnomalyCLR idea, first discussing contrastive learning and then how  this can be modified for use in anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The specifics of the application to event-level  collider data such as the CMS ADC dataset is given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The discussion on how we estimate  anomaly scores is given in Section 5, where the architecture and optimisation of the AutoEncoder  we use is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The results are presented in Section 6, along with an analysis of how different  anomaly-augmentations and different representation dimensions affect the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We conclude  in Section 7 with a discussion of the results and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  3  SciPost Physics  Submission  2  Dataset  To test the performance of the AnomalyCLR representations compared to raw data in an anomaly  detection task we use the CMS anomaly detection challenge dataset [65], which contains simu-  lated proton-proton collisions with a 13 TeV centre-of-mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The events are selected to  have at least one e or µ with transverse momenta pT >23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The pseudo-rapidity (|η|) is required  to be <3 and <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='1 respectively for e and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Further, the events are allowed to have up to 10 jets  with pT > 15 GeV and |η| < 4, up to 4 muons pT > 3 GeV and |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='1, up to 4 electrons pT > 3  GeV and |η|<3 and missing transverse energy (MET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The dataset is generated with Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='240  generator [66] with a fast detector simulation using Delphes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='2 [67] with the Phase-II CMS  detector card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The jets are reconstructed using anti-kt algorithm [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In the provided dataset  each event is formatted such that the first entry is assigned for MET, next eight are assigned for  electrons and muons respectively and, the final 10 entries are for jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For each particle object  the data set contains information of pT, η, φ and particle id such that the shape of an event in  the data frame is [N,19,4] where N is the total number of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Note that if an event has less  than the maximum allowed of a type of object, the remaining entries in that case are zero padded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  The background dataset consists of a number of Standard Model processes and to determine the  performance of the anomaly detection algorithm four light BSM scenarios are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Backgrounds  For the SM background a collection of events are generated from production channels with at  least a single lepton in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The fraction of events to be included in the SM for each  process is fixed by its trigger efficiency and the LO cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Thus, four leading processes are  considered: W and Z inclusive productions, QCD multijet contributions, and t¯t production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The  proportions between the four processes are given in [69] as:  pp → W ± + jets → ℓ±νℓ + jets  (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='2%)  pp → Z + jets → ℓ+ℓ− + jets  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='7%)  pp → t¯t + jets  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3%)  pp → jets  (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='8%) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  (1)  with ℓ = e,µ,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The QCD multijet production is by far the largest production process at the  LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Although leptons in QCD multijet backgrounds are rarely present and mainly originate from  decays of unstable hadrons, the sheer volume of QCD multijet production makes it one of the  largest processes in the data stream for the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  New physics signals  The signal datasets provided by the challenge consist of events simulated from the following signal  models:  Leptoquark (LQ): A 80 GeV LQ decaying in to a b and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Neutral scalar boson A: A 50 GeV neutral scalar boson A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The production mechanism  pp → A+X → Z∗Z∗+X (with X is inclusive activity) followed by both Z∗ decaying into charged  leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Scalar boson h0: A scalar boson 60 GeV h0 with pp → h0 + X → τ+τ− + X production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  4  SciPost Physics  Submission  A charged scalar h±: Charged scalar with 60 GeV mass and pp → h±+X → τν+X production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  The most distinguishing high-level features of these signals when compared with the background  processes are the electron, muon, and jet multiplicities and the pT and MET distributions † .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  3  AnomalyCLR  In this section we describe the AnomalyCLR method ‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Contrastive learning of representations  (CLR) [52] is a technique used to construct highly-expressive representations of data for use in  downstream tasks, in our case this task is anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' It is self-supervised in that the  technique does not require any ‘truth’ labels for the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The advantage of this from the  collider physics perspective is that the technique could be run directly on experimental data rather  than on simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Due to the ability of deep learning methods to learn non-trivial correlations  in data that is not expected to be well-modelled by simulation, this is an important aspect of CLR  for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='1  Contrastive learning  The basic idea is that some function f (·) (typically a neural network) is used to map from the data  space D to a representation space R, with the function being optimised to solve some auxiliary  task which does not require truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This auxiliary task is framed as an optimisation problem  using ‘pseudo-labels’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In the anomaly detection scenario addressed in this work, the function that  performs the mapping from D to R is optimised only on background data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Given that the collider  events or objects such as jets typically consist of unordered sets of particles reconstructed by the  experiment, we opt for a permutation-invariant function to perform the mapping from D to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Specifically, we use a transformer encoder neural network, there are more details on this later in  the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  The auxiliary task that our function is optimised to solve uses augmentations of the collider  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In the traditional contrastive learning approach these augmentations are used to define two  types of pseudo-labels:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Positive-pair labels  These labels match each data point in the sample to an augmented version of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Negative-pair labels  These labels match each data point in the sample to every other data point which is not itself  or an augmented/transformed version of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  The function f (·) is then trained to map from the raw data to the representation space such that  positive-pairs are close together in R and negative-pairs are far apart in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' These two optimi-  sation goals are referred to as alignment (of positive-pairs) and uniformity (of negative-pairs),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The augmentations are chosen to be modifications of the data that should leave the  underlying physics unchanged, for example a symmetry in the physical system or an augmentation  that could mimic a detector resolution effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  †We note that since the publication of previous papers using this dataset, a bug fix in the simulation has resulted in  a new dataset, and so it is difficult to make direct comparisons between new and old results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  ‡The code will be made available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='com/bmdillon/AnomalyCLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  5  SciPost Physics  Submission  Each data point is described by an array of data xi with the subscript labelling the specific data  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We denote an augmentation of a data point as x′  i, with the positive-pairs and negative-pairs  being defined as the sets {(xi, x′  i)} and {(xi, x j)}∪{(xi, x′  j)} for i ̸= j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The contrastive  loss function that the network is trained to minimise then is  LCLR = −log  es(zi,z′  i)/τ  �  j̸=i∈batch  �  es(zi,zj)/τ + es(zi,z′  j)/τ� ,  (2)  where zi = f (xi) and z′  i = f (x′  i) are the outputs of the mapping function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The cosine similar-  ity measure s(·,·) is used to compare events and measure distances between them in the new  representation space,  s(zi,zj) =  zi · zj  |zi||zj| = cosθi j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  (3)  In this way, s(·,·) projects each vector zi to the surface of a unit hypersphere and computes the  cosine distance between each pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' As it stands, s(·,·) is not a proper distance metric, however we  could form one by taking di j = θi j/π as the distance between each event in the representation  space, although we do not explore this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The numerator of the contrastive loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' (2)  accounts for the positive-pair and alignment, where distances between events and their augmented  counter-parts enter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' While the denominator accounts for the negative-pairs and uniformity, where  distances between completely different events are accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The degree to which we trade  off between the different tasks is determined by the temperature hyper-parameter τ in the loss  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='2  CLR for anomaly detection  While contrastive learning has been shown to be very useful in generating representations for  downstream classification tasks [53], there is a potential issue when using this approach for down-  stream anomaly detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For the classification task, for example in [53], the function f (·)  is optimised on data from both the background and signal classes, despite not using their truth-  labels explicitly in the optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Through the contrastive learning this allows the function to  encode non-trivial features of both the background and signal data in the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' When  using contrastive learning for a downstream anomaly detection task however, the function f (·) is  optimised on just the background data (or at least a significantly background-dominated dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  This means that the representation learned by the function f (·) will focus solely on features rele-  vant for the background data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This could mean that anomalous data is not out-of-distribution and  so may not lead to competitive performance in downstream anomaly detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This will  become evident when we look at the results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' To remedy this we introduce Anoma-  lyCLR, a modified approach to contrastive learning for anomaly detection in particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' At  the core of this approach is the introduction of ‘anomaly-augmentations’, such that we now have  two categories for augmentations:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Physical augmentations  These are augmentations of the data that we would like the mapping to be invariant to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Anomaly-augmentations  These are unphysical augmentations of the data that are supposed to mimic potential anoma-  lies, we want the representations to be highly discriminative towards these augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  6  SciPost Physics  Submission  We add a third pseudo-label:  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Anomaly-pair labels  These labels match each data point in the sample to an anomaly-augmented version of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  The advantage of anomaly-augmentations is that we can increase the sensitivity of the anomaly  detection tools to anomalies using just the background data, potentially the data directly measured  at colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This keeps the approach in line with the original data-driven CLR idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We can then  define the anomaly-augmented contrastive loss function as  LAnomCLR = −log  e[s(zi,z′  i)−s(zi,z∗  i )]/τ  �  j̸=i∈batch  �  es(zi,zj)/τ + es(zi,z′  j)/τ� ,  (4)  where we denote the representations of the anomaly-augmented events by z∗, and so the anomaly-  pair is defined as {(xi, x∗  i )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Note that the anomaly-augmentations only enter in the numerator  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' (4), and without these the loss function becomes the regular contrastive loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Introducing the anomaly-pairs we expose the network to data features that are outside of the  background distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The CLR portion of the loss function still optimises for alignment and  uniformity, however this uniformity is now disrupted by the anomaly-pair term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' As a result the  background data will not be uniformly distributed in the representation space, with some regions  encoding features of the anomaly-augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This means that anomalous data with features  similar to those generated by the anomaly-augmentations should be out-of-distribution in this  representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  We did some minor testing on alternative forms of this loss function, for example including  the anomaly-augmentations in the denominator of the loss function with the negative-pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' How-  ever since the anomaly-augmentations compute distances between a data point and its augmented  counter-part, and not between other data points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' i ̸= j), it is more intuitive to include this  term in the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' (4) is used to encode features in the represen-  tation space that discriminate between the different data points used during training, which for  anomaly detection is the background data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This is not necessary for anomaly detection, and the  anomaly-pairs should provide the representations with all the discriminative power they need,  so we experimented with removing the denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' (4) altogether, and found that this is  sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In this case the loss function is written as  L+  AnomCLR = −log e[s(zi,z′  i)−s(zi,z∗  i )]/τ =  s(zi,z∗  i ) − s(zi,z′  i)  τ  ,  (5)  where the plus sign in L+  AnomCLR indicates that only positive-pairs are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This results in a much  less computationally expensive loss function, since we no longer need to compute pair-wise cor-  relations between each entry in a batch the complexity scales as Nbatch rather than N 2  batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We also  remove the dependence on τ in L+  AnomCLR, since there is no longer a trade-off between positive-  and negative-pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We could of course introduce a term to control the trade-off between the  physical and anomaly-augmentation terms, but we do not explore that here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In our results we will  compare the performance of both loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  7  SciPost Physics  Submission  4  Application to event-level anomalies  The application of AnomalyCLR to different physical scenarios requires an understanding of the  data and the physics in order to construct the physical and anomaly-augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For the event-  level dataset discussed in Section 2 we consider three physical augmentations the data:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Azimuthal rotations  The whole final state is rotated by an angle φ randomly sampled from [0,2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' η − φ smearing  The (η,φ) coordinate of every object in the event is resampled according from a Normal  distribution centred on the original coordinate and with a variance inversely proportional to  the pT, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' η′ ∼ N (η,σ(pT)) and φ′ ∼ N (φ,σ(pT)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Energy smearing  The pT of every object in the event is re-sampled according to p′  T ∼ N (pT, f (pT)) with f (pT)  determining the strength of the smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  These augmentations reflect both the symmetries in the data and the experimental resolution of  the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Detectors are imperfect, especially in measuring jet energies, and we encode this in  the representations of the data through the energy-smearing augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Here we re-sample  the jet pT’s as p′  T ∼ N (pT, f (pT)), where f (pT) =  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='052p2  T + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='502pT is the energy smearing  applied by Delphes (the pT’s are normalised by 1GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' If not explicitly mentioned, we always  assume units of GeV for energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For the anomaly-augmentation we consider some very simple  scenarios:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Multiplicity shift, x′  i = m(xi)  For each event m(·) adds a random number of electrons, muons, and jets to the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The  number is chosen randomly within the limits (ne,4− ne), (nµ,4− nµ), and (nj,10− nj) for  electrons, muons, and jet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The azimuthal angle and pseudo-rapidities are also  chosen randomly within the limits allowed, and the pT for each object is chosen as a random  fraction of the maximum pT in the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Once the objects have been added, the MET of the  event is recalculated and updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Multiplicity shift, keeping MET and the total pT constant, , x′  i = m(xi)  This is similar to the above augmentation, but now m(·) generates the extra objects by splitting  the existing objects and smearing the η−φ coordinates using the function used in the physical  augmentations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' pT and MET shifts, x′  i = spT (xi)  Here spT (·) shifts the pT’s in the event by the same random factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We randomly choose whether  we shift just the MET, just the reconstructed object pT’s, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' And we ensure that the the  trigger selection is not spoiled by these shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  With the physical augmentations we apply all of them simultaneously, whereas for the anomaly-  augmentations we apply just one augmentation to each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The augmentation that is applied is  selected randomly and uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We do not apply both a physical augmentation and an anomaly-  augmentation to the events in s(zi,z∗  i ), since this would conflict with the optimisation goal of the  s(zi,z′  i) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' It would also be possible to have an anomaly-augmentation that removes objects  from the event, however this effect is already captured by the augmentation that adds objects to  the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Many of the events in the background have the minimal multiplicity allowed by the  applied cuts, so the effect of an anomaly-pair with a low multiplicity background event and the  same event augmented to have more objects is the exact same as the effect of an anomaly-pair  8  SciPost Physics  Submission  with a high-multiplicity background event augmented to have less objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This is because of the  symmetry in the distance function s(zi,z∗  i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' So the anomaly-augmentations here are as general  as can be, and do not target any specific new physics scenario, therefore the technique should be  model-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Architecture and training  The collider event data being used has a well-defined structure:  MET: one entry with (pT,η,φ)  Electrons: four entries, each with (pT,η,φ)  Muons: four entries, each with (pT,η,φ)  Jets: ten entries, each with (pT,η,φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  This amounts to a 19 × 3 array, with the electrons, muons, and jets being ordered by pT and hav-  ing zero-padded entries where there is less than the maximum allowed number of reconstructed  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The multiplicity is typically much less than the maximum allowed, so the data for a  single collider event can have many zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The transformer allows us to avoid this by having a  permutation-invariant and variable length input format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Because the data is now processed in  a permutation-invariant way, the information on which entry corresponds to which object (MET,  electron, muon, or jet) is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We reinstate this information by adding a one-hot encoded ID vec-  tor to (pT,η,φ), with a 1 indicating the correct ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This means that each reconstructed object is  now represented by a 7D vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Before passing the kinematic data to the transformer we do some  very minor preprocessing to make sure that the numbers the networks see are O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Specifically,  we divide all MET and pT values by the average pT of all objects (electrons, muons, jets) in the  background dataset, we do not shift the values to be centred on zero because the distribution is  highly peaked at zero and we want the preprocessed data to have the same sparsity as the original  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We then divide all η and φ values by 4 and π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' When training the AutoEncoder  networks discussed in the next section we use the same preprocessing of the data, this ensures  that any difference in the results can be attributed to AnomalyCLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  The transformer starts by projecting each object to a larger vector whose dimension is deter-  mined by the embedding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The embeddings for each object are then passed through the  transformer, with a feed-forward network between each transformer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The output from the  transformer has a dimension of (n× model dimension) with n being the number of objects in the  event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The last steps are to sum over the n vectors in this output, which enforces the permutation-  invariance, and to pass this vector through a fully-connected head network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The output of this  head network is what is passed to the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For more details on the architecture we re-  fer the reader to [53], here we only list the hyper-parameters used in training the network in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The representation used in the anomaly detection task is taken from the output of the  transformer network, before being passed through the head network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' It is well documented in the  machine learning literature that these intermediate representations from self-supervised networks  generally contain more discriminating features, for example in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  9  SciPost Physics  Submission  hyper-parameter  model (embedding) dimension 160  feed-forward hidden dimension 160  output dimension  160  # self-attention heads  4  # transformer layers (N)  4  # layers  2  dropout rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='1  hyper-parameter  optimiser  Adam(β1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='9, β2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='999)  learning rate  5 × 10−5  batch size  128  # epochs  500  Table 1: Default setup of the transformer-encoder network and the AnomalyCLR train-  ing, unless noted explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  5  Anomaly scores  The basic flow in an AutoEncoder involves two steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' (i) mapping high-dimensional input data  to a compressed latent space using a neural network called an encoder, and (ii) mapping the  compressed latent space representation to a reconstructed version of the input data using a neural  network called a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We refer to the encoder network as e(·) and the decoder network as d(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  With input data of dimension D, and a bottleneck of dimension B, the encoder maps e : �D → �B,  while the decoder maps d : �B → �D, with the AutoEncoder defined as h = e ◦ d : �D → �D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Acting on a single input ⃗x, the AutoEncoder returns ⃗x′ = h(⃗x), and is optimised to minimise the  mean-squared-error (MSE) loss function between the input and reconstructed input,  L(⃗x,θ) =  �  ⃗x − ⃗x′�2 ,  (6)  where θ represents the learnable parameters of the AutoEncoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In the limit where the AutoEn-  coder is able to reconstruct inputs perfectly, which is guaranteed to be possible when D = B, the  function hθ(·) is simply the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' But with B < D the AutoEncoder may not be able to perfectly  reconstruct all features in the data, and therefore it should learn to reconstruct only the most  common or prominent features in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This means that events containing rare or anomalous  features should have a larger ‘reconstruction loss’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' L(⃗x,θ), and this can then be used as the  anomaly score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  The encoder and decoder networks have 5 feed forward layers each with 256, 128, 64, 32, and  16 neurons, connected by a 5-dimensional bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The activation function between layers is  a LeakyReLU with default slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The decoder is a mirrored version of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We don’t apply  regularization techniques during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The training is performed using Adam optimiser with  learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='001 for 100 epochs, the batch size is 4096, and the number of SM events used is  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Note that we have not optimised the AutoEncoder architecture, simply choosing the same  architecture used in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Instead we have only ensured that they are trained to convergence and  that the training is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The AutoEncoder is trained on both the representations obtained from  contrastive learning and the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In the case of the raw data we apply the same preprocessing  to the data as is applied to the data in the contrastive learning network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In this way we ensure  that any differences in the anomaly detection performance can be attributed to the contrastive  learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  10  SciPost Physics  Submission  6  Results  In this section we present some results using the different techniques discussed in the preceding  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The results here are three-fold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' we first compare the different methods based on anomaly  detection performance, we then study the effects of the different anomaly-augmentations on the  AnomalyCLR performance, and lastly we look at the effect of the representation dimension on the  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='1  Comparison of methods  We compare the methods using the ROC (Receiver Operating Characteristic) curves, the SI (Signif-  icance Improvement) curves, and the AUC (Area Under the ROC Curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The baseline we compare  to is the AutoEncoder trained on raw kinematic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We present results using the CLR method  without anomaly-augmentations (LCLR), and the CLR method with anomaly-augmentations (both  LAnomCLR and L+  AnomCLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' So we have 4 methods in total to compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For all results on the raw  data we have trained 5 AutoEncoder networks and taken the central value and the error estimation  from the mean and standard deviation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For the CLR methods we also aggregate over  5 different runs, where in each run we train a different transformer network and a different Au-  toEncoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The CLR representations have a dimension of 160 and where anomaly-augmentations  are used we have used them all as outlined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' 1 we present AnomalyCLR results  using L+  AnomCLR and see that it leads to significant improvements over the raw data representations,  not only in the AUC but also at all signal efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' In the Significance Improvement (SI) curves  we also see large improvements, with the SI being between ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='5−4 for A → 4l and h+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We can see  from Table 2 that the L+  AnomCLR loss function is clearly advantageous over LAnomCLR, beating it on  all signals with the exception of A → 4l, where LAnomCLR achieves better performance at εs =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  A point of interest here is that the AutoEncoder on raw data outperforms the AutoEncoder on the  CLR representations in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This is likely due to the fact that traditional CLR optimises  for uniformity, and since it is trained on background only, the mapping is not optimised to sepa-  rate SM-like background events from any event which may look different to that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The benefit of  Signal  AE-Raw  CLR  AnomCLR  AnomCLR+  AUC  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='885(2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='880(7)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='907(6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='909(3)  h0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='755(2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='740(5)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='765(4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='776(2)  h+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='900(4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='87(1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='913(2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='930(1)  LQ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='856(2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='841(9)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='854(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='880(1)  ε−1  b (εs =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3)  A  47(2)  80(22)  156(34)  139(20)  h0  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='9(7)  11(1)  18(1)  23(1)  h+  60(10)  28(6)  98(9)  171(7)  LQ  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='4(6)  18(2)  28(3)  39(1)  SI(εs =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3)  A  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='05(5)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='7(4)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='7(4)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='5(2)  h0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='16(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='99(4)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='26(4)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='44(3)  h+  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3(2)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='6(2)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='0(1)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='9(1)  LQ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='48(2)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3(1)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='6(1)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='88(2)  Table 2: Comparison of the different CLR loss functions, with and without anomaly-  augmentations, and the AE trained on raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  11  SciPost Physics  Submission  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='0  ϵs  100  101  102  103  ϵ−1  b  AE-Raw  A, AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='885(2)  h0, AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='755(2)  h+, AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='900(4)  LQ, AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='856(2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+page_content='0  ϵs  100  101  102  103  ϵ−1  b  AnomCLR+  A, AUC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='909(3)  h0, AUC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='776(2)  h+, AUC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='930(1)  LQ, AUC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='880(1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+page_content='0  ϵs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+page_content='0  ϵs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+page_content='0  SI  AnomCLR+  A  h0  h+  LQ  Figure 1: Comparison between the AE on raw data and the AE on the CLR representa-  tions trained with the L+  AnomCLR loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  anomaly-augmentations here is strikingly clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='2  The effect of anomaly-augmentations  We now want to study how the addition of the individual anomaly-augmentations affects the  anomaly detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For this we use just L+  AnomCLR , however we expect the results  with LAnomCLR to be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We use a representation dimension of 160 and obtain the error  estimate from just 2 runs due to the computational cost of the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  We can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' 2 that the affect of the augmentations together results in the best over-  all performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' One thing we noticed is that it can be difficult to determine from the affect of  individual augmentations, or subgroups of them, what the performance of all of them together  will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For example, in most cases if we take just the m(x) augmentation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' the multiplicity  augmentation that simply adds reconstructed objects, we see that it alone decreases performance  below baseline for three out of four signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' However when used in combination with the others  it increases the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This is most clear for the leptoquark signal, where all augmenta-  12  SciPost Physics  Submission  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='850  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='900  A  AUC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='700  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='750  h0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+page_content='900  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='950  h+  m(x)  m(x)  spT(x) m(x)/m(x)  all  anomaly augmentations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='750  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='850  LQ  Figure 2: Results of a scan on the anomaly-augmentations used with the L+  AnomCLR loss  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The augmentations are defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The dashed lines here correspond  to the AutoEncoder on raw data baseline performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  tions taken individually result in a performance which is at or below baseline, but when taken  together we get a significant boost in the AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We also see the interplay between the m(x) and  m(x) augmentations, since individually these augmentations do not seem to help much, but when  they are both applied in the same optimisation we see a reduced error and in most cases better  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' When drawing conclusions here we should keep in mind that only two runs for each  combination have been used to compute the mean and error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3  The effect of representation dimension  With CLR we can project our raw data from D to a representation of any dimension we like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  We would expect that the larger the representation dimension the more information that can be  encoded in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' However we also expect that this would plateau or even peak at some point,  and this what we want to investigate here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' For this we use just L+  AnomCLR , however we expect the  results with LAnomCLR to be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Here we also obtain the error estimate from just 2 runs due  to the computational cost of the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' 3 we see that increasing the representation dimension certainly improves the perfor-  mance of the anomaly detection, at least up until a certain point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The A, h0, and LQ signals  all appear to achieve peak performance somewhere between dimensions 120 and 200, while the  h+ signals performance increases right up until 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' There is no fundamental limitation related  to the representation size which we would expect to cause a degradation at larger dimensions,  however there are two points we should keep in mind here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The first is simple, these means and  variances are calculated with only two runs, so more runs might present a clearer picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The  13  SciPost Physics  Submission  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='850  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='875  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='900  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='925  A  AUC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='740  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='760  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='780  h0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='900  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='920  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='940  h+  4  8  12  20  40  80  120  160  200  300  400  500  representation dimension  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='840  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='860  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='880  LQ  Figure 3: Results of a scan on the representation dimension used with the L+  AnomCLR loss  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The dashed lines here correspond to the AutoEncoder on raw data baseline  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  second point is that we have not optimised the AutoEncoder architecture or hyper-parameters as  the representation size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' While it is beyond the scope of this paper, it is possible that  an independent hyper-parameter optimisation for each representation dimension would improve  these results, particularly at larger dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' What these results show is that there is a clear  tendancy for the results to improve as we increase from dimensions of ∼ 4 to ∼ 100, as we would  naturally expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  7  Summary & conclusions  In this paper we have introduced AnomalyCLR§, a new method for density-based anomaly detec-  tion in high-energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' It makes use of anomalous augmentations of collider data to build a  representation space from which to construct anomaly scores with a range of methods, for exam-  ple using AutoEncoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' It is a self-supervised method, based on the contrastive learning idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We  tested this method on the CMS ADC dataset, and compared to the raw data baselines we find large  improvements on all signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' At a fixed signal efficiency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='3 and a fixed representation dimen-  sion of 160 we find significance improvements for the different signals in the range of 14−70%,  and a decreased relative error on the significance improvement in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Allowing for varying  signal efficiencies and representation dimensions would improve these performance markers even  §The AnomalyCLR code, along with the event-level anomaly detection application, will be made available at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='com/bmdillon/AnomalyCLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  14  SciPost Physics  Submission  further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Density-based anomaly detection, using AutoEncoders or normalising flows, suffer from the  ambiguity that a change in the ‘coordinate system’ or representation of the data results in a fun-  damental change in how the anomaly score is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This makes it difficult to choose a suitable  representation by hand, for example a simple re-mapping of pT’s along with some re-scaling of  numerical inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' These simple choices are difficult to motivate from a physics perspective and  can drastically change the results of the anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This change can be for better or for  worse, and typically depends on the signal models used to test the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  AnomalyCLR addresses this by constructing a representation of the data using self-supervised  contrastive learning with the addition of anomaly-augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The anomaly-augmented data  is constructed from the background data through feature augmentation, designed to emulate a  generic anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We have discussed in detail how we do this for the event-level anomalies in  the CMS ADC dataset, however this would of course be different in different physics cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We  proposed a new loss function which we use to train a deep transformer-based neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This  network projects the events to a new representation, in which the anomaly-augmented events are  far from their original counterparts, while being close to events which are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' The transformer  network then learns a highly discriminative representation of the events which is sensitive to  the presence of potential anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' We have seen that the choice of these augmentations is  quite model agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This model-agnostic nature of the approach can be seen in how the results  improve across all four signals considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  We have shown the effectiveness of self-supervision and the idea of anomaly-augmentations  in significantly enhancing anomaly detection performance in a model-agnostic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' This opens  the door to further studies, such as improving the density-estimation portion of the method with a  more sophisticated hyper-parameter optimisation of the AutoEncoders, using normalising flows,  or even using the Normalised AutoEncoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' More generally, the use of anomaly-augmented data  could be explored further in other anomaly detection approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  Acknowledgements  We would like to thank Jernej Kamenik and Ben Nachman for their helpful comments on the  manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' BMD acknowledges funding from the Alexander von Humboldt Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' LF, TM,  and TP are funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)  under grant 396021762 – TRR 257: Particle Physics Phenomenology after the Higgs Discovery  and Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Ex-  cellence Cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  References  [1] A general search for new phenomena with the ATLAS detector in pp collisions at �s = 8  TeV, ATLAS-CONF-2014-006 (ATLAS-CONF-2014-006) (2014), https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+page_content=' Butter, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Dillon and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+page_content=' Nachman and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Thaler, Classification without labels: Learning from mixed  samples in high energy physics,  JHEP 10, 174 (2017),  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+page_content='02949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='  [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Nachman and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Shih, Anomaly Detection with Density Estimation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content=' D 101,  075042 (2020), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
+page_content='075042, arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfrgpm/content/2301.04660v1.pdf'}
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+On the accuracy of one-way approximate models
+for nonlinear waves in soft solids
+Harold Berjamin a
+aSchool of Mathematical and Statistical Sciences, University of Galway, University Road, Galway, Republic of Ireland
+Abstract
+A simple strain-rate viscoelasticity model of isotropic soft solid is introduced. The constitutive equations account for
+finite strain, incompressibility, material frame-indifference, nonlinear elasticity, and viscous dissipation. A nonlinear
+viscous wave equation for the shear strain is obtained exactly, and a corresponding one-way Burgers-type equation
+is derived by making standard approximations. Analysis of the travelling wave solutions shows that the two partial
+differential equations produce distinct solutions, and that deviations are exacerbated when wave amplitudes are not
+arbitrarily small. In the elastic limit, the one-way approximate wave equation can be linked to simple wave theory,
+thus allowing direct error measurements.
+1
+Introduction
+In nonlinear acoustics, the Burgers equation is often viewed as the simplest model equation that includes nonlinear
+wave propagation and diffusion effects (Witham, 1999). This partial differential equation in space and time can be
+derived directly from the one-dimensional Navier–Stokes equation by dropping the pressure term, or as a special case
+of the Westerwelt equation. Besides Burgers’ equation, other one-way wave equations have been derived to describe
+wave propagation in fluids and solids at large amplitudes (Hamilton and Blackstock, 1998; Naugolnykh and Ostro-
+vsky, 1998). Based on an appropriate scaling of the wave amplitude, such approximate partial differential equations
+describe unidirectional wave motion for slowly-varying wave profiles of moderate amplitude.
+One-way approximate wave equations have found applications in various areas of nonlinear acoustics. For in-
+stance, works by Radostin et al. (2013) and Nazarov et al. (2017) describe compression wave propagation in solids with
+bimodular elastic behaviour. Another example is the Zabolotskaya equation that describes unidirectional plane shear
+wave propagation in soft solids such as gels and brain tissue (Zabolotskaya et al., 2004), see also Cormack and Hamil-
+ton (2018). In these latter cases, the underlying three-dimensional constitutive theories were revisited by Destrade
+et al. (2013) as well as Saccomandi and Vianello (2021) to enforce objectivity (i.e., invariance by change of observer),
+leading to slight modifications of the equations of motion.
+For these partial differential equations, not many analytical solutions are known. Nevertheless, it is sometimes
+possible to derive exact stationary wave solutions that keep an invariant wave profile throughout the motion, which
+occurs at a suitable constant speed. Those permanent waveforms result from the interaction between nonlinearity
+and dispersion (here of dissipative nature), a common feature that they share with solitary waves.
+One might wonder whether it is preferable to seek closed-form travelling wave solutions by using directly the
+full equations of motion, or by using their one-way approximation. As a matter of fact, both approaches have been
+considered separately in the above literature. The present study aims to provide evidence to advocate for a derivation
+of travelling waves based on the complete equations of motion, thus supporting a remark by Jordan and Puri (2005)
+in relation with the study by Catheline et al. (2003) — this remark led to the publication of an erratum that briefly
+discusses the validity of a particular one-way wave equation (Catheline et al., 2005).
+For this purpose, we consider the case of shear wave propagation in soft viscoelastic solids of strain rate type. We
+derive the simplest three-dimensional constitutive theory that accounts for finite strain, incompressibility, material
+frame-indifference, and viscous dissipation (Section 2). Then, this theory is applied to simple shear deformations, aka.
+transverse plane waves (Section 3), including the reduction to a one-way model described by a Burgers-type equation
+with cubic nonlinearity. Finally, we investigate the travelling wave solutions deduced from the full equations of motion
+as well as from the reduced wave equation (Section 4). Results show non-negligible discrepancies introduced by the
+reduction to unidirectional motion as soon as wave amplitudes are no longer infinitesimal. These comparisons are
+reconsidered in the lossless elastic limit where connections between the one-way model and simple wave theory are
+established (Section 5).
+1
+arXiv:2301.03284v1  [cond-mat.soft]  9 Jan 2023
+
+2
+Strain-rate model
+2.1
+Basic equations
+In what follows, we present the basic equations of Lagrangian dynamics for incompressible solids (Holzapfel, 2000).
+We consider a homogeneous and isotropic solid continuum on which no external body force is applied. Its motion
+in the Euclidean space is described by using an orthonormal Cartesian coordinate system (O,x, y,z). Thus, a particle
+initially located at some position X of the reference configuration moves to a position x of the current configuration.
+The deformation gradient is the second-order tensor defined as F = ∂x/∂X . Introducing the displacement field u =
+x − X and the identity tensor I = [δi j ] whose components are represented by Kronecker’s delta, we therefore have
+F = I +Gradu where Grad denotes the gradient operator with respect to the material coordinates X = (x, y,z).
+In incompressible solids, isochoricity
+J = detF ≡ 1
+(1)
+is prescribed. Thus, the mass density ρ is constant in time. It follows also that ˙J = JF −⊺ : ˙F ≡ 0, where the dot de-
+notes the material time derivative ∂/∂t and the colon indicates double contraction. Introducing the Eulerian velocity
+gradient L = ˙FF −1, this condition can be rewritten as trL = 0.
+Various strain tensors are defined as functions of F. Here, constitutive laws are expressed in terms of the Green–
+Lagrange strain tensor E = 1
+2(F ⊺F − I), which is often a preferred choice in physical acoustics. We introduce also its
+rate ˙E = F ⊺DF obtained by differentiation with respect to time, where D = 1
+2(L +L⊺) is the strain rate tensor. We note
+that D is trace-free due to incompressibility (1).
+The motion is governed by the conservation of linear momentum equation ρ ˙v = DivP, where v = ˙x is the velocity
+field and ρ is the mass density. The equation of motion involves the Lagrangian divergence of the first Piola–Kirchhoff
+stress tensor P = FS where S is the second Piola–Kirchhoff stress tensor. Those stress tensors are specified later on by
+the provision of a constitutive law.
+The present definitions are consistent with notations and conventions used in the monograph by Holzapfel (2000).
+In particular, the divergence of the tensor P reads [DivP]i = Pi j,j componentwise, where indices after the coma de-
+note spatial differentiation. In some other texts, a transposed definition of the divergence is used. Then, the equation
+of motion involves the material divergence of the nominal stress tensor P⊺ instead of P.
+2.2
+Generalities
+In the present study, we consider deformable solids whose constitutive behaviour is described by the state vari-
+ables S = {s,E}, where s is the specific entropy. The choice of variables S is coherent with the postulate of frame-
+indifference of the internal energy (Holzapfel, 2000). In fact, a change of observer specified by a superimposed rigid-
+body motion leaves S invariant, as well as the internal energy U. Note that the internal energy does not depend on
+rates of strain.
+The internal energy per unit volume U is a function of state to be specified. The thermodynamic temperature
+is defined as the conjugate variable of s in the partial Legendre transform of U/ρ with respect to s (Berjamin et al.,
+2021). However, the explicit dependence of U with respect to s is usually omitted in the definition of a strain energy
+density function W e such that U = W e(E). The strain energy W e is regarded as a scalar-valued isotropic function of
+its arguments. Thus, its dependence with respect to E can be reduced to a dependence with respect to three scalar
+invariants
+I1 = tr(E),
+I2 = tr(E2),
+I3 = tr(E3).
+(2)
+They can be used directly, or other physically meaningful scalar quantities might be defined from them.
+The first and second principles of thermodynamics yield the Clausius–Duhem inequality
+D = (S −Se) : ˙E = Sv : ˙E ≥ 0,
+(3)
+where D is the dissipation, S = Se +Sv is the total second Piola–Kirchhoff stress,
+Se = −pC −1 + ∂W e
+∂E
+(4)
+denotes the elastic part, and Sv is a viscous contribution to be specified subsequently. The scalar p is an arbitrary
+Lagrange multiplier for the incompressibility constraint (1), see Sec. 6.3 of Holzapfel (2000), and C = I +2E is the right
+Cauchy–Green strain tensor F ⊺F. Therefore, no dissipation occurs in the elastic case S = Se where the viscous stress
+tensor Sv is equal to zero.
+According to the dissipation inequality (3), the viscous stress Sv is a function of state and evolution variables, e.g.
+the set S ∪ { ˙E} which is a consistent choice to enforce frame-invariance (Antman, 1998; Ball, 2002). We introduce a
+dissipation potential W v(E, ˙E) such that
+Sv = ∂W v
+∂ ˙E
+(5)
+2
+
+defines the viscous stress (Maugin, 1999). In general, the dissipation potential is described by additional invariants
+(Pioletti and Rakotomanana, 2000)
+I4 = tr( ˙E),
+I5 = tr( ˙E2),
+I6 = tr( ˙E3),
+I7 = tr( ˙EE),
+I8 = tr( ˙EE2),
+I9 = tr( ˙E2E),
+I10 = tr( ˙E2E2).
+(6)
+In the present study, we consider Newtonian-type viscosity models whose dissipation potential is as simple as possi-
+ble.
+2.3
+Consequences of incompressibility
+First, let us investigate the consequences of the incompressibility constraint (1). As noted in Jacob et al. (2007), the
+invariants (2) of E are linked through
+I1 = I2 − 4
+3 I3 − I 2
+1 +2I1I2 − 2
+3 I 3
+1 ,
+(7)
+by virtue of incompressibility. This identity follows from the expression of the principal invariants of the unimodular
+tensor C = I +2E in terms of the invariants Ik, see the Appendix of Destrade et al. (2010). Using the differential version
+of the incompressibility constraint, the invariants (2)-(6) of E, ˙E satisfy the particular relationship
+1
+2 I4 = I7 −2I8 +2I1I7 −
+�
+I1 − I2 + I 2
+1
+�
+I4
+(8)
+deduced from the identity trD = 0, see Appendix.
+The relationship (7) means that the invariant I1 = tr(E) is no longer linear with respect to the components of the
+strain tensor E; instead, Eq. (7) shows that it has terms of polynomial order two and three with respect to the strain.
+Furthermore, due to the relationship (8), the invariant I4 = tr( ˙E) is still linear with respect to the components of the
+strain-rate tensor ˙E. However, Eq. (8) shows that I4 is no longer invariant on the strain tensor E; instead, it has terms
+of polynomial order one, two and three with respect to the Green–Lagrange strain.
+2.4
+Constitutive assumptions
+In weakly nonlinear elasticity, the strain energy density function is sought in the form of a polynomial of the invariants
+Ik with constant coefficients. Similarly to Zabolotskaya et al. (2004), we assume that the internal energyU has a fourth-
+order polynomial expression with respect to the components of the strain tensor E of the form
+W e = µI2 + 1
+3 AI3 +DI 2
+2,
+(9)
+where µ ≥ 0 is the shear modulus (in Pa), and the coefficients A, D are higher-order elastic constants.
+Now, let us propose an expression for the dissipation potential. To end up with a linear viscosity model similar to
+that by Destrade et al. (2013), we assume that the dissipation potential is a second-order polynomial expansion of the
+strain rate tensor ˙E, and a zeroth-order polynomial of E. This assumption amounts to selecting W v of second order
+in (E, ˙E), and to ignore the terms proportional to ˙E that produce elastic stresses. Due to the relationships (7)-(8), we
+therefore keep
+W v = ηI5,
+(10)
+where η ≥ 0 is the shear viscosity (in Pa.s). In the above expression, the absence of bulk viscosity “ζ” is due to the
+assumption on polynomial orders for the viscous part, and to the incompressibility property (8). Setting the bulk
+viscosity ζ = 2
+3η in Destrade et al. (2013) yields the same expressions as above.
+Computation of the tensor derivatives of the potentials (9)-(10) by means of the chain rule for W •(Ik,...) yields the
+following elastic (4) and viscous stress contributions (5)
+Se = −pC −1 +2(µ+2DI2)E + AE2,
+Sv = 2η ˙E.
+(11)
+Thermodynamic consistency (3) is ensured provided that the dissipation D = 2W v is non-negative. In fact, the present
+dissipation potential W v is a homogeneous function of degree two with respect to ˙E (Maugin, 1999). A sufficient
+condition for the restriction D ≥ 0 to be always satisfied is that the viscosity η is non-negative.
+3
+
+3
+Plane shear waves
+3.1
+Nonlinear viscous wave equation
+Similarly to Destrade et al. (2013), we consider simple shear deformations described by the displacement field u =
+[u,0,0]⊺ where u = u(z,t) denotes the particle displacement along the x-direction. Thus, the deformation gradient
+tensor reads
+F =
+�
+�
+1
+0
+γ
+0
+1
+0
+0
+0
+1
+�
+�,
+(12)
+where γ = ∂u/∂z is the shear strain. The velocity field takes the form v = [v,0,0]⊺ where v = ∂u/∂t is the shear velocity.
+In the equation of motion ρ ˙v = DivP, the relevant first Piola–Kirchhoff stress component P13 is deduced from
+the expression of the elastic and viscous parts Pe
+13 = µγ+Γγ3 and Pv
+13 = η(1+2γ2) ˙γ, where only terms up to order γ3
+have been kept. The non-negative constant Γ = µ+ A/2+D is a parameter of nonlinearity (Zabolotskaya et al., 2004).
+Hence, upon division by the shear modulus µ, the x-component of the equation of motion produces the nonlinear
+wave equation
+1
+c2
+∂2u
+∂t2 = ∂2u
+∂z2 + 2
+3β ∂
+∂z
+�∂u
+∂z
+�3
++τ ∂
+∂z
+��
+1+2
+�∂u
+∂z
+�2� ∂2u
+∂z∂t
+�
+,
+(13)
+describing transverse wave propagation along the z-direction, where we have introduced the notations
+c =
+�
+µ
+ρ ,
+β = 3
+2
+Γ
+µ,
+τ = η
+µ.
+(14)
+Spatial differentiation of Eq. (13) allows to write a similar wave equation for the strain
+1
+c2
+∂2γ
+∂t2 = ∂2γ
+∂z2 + 2
+3β ∂2
+∂z2 γ3 +τ ∂2
+∂z2
+��
+1+2γ2� ∂γ
+∂t
+�
+,
+(15)
+which will be used later on.
+According to the wave equations (13)-(15), shear waves of infinitesimal amplitude propagate at the shear wave
+speed c =
+�
+µ/ρ in the absence of nonlinearity and viscosity (β = 0, τ = 0). Typically, this sound velocity equals
+c ≈ 2 m/s in gels (Jacob et al., 2007), whereas β ≈ 10 and τ ≈ 0.12 ms at a loading frequency of 100 Hz. Here, we have
+obtained the same wave equations than those derived in Destrade et al. (2013) for the particular bulk viscosity ζ = 2
+3η.
+Note in passing the presence of a nonlinear viscous term which is absent in Zabolotskaya et al. (2004).1
+3.2
+Slow scale approximations
+Similarly to Zabolotskaya et al. (2004) and Pucci et al. (2019), we proceed now to a reduction of the above wave equa-
+tion (13) for one-way wave propagation with slowly varying profile. We present two approximations based either on a
+slow space variable or a slow time variable.
+Slow space
+Let us follow the scaling procedure in Zabolotskaya et al. (2004). For this purpose, we introduce the
+following scaling defined by the change of variables {˜z = ϵ2z, ˜t = t − z/c,u = ϵ ˜u}, where ϵ is a small parameter and
+˜u = ˜u(˜z, ˜t). Furthermore, we assume that τ is of order ϵ2. Note that this set of assumptions corresponds to a slowly-
+varying profile in space.
+This Ansatz is then substituted in the equation of motion (13). At leading (cubic) order in ϵ, the motion of soft
+viscous solids is governed by the scalar equation
+ϵ3c ∂2 ˜u
+∂˜z∂˜t = ϵ3 β
+c2
+�∂ ˜u
+∂˜t
+�2 ∂2 ˜u
+∂˜t2 +ϵτ
+2
+∂3 ˜u
+∂˜t3 .
+(16)
+Transforming back to the initial displacement u and physical coordinates (z,t) leads to a reduced wave equation
+c ∂v
+∂z +
+�
+1−βv2/c2� ∂v
+∂t = τ
+2
+∂2v
+∂t2 ,
+(17)
+for the velocity v = ∂u/∂t.
+1The wave equation proposed by Catheline et al. (2003) and analysed by Jordan and Puri (2005) cannot be obtained rigorously from the equations
+of motion unless time derivatives are (questionably) replaced by spatial derivatives.
+4
+
+Up to the choice of time variable used here (i.e., the physical time t instead of the retarded time ˜t), the partial
+differential equation (17) is identical to the cubic Burgers-type equation of Zabolotskaya et al. (2004). However, the
+underlying modelling assumptions are not equivalent, since the initial wave equation (13) includes the extra nonlinear
+viscosity term 2τ∂(γ2 ˙γ)/∂z. This additional term is lost in the rescaling procedure given that it is of higher order in ϵ
+than the leading-order viscous term τ∂ ˙γ/∂z. In the end, while the modelling efforts by Destrade et al. (2013) aimed
+at enforcing objectivity lead to a slight modification of the wave equation (more precisely, the addition of a nonlinear
+viscous term), they do not induce any modification of the transport equation (17).
+Slow time
+For later comparisons, let us derive a similar Burgers-type equation governing the evolution of the strain
+instead of the velocity by following Pucci et al. (2019). To do so, we introduce the slow-time scaling based on the
+change of variables {˜t = ϵ2t, ˜z = z −ct,u = ϵ ˜u} where ϵ is a small parameter. Proceeding in a similar fashion to above,
+we end up with the nonlinear transport equation
+∂γ
+∂t +c
+�
+1+βγ2� ∂γ
+∂z = τc2
+2
+∂2γ
+∂z2 ,
+(18)
+where γ = ∂u/∂z is the shear strain. Here too, after keeping leading order terms, we have transformed back to the
+initial physical coordinates (z,t). Therefore, the above partial differential equation may be viewed as a one-way ap-
+proximation of the wave equation (15). Their travelling wave solutions are compared in the next section.
+4
+Travelling wave solutions
+4.1
+Nonlinear viscous wave equation
+Let us seek travelling wave solutions to the wave equation (15), i.e. specific smooth waveforms that propagate at a
+constant velocity with a steady profile. In a similar fashion to Destrade et al. (2013), we first introduce the following
+rescaled dimensionless variables and coordinates
+g(¯z, ¯t) =
+�
+2
+3βγ(z,t),
+¯t = t/τ,
+¯z = z/(cτ),
+(19)
+in Eq. (15), such that
+∂2g
+∂¯t2 = ∂2g
+∂¯z2 + ∂2
+∂¯z2 g 3 + ∂2
+∂¯z2
+��
+1+ 3
+β g 2
+� ∂g
+∂¯t
+�
+.
+(20)
+Next, we seek travelling wave solutions of the form g =
+�
+ν2 −1G(ξ) where ξ = (ν2 − 1)(¯t − ¯z/ν) involves the dimen-
+sionless wave velocity ν ≥ 1. Injecting this Ansatz in the above partial differential equation and integrating twice with
+respect to ξ with vanishing integration constants yields a nonlinear differential equation for the strain:
+G = G3 +
+�
+1+αG2� d
+dξG,
+(21)
+where α = 3(ν2 −1)/β is a parameter.
+From the above differential equation, one observes that travelling wave solutions to the wave equation (15) should
+connect the equilibrium strains G = 0 and G = ±1 by following a smooth transition that depends on the parameter α.
+Solutions read (Destrade et al., 2013)
+ξ = −ln
+�
+1
+2G
+�4
+3(1−G2)
+� 1+α
+2
+�
+(22)
+in implicit form, where we have enforced G(0) = 1/2 without loss of generality. Illustrations are provided later on.
+4.2
+Slow time approximation
+In a similar fashion, let us now seek travelling wave solutions to the reduced wave equation (18). Thus, we first perform
+the substitutions (19) to get
+∂g
+∂¯t + ∂
+∂¯z
+�
+g + 1
+2 g 3
+�
+= 1
+2
+∂2g
+∂¯z2 .
+(23)
+In order to obtain wave solutions that correspond to the same strain values at infinity as in Sec. 4.1, we introduce a
+slightly different scaling. Indeed, let us inject the Ansatz g =
+�
+ν2 −1G(χ) with χ = (ν2 − 1)(ϑ¯t − ¯z) in Eq. (23), where
+ϑ = 1+ 1
+2(ν2 −1) is the new dimensionless velocity (Fig. 1). Thus, we arrive at the differential equation
+G = G3 + d
+dχG
+(24)
+5
+
+1
+1.2
+1.4
+1.6
+1.8
+1
+1.5
+2
+ν
+ϑ
+Figure 1: Scaled velocity ϑ = 1+ 1
+2(ν2 −1) for the ‘slow-time’ reduced model in terms of the scaled velocity ν for the full
+wave equation.
+of which the strain values 0 and 1 are steady states. Enforcing the initial value G = 1/2 at χ = 0 gives
+G =
+1
+�
+1+3e−2χ ,
+(25)
+which does not involve any extra parameter. One observes that this expression corresponds to the case α = 0 in
+Eqs. (21)-(22).
+Remark. One might proceed in a similar fashion with the Burgers-type equation (17) corresponding to the slow space
+approximation. Similarly to (19), we perform the substitutions r(¯z, ¯t) =
+�
+2β/3v(z,t)/c in Eq. (17) to get
+∂r
+∂¯z + ∂
+∂¯t
+�
+r − 1
+2r 3
+�
+= 1
+2
+∂2r
+∂¯t2 .
+(26)
+Next, we introduce r =
+�
+ν2 −1V (ψ) where ψ = (ν2 −1)(¯t − ¯z/κ) involves the dimensionless velocity κ defined by the
+relationship κ−1 = 1− 1
+2(ν2 −1). This way, we obtain the same differential equation V = V 3 + d
+dψV for the dimension-
+less velocity V as previously for the strain (24). Therefore, within the scope of the present study, the slow time and
+slow space approximations lead to related travelling wave solutions that describe the evolution of distinct kinematic
+variables (strain and velocity, respectively).
+4.3
+Comparison
+Let us compare the solutions (22)-(25) obtained for the full wave equation (15) and the one-way model (18). First, one
+observes that these travelling waves of same amplitude do not propagate at the same speed, as illustrated in Fig. 1.
+Indeed, given the expression of ϑ, we can express the relative error E = ϑ/ν−1 on the scaled velocity as a function of
+ν. To ensure that the latter remains less than 5% (respectively 1%), we obtain the requirement ν ≤ 1.3 (resp. ν ≤ 1.1)
+marked by dotted lines in the figure.
+Now, let us observe that for a unit kink covering the range 0 ≤ G ≤ 1, the corresponding shear strains satisfy
+0 ≤ γ ≤
+�
+α/2
+(27)
+where α = 3(ν2 − 1)/β was introduced earlier on, see Eq. (19). In other words, the coefficient α in the differential
+equation (21) is related to the maximum strain of travelling waves, and these bounds are valid for both models at hand
+due to application of the rescaling procedure (19). Thus, restrictions of the wave speed ν can be expressed in terms of
+the strain. To ensure that the velocity error E remains less than 5% (respectively 1%), we therefore require γ
+�
+β ≤ 1.0
+(resp. γ
+�
+β ≤ 0.56). Note that the parameter of nonlinearity can take such values as β ≈ 10 for gels (Jacob et al., 2007).
+Therefore, the slow scale approximation has a very restricted validity for such a soft viscoelastic material.
+This property is further illustrated in Fig. 2, where we have represented the evolution of the relative velocity ν−1
+(or ϑ−1) in terms of the maximum strain amplitude, both for the full wave equation and its one-way approximation.
+According to the expression of α above, we have the relationship ϑ−1 = 1
+3β(
+�
+α/2)2 in the case of the one-way approx-
+imate model, which produce lines of slope two in log-log coordinates (dashed lines in the figure). However, for the full
+wave equation, this relationship between the wave speed ν and the strain amplitude is not satisfied. For fixed values
+of the nonlinearity parameter β, differences between the one-way model and the full wave equation become visible
+at large strains.
+In Fig. 3, we display the evolution of the waveforms (22)-(25) in terms of the scaled coordinates ξ, χ. In the case
+of the full wave equation (21), the parameter α takes the values {0,1.2,3}. It appears that the waveforms so-obtained
+follow a drastically different evolution when parameters are modified. In particular, the wavefront deduced from
+the full wave equation (solid lines) does not exhibit the same invariance and symmetry properties as the wavefront
+deduced from the one-way model (dashed line).
+6
+
+10−1
+100
+10−3
+10−2
+10−1
+100
+β = 3
+β = 1
+Strain amplitude
+Relative velocity
+one-way
+wave eq.
+Figure 2: For the full wave equation (solid line) and the ‘slow-time’ reduced model (dashed line), we represent the
+evolution of the relative velocity ν − 1 (respectively, ϑ − 1) in terms of the strain amplitude
+�
+α/2. The axes have a
+logarithmic scale.
+−2
+0
+2
+0
+0.5
+1
+α
+ξ, χ
+G
+wave eq.
+one-way
+Figure 3: Steady waveforms deduced from Eqs. (22)-(25) for increasing values of the parameter 0 ≤ α ≤ 3 (arrow).
+Evolution of the scaled shear strain G in terms of the related dimensionless coordinate ξ or χ.
+7
+
+5
+Simple waves
+In the lossless case, exact one-way wave equations can be derived by using the method of Riemann invariants, see
+for instance the introductory example by John (1976). Such particular wave solutions called simple waves keep one
+Riemann invariant constant. In other words, the particle velocity v = R−−Q(γ) withQ(γ) = c
+�γ
+0
+�
+1+2βg 2 dg depends
+explicitly on the strain γ. The scalar R− is an arbitrary constant, for instance R− ≡ 0 in some specific boundary-value
+problems (Berjamin and Chockalingam, 2022), which will be assumed satisfied from now on. Spatial differentiation
+of the velocity then produces
+∂γ
+∂t +c
+�
+1+2βγ2 ∂γ
+∂z = 0,
+(28)
+where we have used the equality of mixed partials ∂v/∂z = ∂γ/∂t. Obviously, the lossless one-way wave equation (18)
+with τ = 0 is an approximation of (28) for 2βγ2 ≪ 1.
+Let us analyse this requirement in a more quantitative manner. To ensure that the relative error on the advection
+velocity E =
+1+a
+�
+1+2a −1 for a = βγ2 remains less than 5% (respectively 1%), we obtain the requirement a ≤ 0.44 (resp.
+a ≤ 0.16). Application of the square root leads to the restriction γ
+�
+β ≤ 0.66 (resp. γ
+�
+β ≤ 0.40) which is slightly more
+constraining than in the case of viscoelastic travelling waves (Sec. 4.3).
+Along a simple wave, computation of the partial derivative of the velocity v = R− −Q(γ) with respect to time pro-
+duces
+c ∂v
+∂z +
+�
+1+2βγ2�−1/2 ∂v
+∂t = 0,
+(29)
+where the strain γ = Q−1(−v) can be expressed formally as a function of the velocity, despite no analytical expression
+of the inverse function Q−1 of Q is known in the present case. If |v| is small, then we can use the approximation
+γ ≃ −v/c of the strain which follows from the asymptotic equivalence of Q ∼ cγ at small strains. Next, the (·)−1/2-
+factor in Eq. (29) can be approximated by the polynomial expression 1−βγ2 as long as 2βγ2 ≪ 1. This way, we have
+shown that the one-way wave equation (17) is an approximation of Eq. (29) obtained for R− = 0 and 2βv2/c2 ≪ 1 in
+the elastic limit τ = 0. This observation is consistent with the discussions in Catheline et al. (2005). In summary, the
+lossless ‘slow-space’ and ‘slow-time’ reductions (17)-(18) with τ = 0 are approximate governing equations for simple
+waves with small values of βv2/c2 and of βγ2, respectively.
+6
+Conclusion
+For a specific strain-rate viscoelasticity theory of soft solids, we have shown that one-way approximate wave propaga-
+tion models produce significantly different travelling wave solutions than the full equations of motion as soon as the
+wave amplitude is not infinitesimal. Similar observations are reported in the literature in relation with shear shock
+formation (Berjamin and Chockalingam, 2022). In the elastic limit, we have examined the validity of one-way approx-
+imations in relation with simple wave theory, thus leading to dedicated criteria of validity involving small velocity and
+strain amplitudes. We conclude that these approximations should be used with care given their limited accuracy, in
+general. Nevertheless, they might remain useful for the interpretation of experimental results where their validity is
+not always severely penalised (Catheline et al., 2003, 2005).
+Acknowledgments
+The author is grateful to Michel Destrade (Galway, Ireland) for support. This project has received funding from the
+European Union’s Horizon 2020 research and innovation programme under grant agreement TBI-WAVES — H2020-
+MSCA-IF-2020 project No. 101023950.
+A
+Consequence of incompressibility
+This Appendix is devoted to the derivation of Eq. (8). We start with the Cayley–Hamilton identity for the right Cauchy–
+Green tensor C = F ⊤F, which reads
+C 3 −I C 2 +II C −III I = 0,
+(30)
+where I, II, III are the principal invariants of C. In the case of volume-preserving motions (1), the tensor C is uni-
+modular, i.e. we have III = 1. Next, multiplication of (30) by C −1 ˙E on the right side, substitution of C = I + 2E and
+computation of the trace entails the relationship
+(I4 +4I7 +4I8)−(3+2I1)(I4 +2I7)
++(3+4I1 +2I 2
+1 −2I2)I4 = 0,
+(31)
+8
+
+where we have used the incompressibility property trD = tr(C −1 ˙E) = 0, the definition of the invariants (2)-(6), and the
+relationship between I, II and the invariants Ik used here (Destrade et al., 2010). Rearranging terms, we get the desired
+identity (8).
+References
+S. S. Antman.
+Physically unacceptable viscous stresses.
+Z. angew. Math. Phys.,
+49(6):980–988,
+1998.
+doi:10.1007/s000330050134.
+J. M. Ball. Some open problems in elasticity. In P. Newton, P. Holmes, and A. Weinstein, editors, Geometry, Mechanics,
+and Dynamics, pages 3–59. Springer, New York, 2002. doi:10.1007/0-387-21791-6_1.
+H. Berjamin and S. Chockalingam. Shear shock formation in incompressible viscoelastic solids. Wave Motion, 110:
+102899, 2022. doi:10.1016/j.wavemoti.2022.102899.
+H. Berjamin, M. Destrade, and W. J. Parnell. On the thermodynamic consistency of quasi-linear viscoelastic models
+for soft solids. Mech. Res. Commun., 111:103648, 2021. doi:10.1016/j.mechrescom.2020.103648.
+S. Catheline, J.-L. Gennisson, M. Tanter, and M. Fink. Observation of shock transverse waves in elastic media. Phys.
+Rev. Lett., 91(16):164301, 2003. doi:10.1103/PhysRevLett.91.164301.
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+J. M. Cormack and M. F. Hamilton. Plane nonlinear shear waves in relaxing media. J. Acoust. Soc. Am., 143(2):1035–
+1048, 2018. doi:10.1121/1.5023394.
+M. Destrade, M. D. Gilchrist, and J. G. Murphy. Onset of nonlinearity in the elastic bending of blocks. J. Appl. Mech.,
+77(6), 2010. doi:10.1115/1.4001282.
+M. Destrade, G. Saccomandi, and M. Vianello. Proper formulation of viscous dissipation for nonlinear waves in solids.
+J. Acoust. Soc. Am., 133(3):1255–1259, 2013. doi:10.1121/1.4776178.
+M. F. Hamilton and D. T. Blackstock, editors. Nonlinear Acoustics. Academic Press, 1998.
+G. A. Holzapfel. Nonlinear Solid Mechanics: A Continuum Approach for Engineering. John Wiley & Sons Ltd., Chich-
+ester, 2000.
+X. Jacob, S. Catheline, J.-L. Gennisson, C. Barrière, D. Royer, and M. Fink. Nonlinear shear wave interaction in soft
+solids. J. Acoust. Soc. Am., 122(4):1917–1926, 2007. doi:10.1121/1.2775871.
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+Appl. Math., 29(6):649–682, 1976. doi:10.1002/cpa.3160290608.
+P. M. Jordan and A. Puri. A note on traveling wave solutions for a class of nonlinear viscoelastic media. Phys. Lett. A,
+335(2-3):150–156, 2005. doi:10.1016/j.physleta.2004.11.058.
+G. A. Maugin.
+The Thermomechanics of Nonlinear Irreversible Behaviors.
+World Scientific Publishing, 1999.
+doi:10.1142/3700.
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+72–76, 2017. doi:10.1016/j.wavemoti.2017.09.003.
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+(5):749–759, 2000. doi:10.1016/S0997-7538(00)00202-3.
+E. Pucci, G. Saccomandi, and L. Vergori. Linearly polarized waves of finite amplitude in pre-strained elastic materials.
+Proc. R. Soc. A, 475(2226):20180891, 2019. doi:10.1098/rspa.2018.0891.
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+9
+
diff --git a/19E1T4oBgHgl3EQflQTp/content/tmp_files/load_file.txt b/19E1T4oBgHgl3EQflQTp/content/tmp_files/load_file.txt
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@@ -0,0 +1,567 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf,len=566
+page_content='On the accuracy of one-way approximate models  for nonlinear waves in soft solids  Harold Berjamin a  aSchool of Mathematical and Statistical Sciences, University of Galway, University Road, Galway, Republic of Ireland  Abstract  A simple strain-rate viscoelasticity model of isotropic soft solid is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The constitutive equations account for  finite strain, incompressibility, material frame-indifference, nonlinear elasticity, and viscous dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' A nonlinear  viscous wave equation for the shear strain is obtained exactly, and a corresponding one-way Burgers-type equation  is derived by making standard approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Analysis of the travelling wave solutions shows that the two partial  differential equations produce distinct solutions, and that deviations are exacerbated when wave amplitudes are not  arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In the elastic limit, the one-way approximate wave equation can be linked to simple wave theory,  thus allowing direct error measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  1  Introduction  In nonlinear acoustics, the Burgers equation is often viewed as the simplest model equation that includes nonlinear  wave propagation and diffusion effects (Witham, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' This partial differential equation in space and time can be  derived directly from the one-dimensional Navier–Stokes equation by dropping the pressure term, or as a special case  of the Westerwelt equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Besides Burgers’ equation, other one-way wave equations have been derived to describe  wave propagation in fluids and solids at large amplitudes (Hamilton and Blackstock, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Naugolnykh and Ostro-  vsky, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Based on an appropriate scaling of the wave amplitude, such approximate partial differential equations  describe unidirectional wave motion for slowly-varying wave profiles of moderate amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  One-way approximate wave equations have found applications in various areas of nonlinear acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' For in-  stance, works by Radostin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2013) and Nazarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2017) describe compression wave propagation in solids with  bimodular elastic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Another example is the Zabolotskaya equation that describes unidirectional plane shear  wave propagation in soft solids such as gels and brain tissue (Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', 2004), see also Cormack and Hamil-  ton (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In these latter cases, the underlying three-dimensional constitutive theories were revisited by Destrade  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2013) as well as Saccomandi and Vianello (2021) to enforce objectivity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', invariance by change of observer),  leading to slight modifications of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  For these partial differential equations, not many analytical solutions are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Nevertheless, it is sometimes  possible to derive exact stationary wave solutions that keep an invariant wave profile throughout the motion, which  occurs at a suitable constant speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Those permanent waveforms result from the interaction between nonlinearity  and dispersion (here of dissipative nature), a common feature that they share with solitary waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  One might wonder whether it is preferable to seek closed-form travelling wave solutions by using directly the  full equations of motion, or by using their one-way approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' As a matter of fact, both approaches have been  considered separately in the above literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The present study aims to provide evidence to advocate for a derivation  of travelling waves based on the complete equations of motion, thus supporting a remark by Jordan and Puri (2005)  in relation with the study by Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2003) — this remark led to the publication of an erratum that briefly  discusses the validity of a particular one-way wave equation (Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  For this purpose, we consider the case of shear wave propagation in soft viscoelastic solids of strain rate type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' We  derive the simplest three-dimensional constitutive theory that accounts for finite strain, incompressibility, material  frame-indifference, and viscous dissipation (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Then, this theory is applied to simple shear deformations, aka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  transverse plane waves (Section 3), including the reduction to a one-way model described by a Burgers-type equation  with cubic nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Finally, we investigate the travelling wave solutions deduced from the full equations of motion  as well as from the reduced wave equation (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Results show non-negligible discrepancies introduced by the  reduction to unidirectional motion as soon as wave amplitudes are no longer infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' These comparisons are  reconsidered in the lossless elastic limit where connections between the one-way model and simple wave theory are  established (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='03284v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='soft]  9 Jan 2023  2  Strain-rate model  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='1  Basic equations  In what follows, we present the basic equations of Lagrangian dynamics for incompressible solids (Holzapfel, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  We consider a homogeneous and isotropic solid continuum on which no external body force is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Its motion  in the Euclidean space is described by using an orthonormal Cartesian coordinate system (O,x, y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Thus, a particle  initially located at some position X of the reference configuration moves to a position x of the current configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  The deformation gradient is the second-order tensor defined as F = ∂x/∂X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Introducing the displacement field u =  x − X and the identity tensor I = [δi j ] whose components are represented by Kronecker’s delta, we therefore have  F = I +Gradu where Grad denotes the gradient operator with respect to the material coordinates X = (x, y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  In incompressible solids, isochoricity  J = detF ≡ 1  (1)  is prescribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Thus, the mass density ρ is constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' It follows also that ˙J = JF −⊺ : ˙F ≡ 0, where the dot de-  notes the material time derivative ∂/∂t and the colon indicates double contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Introducing the Eulerian velocity  gradient L = ˙FF −1, this condition can be rewritten as trL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Various strain tensors are defined as functions of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Here, constitutive laws are expressed in terms of the Green–  Lagrange strain tensor E = 1  2(F ⊺F − I), which is often a preferred choice in physical acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' We introduce also its  rate ˙E = F ⊺DF obtained by differentiation with respect to time, where D = 1  2(L +L⊺) is the strain rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' We note  that D is trace-free due to incompressibility (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  The motion is governed by the conservation of linear momentum equation ρ ˙v = DivP, where v = ˙x is the velocity  field and ρ is the mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The equation of motion involves the Lagrangian divergence of the first Piola–Kirchhoff  stress tensor P = FS where S is the second Piola–Kirchhoff stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Those stress tensors are specified later on by  the provision of a constitutive law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  The present definitions are consistent with notations and conventions used in the monograph by Holzapfel (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  In particular, the divergence of the tensor P reads [DivP]i = Pi j,j componentwise, where indices after the coma de-  note spatial differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In some other texts, a transposed definition of the divergence is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Then, the equation  of motion involves the material divergence of the nominal stress tensor P⊺ instead of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='2  Generalities  In the present study, we consider deformable solids whose constitutive behaviour is described by the state vari-  ables S = {s,E}, where s is the specific entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The choice of variables S is coherent with the postulate of frame-  indifference of the internal energy (Holzapfel, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In fact, a change of observer specified by a superimposed rigid-  body motion leaves S invariant, as well as the internal energy U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Note that the internal energy does not depend on  rates of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  The internal energy per unit volume U is a function of state to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The thermodynamic temperature  is defined as the conjugate variable of s in the partial Legendre transform of U/ρ with respect to s (Berjamin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' However, the explicit dependence of U with respect to s is usually omitted in the definition of a strain energy  density function W e such that U = W e(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The strain energy W e is regarded as a scalar-valued isotropic function of  its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Thus, its dependence with respect to E can be reduced to a dependence with respect to three scalar  invariants  I1 = tr(E),  I2 = tr(E2),  I3 = tr(E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  (2)  They can be used directly, or other physically meaningful scalar quantities might be defined from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  The first and second principles of thermodynamics yield the Clausius–Duhem inequality  D = (S −Se) : ˙E = Sv : ˙E ≥ 0,  (3)  where D is the dissipation, S = Se +Sv is the total second Piola–Kirchhoff stress,  Se = −pC −1 + ∂W e  ∂E  (4)  denotes the elastic part, and Sv is a viscous contribution to be specified subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The scalar p is an arbitrary  Lagrange multiplier for the incompressibility constraint (1), see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='3 of Holzapfel (2000), and C = I +2E is the right  Cauchy–Green strain tensor F ⊺F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Therefore, no dissipation occurs in the elastic case S = Se where the viscous stress  tensor Sv is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  According to the dissipation inequality (3), the viscous stress Sv is a function of state and evolution variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  the set S ∪ { ˙E} which is a consistent choice to enforce frame-invariance (Antman, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Ball, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' We introduce a  dissipation potential W v(E, ˙E) such that  Sv = ∂W v  ∂ ˙E  (5)  2  defines the viscous stress (Maugin, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In general, the dissipation potential is described by additional invariants  (Pioletti and Rakotomanana, 2000)  I4 = tr( ˙E),  I5 = tr( ˙E2),  I6 = tr( ˙E3),  I7 = tr( ˙EE),  I8 = tr( ˙EE2),  I9 = tr( ˙E2E),  I10 = tr( ˙E2E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  (6)  In the present study, we consider Newtonian-type viscosity models whose dissipation potential is as simple as possi-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='3  Consequences of incompressibility  First, let us investigate the consequences of the incompressibility constraint (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' As noted in Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2007), the  invariants (2) of E are linked through  I1 = I2 − 4  3 I3 − I 2  1 +2I1I2 − 2  3 I 3  1 ,  (7)  by virtue of incompressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' This identity follows from the expression of the principal invariants of the unimodular  tensor C = I +2E in terms of the invariants Ik, see the Appendix of Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Using the differential version  of the incompressibility constraint, the invariants (2)-(6) of E, ˙E satisfy the particular relationship  1  2 I4 = I7 −2I8 +2I1I7 −  �  I1 − I2 + I 2  1  �  I4  (8)  deduced from the identity trD = 0, see Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  The relationship (7) means that the invariant I1 = tr(E) is no longer linear with respect to the components of the  strain tensor E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' instead, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (7) shows that it has terms of polynomial order two and three with respect to the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Furthermore, due to the relationship (8), the invariant I4 = tr( ˙E) is still linear with respect to the components of the  strain-rate tensor ˙E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' However, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (8) shows that I4 is no longer invariant on the strain tensor E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' instead, it has terms  of polynomial order one, two and three with respect to the Green–Lagrange strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='4  Constitutive assumptions  In weakly nonlinear elasticity, the strain energy density function is sought in the form of a polynomial of the invariants  Ik with constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Similarly to Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2004), we assume that the internal energyU has a fourth-  order polynomial expression with respect to the components of the strain tensor E of the form  W e = µI2 + 1  3 AI3 +DI 2  2,  (9)  where µ ≥ 0 is the shear modulus (in Pa), and the coefficients A, D are higher-order elastic constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Now, let us propose an expression for the dissipation potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' To end up with a linear viscosity model similar to  that by Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2013), we assume that the dissipation potential is a second-order polynomial expansion of the  strain rate tensor ˙E, and a zeroth-order polynomial of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' This assumption amounts to selecting W v of second order  in (E, ˙E), and to ignore the terms proportional to ˙E that produce elastic stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Due to the relationships (7)-(8), we  therefore keep  W v = ηI5,  (10)  where η ≥ 0 is the shear viscosity (in Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In the above expression, the absence of bulk viscosity “ζ” is due to the  assumption on polynomial orders for the viscous part, and to the incompressibility property (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Setting the bulk  viscosity ζ = 2  3η in Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2013) yields the same expressions as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Computation of the tensor derivatives of the potentials (9)-(10) by means of the chain rule for W •(Ik,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=') yields the  following elastic (4) and viscous stress contributions (5)  Se = −pC −1 +2(µ+2DI2)E + AE2,  Sv = 2η ˙E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  (11)  Thermodynamic consistency (3) is ensured provided that the dissipation D = 2W v is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In fact, the present  dissipation potential W v is a homogeneous function of degree two with respect to ˙E (Maugin, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' A sufficient  condition for the restriction D ≥ 0 to be always satisfied is that the viscosity η is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  3  3  Plane shear waves  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='1  Nonlinear viscous wave equation  Similarly to Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2013), we consider simple shear deformations described by the displacement field u =  [u,0,0]⊺ where u = u(z,t) denotes the particle displacement along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Thus, the deformation gradient  tensor reads  F =  �  �  1  0  γ  0  1  0  0  0  1  �  �,  (12)  where γ = ∂u/∂z is the shear strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The velocity field takes the form v = [v,0,0]⊺ where v = ∂u/∂t is the shear velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  In the equation of motion ρ ˙v = DivP, the relevant first Piola–Kirchhoff stress component P13 is deduced from  the expression of the elastic and viscous parts Pe  13 = µγ+Γγ3 and Pv  13 = η(1+2γ2) ˙γ, where only terms up to order γ3  have been kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The non-negative constant Γ = µ+ A/2+D is a parameter of nonlinearity (Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Hence, upon division by the shear modulus µ, the x-component of the equation of motion produces the nonlinear  wave equation  1  c2  ∂2u  ∂t2 = ∂2u  ∂z2 + 2  3β ∂  ∂z  �∂u  ∂z  �3  +τ ∂  ∂z  ��  1+2  �∂u  ∂z  �2� ∂2u  ∂z∂t  �  ,  (13)  describing transverse wave propagation along the z-direction, where we have introduced the notations  c =  �  µ  ρ ,  β = 3  2  Γ  µ,  τ = η  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  (14)  Spatial differentiation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (13) allows to write a similar wave equation for the strain  1  c2  ∂2γ  ∂t2 = ∂2γ  ∂z2 + 2  3β ∂2  ∂z2 γ3 +τ ∂2  ∂z2  ��  1+2γ2� ∂γ  ∂t  �  ,  (15)  which will be used later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  According to the wave equations (13)-(15), shear waves of infinitesimal amplitude propagate at the shear wave  speed c =  �  µ/ρ in the absence of nonlinearity and viscosity (β = 0, τ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Typically, this sound velocity equals  c ≈ 2 m/s in gels (Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', 2007), whereas β ≈ 10 and τ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='12 ms at a loading frequency of 100 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Here, we have  obtained the same wave equations than those derived in Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2013) for the particular bulk viscosity ζ = 2  3η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Note in passing the presence of a nonlinear viscous term which is absent in Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='2  Slow scale approximations  Similarly to Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2004) and Pucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2019), we proceed now to a reduction of the above wave equa-  tion (13) for one-way wave propagation with slowly varying profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' We present two approximations based either on a  slow space variable or a slow time variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Slow space  Let us follow the scaling procedure in Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' For this purpose, we introduce the  following scaling defined by the change of variables {˜z = ϵ2z, ˜t = t − z/c,u = ϵ ˜u}, where ϵ is a small parameter and  ˜u = ˜u(˜z, ˜t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Furthermore, we assume that τ is of order ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Note that this set of assumptions corresponds to a slowly-  varying profile in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  This Ansatz is then substituted in the equation of motion (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' At leading (cubic) order in ϵ, the motion of soft  viscous solids is governed by the scalar equation  ϵ3c ∂2 ˜u  ∂˜z∂˜t = ϵ3 β  c2  �∂ ˜u  ∂˜t  �2 ∂2 ˜u  ∂˜t2 +ϵτ  2  ∂3 ˜u  ∂˜t3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  (16)  Transforming back to the initial displacement u and physical coordinates (z,t) leads to a reduced wave equation  c ∂v  ∂z +  �  1−βv2/c2� ∂v  ∂t = τ  2  ∂2v  ∂t2 ,  (17)  for the velocity v = ∂u/∂t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  1The wave equation proposed by Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2003) and analysed by Jordan and Puri (2005) cannot be obtained rigorously from the equations  of motion unless time derivatives are (questionably) replaced by spatial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  4  Up to the choice of time variable used here (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', the physical time t instead of the retarded time ˜t), the partial  differential equation (17) is identical to the cubic Burgers-type equation of Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' However, the  underlying modelling assumptions are not equivalent, since the initial wave equation (13) includes the extra nonlinear  viscosity term 2τ∂(γ2 ˙γ)/∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' This additional term is lost in the rescaling procedure given that it is of higher order in ϵ  than the leading-order viscous term τ∂ ˙γ/∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In the end, while the modelling efforts by Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2013) aimed  at enforcing objectivity lead to a slight modification of the wave equation (more precisely, the addition of a nonlinear  viscous term), they do not induce any modification of the transport equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Slow time  For later comparisons, let us derive a similar Burgers-type equation governing the evolution of the strain  instead of the velocity by following Pucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' To do so, we introduce the slow-time scaling based on the  change of variables {˜t = ϵ2t, ˜z = z −ct,u = ϵ ˜u} where ϵ is a small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Proceeding in a similar fashion to above,  we end up with the nonlinear transport equation  ∂γ  ∂t +c  �  1+βγ2� ∂γ  ∂z = τc2  2  ∂2γ  ∂z2 ,  (18)  where γ = ∂u/∂z is the shear strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Here too, after keeping leading order terms, we have transformed back to the  initial physical coordinates (z,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Therefore, the above partial differential equation may be viewed as a one-way ap-  proximation of the wave equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Their travelling wave solutions are compared in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  4  Travelling wave solutions  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='1  Nonlinear viscous wave equation  Let us seek travelling wave solutions to the wave equation (15), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' specific smooth waveforms that propagate at a  constant velocity with a steady profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In a similar fashion to Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2013), we first introduce the following  rescaled dimensionless variables and coordinates  g(¯z, ¯t) =  �  2  3βγ(z,t),  ¯t = t/τ,  ¯z = z/(cτ),  (19)  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (15), such that  ∂2g  ∂¯t2 = ∂2g  ∂¯z2 + ∂2  ∂¯z2 g 3 + ∂2  ∂¯z2  ��  1+ 3  β g 2  � ∂g  ∂¯t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  (20)  Next, we seek travelling wave solutions of the form g =  �  ν2 −1G(ξ) where ξ = (ν2 − 1)(¯t − ¯z/ν) involves the dimen-  sionless wave velocity ν ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Injecting this Ansatz in the above partial differential equation and integrating twice with  respect to ξ with vanishing integration constants yields a nonlinear differential equation for the strain:  G = G3 +  �  1+αG2� d  dξG,  (21)  where α = 3(ν2 −1)/β is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  From the above differential equation, one observes that travelling wave solutions to the wave equation (15) should  connect the equilibrium strains G = 0 and G = ±1 by following a smooth transition that depends on the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Solutions read (Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', 2013)  ξ = −ln  �  1  2G  �4  3(1−G2)  � 1+α  2  �  (22)  in implicit form, where we have enforced G(0) = 1/2 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Illustrations are provided later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='2  Slow time approximation  In a similar fashion, let us now seek travelling wave solutions to the reduced wave equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Thus, we first perform  the substitutions (19) to get  ∂g  ∂¯t + ∂  ∂¯z  �  g + 1  2 g 3  �  = 1  2  ∂2g  ∂¯z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  (23)  In order to obtain wave solutions that correspond to the same strain values at infinity as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='1, we introduce a  slightly different scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Indeed, let us inject the Ansatz g =  �  ν2 −1G(χ) with χ = (ν2 − 1)(ϑ¯t − ¯z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (23), where  ϑ = 1+ 1  2(ν2 −1) is the new dimensionless velocity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Thus, we arrive at the differential equation  G = G3 + d  dχG  (24)  5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='5  2  ν  ϑ  Figure 1: Scaled velocity ϑ = 1+ 1  2(ν2 −1) for the ‘slow-time’ reduced model in terms of the scaled velocity ν for the full  wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  of which the strain values 0 and 1 are steady states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Enforcing the initial value G = 1/2 at χ = 0 gives  G =  1  �  1+3e−2χ ,  (25)  which does not involve any extra parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' One observes that this expression corresponds to the case α = 0 in  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (21)-(22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' One might proceed in a similar fashion with the Burgers-type equation (17) corresponding to the slow space  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Similarly to (19), we perform the substitutions r(¯z, ¯t) =  �  2β/3v(z,t)/c in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (17) to get  ∂r  ∂¯z + ∂  ∂¯t  �  r − 1  2r 3  �  = 1  2  ∂2r  ∂¯t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  (26)  Next, we introduce r =  �  ν2 −1V (ψ) where ψ = (ν2 −1)(¯t − ¯z/κ) involves the dimensionless velocity κ defined by the  relationship κ−1 = 1− 1  2(ν2 −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' This way, we obtain the same differential equation V = V 3 + d  dψV for the dimension-  less velocity V as previously for the strain (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Therefore, within the scope of the present study, the slow time and  slow space approximations lead to related travelling wave solutions that describe the evolution of distinct kinematic  variables (strain and velocity, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='3  Comparison  Let us compare the solutions (22)-(25) obtained for the full wave equation (15) and the one-way model (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' First, one  observes that these travelling waves of same amplitude do not propagate at the same speed, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Indeed, given the expression of ϑ, we can express the relative error E = ϑ/ν−1 on the scaled velocity as a function of  ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' To ensure that the latter remains less than 5% (respectively 1%), we obtain the requirement ν ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' ν ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='1)  marked by dotted lines in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Now, let us observe that for a unit kink covering the range 0 ≤ G ≤ 1, the corresponding shear strains satisfy  0 ≤ γ ≤  �  α/2  (27)  where α = 3(ν2 − 1)/β was introduced earlier on, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In other words, the coefficient α in the differential  equation (21) is related to the maximum strain of travelling waves, and these bounds are valid for both models at hand  due to application of the rescaling procedure (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Thus, restrictions of the wave speed ν can be expressed in terms of  the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' To ensure that the velocity error E remains less than 5% (respectively 1%), we therefore require γ  �  β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='0  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' γ  �  β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Note that the parameter of nonlinearity can take such values as β ≈ 10 for gels (Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Therefore, the slow scale approximation has a very restricted validity for such a soft viscoelastic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  This property is further illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' 2, where we have represented the evolution of the relative velocity ν−1  (or ϑ−1) in terms of the maximum strain amplitude, both for the full wave equation and its one-way approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  According to the expression of α above, we have the relationship ϑ−1 = 1  3β(  �  α/2)2 in the case of the one-way approx-  imate model, which produce lines of slope two in log-log coordinates (dashed lines in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' However, for the full  wave equation, this relationship between the wave speed ν and the strain amplitude is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' For fixed values  of the nonlinearity parameter β, differences between the one-way model and the full wave equation become visible  at large strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' 3, we display the evolution of the waveforms (22)-(25) in terms of the scaled coordinates ξ, χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In the case  of the full wave equation (21), the parameter α takes the values {0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='2,3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' It appears that the waveforms so-obtained  follow a drastically different evolution when parameters are modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In particular, the wavefront deduced from  the full wave equation (solid lines) does not exhibit the same invariance and symmetry properties as the wavefront  deduced from the one-way model (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  6  10−1  100  10−3  10−2  10−1  100  β = 3  β = 1  Strain amplitude  Relative velocity  one-way  wave eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Figure 2: For the full wave equation (solid line) and the ‘slow-time’ reduced model (dashed line), we represent the  evolution of the relative velocity ν − 1 (respectively, ϑ − 1) in terms of the strain amplitude  �  α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The axes have a  logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  −2  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='5  1  α  ξ, χ  G  wave eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  one-way  Figure 3: Steady waveforms deduced from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (22)-(25) for increasing values of the parameter 0 ≤ α ≤ 3 (arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Evolution of the scaled shear strain G in terms of the related dimensionless coordinate ξ or χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  7  5  Simple waves  In the lossless case, exact one-way wave equations can be derived by using the method of Riemann invariants, see  for instance the introductory example by John (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Such particular wave solutions called simple waves keep one  Riemann invariant constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In other words, the particle velocity v = R−−Q(γ) withQ(γ) = c  �γ  0  �  1+2βg 2 dg depends  explicitly on the strain γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' The scalar R− is an arbitrary constant, for instance R− ≡ 0 in some specific boundary-value  problems (Berjamin and Chockalingam, 2022), which will be assumed satisfied from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Spatial differentiation  of the velocity then produces  ∂γ  ∂t +c  �  1+2βγ2 ∂γ  ∂z = 0,  (28)  where we have used the equality of mixed partials ∂v/∂z = ∂γ/∂t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Obviously, the lossless one-way wave equation (18)  with τ = 0 is an approximation of (28) for 2βγ2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Let us analyse this requirement in a more quantitative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' To ensure that the relative error on the advection  velocity E =  1+a  �  1+2a −1 for a = βγ2 remains less than 5% (respectively 1%), we obtain the requirement a ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='44 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  a ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Application of the square root leads to the restriction γ  �  β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='66 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' γ  �  β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='40) which is slightly more  constraining than in the case of viscoelastic travelling waves (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Along a simple wave, computation of the partial derivative of the velocity v = R− −Q(γ) with respect to time pro-  duces  c ∂v  ∂z +  �  1+2βγ2�−1/2 ∂v  ∂t = 0,  (29)  where the strain γ = Q−1(−v) can be expressed formally as a function of the velocity, despite no analytical expression  of the inverse function Q−1 of Q is known in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' If |v| is small, then we can use the approximation  γ ≃ −v/c of the strain which follows from the asymptotic equivalence of Q ∼ cγ at small strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Next, the (·)−1/2-  factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (29) can be approximated by the polynomial expression 1−βγ2 as long as 2βγ2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' This way, we have  shown that the one-way wave equation (17) is an approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (29) obtained for R− = 0 and 2βv2/c2 ≪ 1 in  the elastic limit τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' This observation is consistent with the discussions in Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In summary, the  lossless ‘slow-space’ and ‘slow-time’ reductions (17)-(18) with τ = 0 are approximate governing equations for simple  waves with small values of βv2/c2 and of βγ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  6  Conclusion  For a specific strain-rate viscoelasticity theory of soft solids, we have shown that one-way approximate wave propaga-  tion models produce significantly different travelling wave solutions than the full equations of motion as soon as the  wave amplitude is not infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Similar observations are reported in the literature in relation with shear shock  formation (Berjamin and Chockalingam, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In the elastic limit, we have examined the validity of one-way approx-  imations in relation with simple wave theory, thus leading to dedicated criteria of validity involving small velocity and  strain amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' We conclude that these approximations should be used with care given their limited accuracy, in  general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Nevertheless, they might remain useful for the interpretation of experimental results where their validity is  not always severely penalised (Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', 2003, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  Acknowledgments  The author is grateful to Michel Destrade (Galway, Ireland) for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' This project has received funding from the  European Union’s Horizon 2020 research and innovation programme under grant agreement TBI-WAVES — H2020-  MSCA-IF-2020 project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' 101023950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  A  Consequence of incompressibility  This Appendix is devoted to the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' We start with the Cayley–Hamilton identity for the right Cauchy–  Green tensor C = F ⊤F, which reads  C 3 −I C 2 +II C −III I = 0,  (30)  where I, II, III are the principal invariants of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' In the case of volume-preserving motions (1), the tensor C is uni-  modular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' we have III = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Next, multiplication of (30) by C −1 ˙E on the right side, substitution of C = I + 2E and  computation of the trace entails the relationship  (I4 +4I7 +4I8)−(3+2I1)(I4 +2I7)  +(3+4I1 +2I 2  1 −2I2)I4 = 0,  (31)  8  where we have used the incompressibility property trD = tr(C −1 ˙E) = 0, the definition of the invariants (2)-(6), and the  relationship between I, II and the invariants Ik used here (Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Rearranging terms, we get the desired  identity (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content='  References  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
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+page_content=' In P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Newton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Holmes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Weinstein, editors, Geometry, Mechanics,  and Dynamics, pages 3–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Springer, New York, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
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+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Berjamin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
+page_content=' Chockalingam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}
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+1
+Can Continuous Aperture MIMO Achieve Much
+Better Performance than Discrete MIMO?
+Zhongzhichao Wan, Jieao Zhu, and Linglong Dai, Fellow, IEEE
+Abstract—The concept of continuous-aperture multiple-
+input multiple-output (CAP-MIMO) technology has been
+proposed recently, which aims at achieving high spectrum
+density by deploying extremely dense antennas or even
+continuous antennas in a given aperture. The fundamental
+question of CAP-MIMO is whether it can achieve much
+better performance than the traditional discrete MIMO
+system. In this paper, to model the CAP-MIMO, we use self-
+adjoint operators to depict the structural characteristics of
+the continuous random electromagnetic fields from physical
+laws. Then, we propose a non-asymptotic performance
+comparison scheme between continuous and discrete MIMO
+systems based on the analysis of mutual information. We
+show the consistency of the proposed scheme by proving
+that the mutual information between discretized transceivers
+converges to that between continuous transceivers. Numeri-
+cal analysis verifies the theoretical results, and suggests that
+the mutual information obtained from the discrete MIMO
+with widely adopted half-wavelength spaced antennas al-
+most achieves the mutual information obtained from CAP-
+MIMO.
+Index Terms—Multiple-input multiple-output (MIMO),
+Continuous-aperture MIMO (CAP-MIMO), mutual infor-
+mation, random fields, Fredholm determinant.
+I. INTRODUCTION
+The spectrum efficiency of wireless communication
+systems has been greatly improved from 3G to 5G
+because of the use of multiple-input multiple-output
+(MIMO) technology [1]–[3]. The MIMO systems utilize
+multiple antennas to exploit the spatial multiplexing gain
+[4], where the antennas are modeled as discrete points
+in the continuous space. Along with the tendency of
+increasing the number of antennas to achieve higher
+spectrum efficiency, people are considering deploying
+extremely dense antennas in a given aperture [5], [6].
+When the number of antennas in a given aperture tends to
+infinity, the traditional MIMO systems with transceivers
+composed of discrete point antennas are equivalent
+to the MIMO systems with continuously controllable
+transceivers. Therefore, the MIMO with extremely dense
+All authors are with the Department of Electronic Engineer-
+ing, Tsinghua University as well as Beijing National Research
+Center
+for
+Information Science
+and
+Technology
+(BNRist), Bei-
+jing 100084, China (E-mails: {wzzc20, zja21}@mails.tsinghua.edu.cn;
+daill@tsinghua.edu.cn).
+This work was supported in part by the National Key Research
+and Development Program of China (Grant No. 2020YFB1807201), in
+part by the National Natural Science Foundation of China (Grant No.
+62031019).
+antennas is called continuous-aperture MIMO (CAP-
+MIMO), and is also called holographic MIMO [7]–[9]
+or large intelligent surface [5], [10] in the recent litera-
+ture1. It has attracted increasing interest in the research
+of MIMO technology. Recent works about CAP-MIMO
+include pattern optimization [6], antenna design [11],
+channel estimation [7], and so on. For CAP-MIMO, the
+fundamental question is whether the CAP-MIMO system
+can achieve much better performance than the traditional
+discrete MIMO system.
+A. Related works
+The structure of CAP-MIMO has been defined in the
+previous part but there are many structures for realizing
+the discrete MIMO. Therefore, we need to choose which
+structure of the discrete MIMO to compare with CAP-
+MIMO. A representative structure of discrete MIMO
+uses half-wavelength spaced antennas to compose the
+transceivers [12]–[14], because half-wavelength sampling
+of the electromagnetic field can reconstruct the original
+field according to the sampling theorem.
+There have been several works discussing the per-
+formance comparison between CAP-MIMO and discrete
+MIMO with half-wavelength spaced antennas. The perfor-
+mance comparison is from the degrees of freedom (DoF)
+perspective. Specifically, when discarding the evanescent
+wave components, the Fourier transform of the received
+field, which is in the wavenumber domain, is concentrated
+in a circle or a segment. This concentration phenomenon
+means that the field is bandlimited in the wavenumber
+domain, thus it can be perfectly recovered from the half-
+wavelength sampling points in the spatial domain [15]
+according to the Nyquist sampling theorem [16]. The
+above conclusion is based on the assumption that we can
+observe the received field in the infinitely large spatial
+domain. However, in practice, the destination where we
+can observe the field is in a finitely large aperture.
+For a rigorous analysis framework of the DoF in a
+finitely large aperture, the prolate spheroidal wave func-
+tion (PSWF) [17] is introduced to perform orthogonal
+expansion on the electromagnetic field. Specifically, to
+1The MIMO with extremely dense antennas can be accurately de-
+scribed by the name CAP-MIMO, while holographic MIMO and large
+intelligent surface do not focus on the continuity of the transceiver
+apertures. Therefore, in the rest part of the paper, we will prefer using
+the name CAP-MIMO rather than using other names like holographic
+MIMO.
+arXiv:2301.08411v1  [cs.IT]  20 Jan 2023
+
+2
+reconstruct the wavenumber-bandlimited electromagnetic
+field observed in a length-l spatial region, the PSWFs
+were used as the basis based on the Slepian’s concentra-
+tion problem [18]. Such an electromagnetic field can be
+perfectly reconstructed from infinite number of PSWFs,
+and approximately reconstructed from a finite number
+of PSWFs. If the reconstruction error can be controlled
+within a given threshold by using N0 PSWFs, the number
+of DoFs of the field can be approximated by N0 [19].
+This analyzing scheme is strict for arbitrary l, but can
+only provide the asymptotic result of the DoF, i.e., the
+quantitative result of N0 can be obtained only when
+the length l or the frequency tends to infinity. However,
+the practical systems are with finitely large aperture and
+finite frequency. The asymptotic result can not provide
+quantitative number of DoFs for practical systems. There-
+fore, a non-asymptotic performance comparison scheme
+between CAP-MIMO and discrete MIMO is required for
+the accurate performance comparison with finitely large
+apertures.
+B. Our contributions
+To solve this problem, in this paper, we provide a non-
+asymptotic performance comparison scheme between
+CAP-MIMO and discrete MIMO, and we further prove
+the rationality of the scheme2. Specifically, the contribu-
+tions of this paper can be summarized as follows:
+• We build models of CAP-MIMO and discrete MIMO
+based on electromagnetic theory. For CAP-MIMO
+with continuous transceivers, we model the structural
+characteristics of the continuous random electro-
+magnetic fields from physical laws by using self-
+adjoint operators. Based on this model, we can
+utilize the spectrum theory of operators to derive the
+information that can be obtained from the received
+field. The existing models of MIMO with discrete
+transceivers are spatially discretized from the contin-
+uous model. Moreover, signal-to-noise ratio (SNR)
+control schemes are introduced to ensure the fairness
+of the comparison between CAP-MIMO and discrete
+MIMO.
+• Then, before comparing the performance between
+CAP-MIMO with continuous transceivers and tra-
+ditional MIMO with discrete transceivers, we first
+utilize the simplified model with continuous trans-
+mitter and discrete receiver. Under this simplified
+model, the transmitter is continuous, which is the
+same as that in the CAP-MIMO system. By theo-
+retically analyzing the mutual information that can
+be obtained from the discrete receiver in this sim-
+plified model, we can obtain some insights about
+how the discretization of the receiver affects the
+mutual information. Moreover, the theoretical proof
+2Simulation
+codes
+will
+be
+provided
+to
+reproduce
+the
+re-
+sults in this paper: http://oa.ee.tsinghua.edu.cn/dailinglong/publications/
+publications.html.
+of the convergence of the mutual information in the
+simplified model can inspire the analysis of a more
+practical scenario, i.e., the discrete transceivers.
+• Finally, we extend the convergence proof from the
+model with discrete receiver to the model with
+discrete transceiver. We prove that the mutual in-
+formation between the discrete transceivers con-
+verges to the mutual information between continuous
+transceivers when the number of antennas of the
+discretized transceivers tends to infinity. Therefore,
+the fairness of the performance comparison is guar-
+anteed. Numerical results are provided to verify the
+theoretical analysis. Moreover, it shows the near-
+optimality of the half-wavelength sampling of the
+transceivers in traditional discrete MIMO.
+C. Organization and notation
+Organization: The rest of our paper is organized as
+follows. Section. II introduces the basic model of EIT and
+proposes models with continuous or discrete transceivers.
+The mutual information between the transceivers is also
+derived. Section. III proves the convergence of the mutual
+information between continuous transmitter and discrete
+receiver when the number of discrete antennas increases.
+Then, the convergence of the mutual information between
+discrete transceivers is illustrated in Section IV. Finally,
+we conclude the paper in Section V.
+Notation: bold characters denote matrices and vectors;
+j is the imaginary unit; E [x] denotes the mean of random
+variable x; x∗ denotes the conjugation of a number or
+a function x; XH denotes the conjugate transpose of
+a vector or a matrix X; µ0 is the permeability of a
+vacuum, Z0 is the free-space intrinsic impedance and
+c is the speed of light in a vacuum; ∇ is the nabla
+operator, and ∇× is the curl operator; |φ⟩ is the quantum
+mechanical notation of a function φ, where the inner
+product is denoted by ⟨ψ| φ⟩; det(·) denotes the matrix
+determinant or the Fredholm determinant; tr(·) denotes
+the trace of a matrix or an operator. Im denotes the m×m
+identity matrix, 1 denotes the indentity operator, δ(x)
+denotes the delta function, and 1i=j denotes the indicator
+function; |x| denotes the modulus of a complex variable,
+and ∥f(x)∥L∞(a,b) is the uniform norm of the function
+f(x) over the interval [a, b]. C∞(K) denotes the set of
+smooth functions supported on a compact set K.
+II. MODELS OF CONTINUOUS AND DISCRETE
+SYSTEMS
+In this section, we introduce the models of contin-
+uous and discrete systems for performance comparison
+between CAP-MIMO and discrete MIMO. We control
+the SNR at the receiver side to ensure the fairness of the
+comparison. The information obtained from these models
+is derived from operators and matrices.
+
+3
+A. Basic model of electromagnetic information theory
+To model the transceivers and the channel, we follow
+the approach of electromagnetic information theory (EIT).
+The EIT is an interdisciplinary subject that integrates the
+classical electromagnetic theory and information theory to
+build an analysis framework for the ultimate performance
+bound of wireless communication systems [20]. The anal-
+ysis framework of EIT is based on spatially continuous
+electromagnetic fields, which provides us the tool to
+model and analyze the continuous transceivers. Then, for
+the consistency, the model of discrete transceivers are
+viewed as the discretization of the continuous model from
+EIT.
+The model of EIT is built on the vector wave equa-
+tion [21] without boundary conditions, which is expressed
+by
+∇×∇×E (r)−κ2
+0E (r) = jωµ0J (r) = jκ0Z0J (r) , (1)
+where κ0 = ω√µ0ε0 is the wavenumber, and Z0 =
+µ0c = 120π [Ω] is the free-space intrinsic impedance.
+We assume that the transceivers are confined in two
+regions Vs and Vr, separately. The current density at the
+source is J(s), where s ∈ R3 is the coordinate of the
+source. The induced electric field at the destination is
+E(r), where r ∈ R3 is the coordinate of the field observer.
+To solve the linear partial differential equation (1), a
+general theoretical approach is to introduce the dyadic
+Green’s function G(r, s) ∈ C3×3. According to the
+linearity of (1), the electric field E(r) can be expressed
+by
+E(r) =
+�
+Vs
+G(r, s)J(s)ds,
+r ∈ Vr.
+(2)
+By exploiting the symmetric properties of the free space,
+the Green’s function in unbounded, homogeneous medi-
+ums at a fixed frequency point is [22]
+G(r, s) = jκ0Z0
+4π
+�
+I + ∇r∇H
+r
+κ2
+0
+� ejκ0∥r−s∥
+∥r − s∥
+= jκ0Z0
+4π
+ejκ0∥r−s∥
+∥r − s∥
+�
+�
+I − ˆpˆpH�
++
+j
+2π ∥r − s∥ /λ
+�
+I − 3ˆpˆpH�
+−
+1
+(2π ∥r − s∥ /λ)2
+�
+I − 3ˆpˆpH�
+�
+[Ω/m2],
+(3)
+where ˆp =
+p
+∥p∥ and p = r − s.
+Since there are some non-ideal factors at the receiver
+that corrupts the recieved field, we call them the noise
+field N(r). The received electric field can be expressed
+by Y(r) = E(r) + N(r). The above equations represent
+the deterministic model in the electromagnetic theory. To
+satisfy the demand of wireless communication, we need
+to convey information through the electromagnetic field.
+Specifically, the wireless communiation system encodes
+the information in the current J(s), and decodes the
+information from the noisy electric field Y(r). Due to
+the randomness of the transmitted bit source, the electro-
+magnetic fields are randomly excited by the transmitter
+equipments before being radiated into the propagation
+media. Therefore, the electromagnetic fields should be
+modeled as random fields [23], which are random func-
+tions with several arguments. We denote the autocorre-
+lation function of the current and the electric field as
+matrix-valued functions RJ(s, s
+′) = E[J(s)JH(s
+′)] and
+RE(r, r
+′) = E[E(r)EH(r
+′)]. The relationship between
+RJ and RE is determined by the Green’s function, which
+is
+RE(r, r′) =
+�
+Vs
+�
+Vs
+G(r, s)RJ(s, s′)GH(r, s)dsds′.
+(4)
+Similar definitions of the autocorrelation functions for
+the noise field and the noisy electric field are repre-
+sented as RN(r, r
+′) = E[N(r)NH(r
+′)] and RY(r, r
+′) =
+E[Y(r)YH(r
+′)].
+B. Continuous trasceivers
+In this part, we will build the model of CAP-MIMO
+with continuous transceivers based on the EIT model
+in the above subsection, and then derive the mutual
+information between the continuous transceivers. For sim-
+plicity, in the rest part of the paper, we assume that the
+transceivers are linear along the ˆz-direction. Moreover,
+since the current J can only exist on the linear source
+and we only observe the electric field on the linear
+receiver, we express all the physical quantities in a
+Cartesian coordinate system that satisfies s = (0, 0, s) and
+r = (d, 0, r), where d is the distance between the parallel
+source and destination line. This model corresponds to
+single-polarized linear antennas. Through this simplifica-
+tion scheme, we use J(s) and E(r) instead of J(s) and
+E(r). The relationship between them can be expressed by
+E(r) =
+� l
+0 G(r, s)J(s)ds, where G(r, s) is the upper left
+element of the matrix G(r, s). We can derive G(r, s) as
+G(r, s) =jZ0ej2π
+√
+x2+d2/λ
+2λ
+√
+x2 + d2
+�
+j
+2π
+√
+x2 + d2/λ
+d2 − 2x2
+x2 + d2
++
+d2
+x2 + d2 −
+1
+(2π/λ)2(x2 + d2)
+d2 − 2x2
+x2 + d2
+�
+,
+(5)
+where x = r − s and λ = 2π/κ0 is the wavelength.
+Here we consider the scenario with no channel state
+information, which means that the signals on the source
+are under equal power allocation. The second moments
+(autocorrelation) of J are denoted by RJ(s, s′) = Pδ(s−
+s′), s, s′ ∈ [0, l].
+Since the noiseless received field is uniquely deter-
+mined by the source and the deterministic channel, the
+autocorrelation function of the electric field is expressed
+
+4
+by the source autocorrelation RJ(s, s′) and the Green’s
+function G(r, s), written as
+RE(r, r′) =
+� l
+0
+� l
+0
+G(r, s)RJ(s, s′)G∗(r′, s′)dsds′
+= P
+� l
+0
+G(r, s)G∗(r′, s)ds.
+(6)
+The received field on the destination is Y (r)
+=
+E(r) + N(r), where N(r) is the noise field at the
+receiver. In this paper, we consider thermal noise model
+E [N(r)N ∗(r′)] =
+n0
+2 δ(r − r′). According to [24], we
+can perform Mercer expansion on the electric field E(r)
+to obtain a set of mutually independent random variables
+ξk. The expansion can be written as E(r) = �
+k ξkφk(r),
+where E[ξkiξ∗
+kj] = λki1i=j and ⟨φki(r), φkj(r)⟩ = δkikj.
+This expansion scheme has split the continuous field
+into independent components. Since the white noise field
+can be expanded under arbitrary orthogonal bases, the
+continuous channel is also decomposed into independent
+subchannels, which makes the mutual information of the
+subchannels summable.
+Next we will show that for the operator TE := φ(r) →
+� l
+0 KE(r, r′)φ(r′)dr′, where KE(r, r′) = RE(r, r′) =
+P
+� l
+0 G(r, s)G∗(r′, s)ds, all of its eigenvalues are real
+and nonnegative. Moreover, the sum of its eigenval-
+ues �∞
+i=1 λi equals P
+� l
+0
+� l
+0 G(r, s)G∗(r, s)drds. Notice
+that TE can be decomposed to T ∗T, where T
+:=
+φ(r) →
+√
+P
+� l
+0 G(r, s)φ(r)ds and T ∗
+:= φ(r) →
+√
+P
+� l
+0 G∗(r, s)φ(r)ds. This decomposition means that
+TE = T ∗
+E is a self-adjoint operator. We assume that λ
+is an eigenvalue of TE and φ(r) is the corresponding
+eigenfunction. Since
+λ = λ⟨φ(r), φ(r)⟩ = ⟨T ∗
+Eφ(r), φ(r)⟩
+= ⟨φ(r), TEφ(r)⟩ = λ∗,
+(7)
+we know that λ is real. From
+λ = λ⟨φ(r), φ(r)⟩ = ⟨T ∗Tφ(r), φ(r)⟩
+= ⟨Tφ(r), Tφ(r)⟩ ⩾ 0,
+(8)
+we know that λ is nonnegative.
+From [25] we know that an integral operator on [a, b]
+is a trace class operator if its kernel K(x, y) satisfies
+K(x, y) and ∂yK(x, y) are continuous on [a, b]2. There-
+fore TE is a trace class operator, which means that the
+sum of its eigenvalues is finite and can be expressed
+by [26]
+tr(TE) =
+� l
+0
+KE(r, r)dr = P
+� l
+0
+� l
+0
+G(r, s)G∗(r, s)drds.
+(9)
+Corollary 1. The non-negative values
+λk
+n0/2 represent
+the SNR of the independent subchannels. The mutual
+information between the noisy received field and the
+current on the source can be expressed by
+I0(J; Y ) =
++∞
+�
+k=1
+log
+�
+1 +
+λk
+n0/2
+�
+.
+(10)
+By introducing the Fredholm determinant which is the
+determinant of operators, we can express (10) by
+I0(J; Y ) = log det
+�
+1 + TE
+n0/2
+�
+,
+(11)
+where (TEφ)(r) :=
+� L
+0 RE(r, r′)φ(r′)dr′ and λk are the
+eigenvalues of TE.
+Remark 1. Our analysis here is based on the sim-
+plified model with uni-polarized linear antennas as the
+transceiver. This simplification reduces the dimension of
+the problem, where random fields degenerate to one-
+dimensional random processes. For the more general
+scenarios, such analyzing schemes are still effective. If
+the random field is defined in a region X, we can expand
+E(r) by E(r) = �
+k ξkΦk(r) and its autocorrelation
+function RE(r, r
+′) by RE(r, r
+′) = �
+k λkΦk(r)ΦH
+k (r
+′)
+[27]. The expansion satisfies that λk and Φk(r) are
+eigenvalues and eigenfunctions of the integral equation
+�
+X RE(r, r
+′)Φ(r
+′)dr
+′ = λkΦ(r). Similar expressions of
+the mutual information in (10) and (11) can be derived.
+C. Continuous transmitter and discrete receiver
+Before building the model with discrete transceivers,
+in this subsection, we will first build a simplfied model
+with continuous transmitter and discrete receiver. The
+simplfied model analyzed here can bring some insights
+about the discretization of both transceivers and the
+SNR control schemes. For the continuous transmitter,
+we still use the length-l linear transmitter along the ˆz-
+direction. For the discrete receiver, we build a model
+with m point antennas on a segment parallel to the
+linear transmitter in the destination region. The ith point
+antenna is placed on ri ∈ [0, l]. The correlation matrices
+of the received signals and received noise are denoted
+by K
+′
+E and K
+′
+N. For the received signals, we assume
+that it is the sampling of the continuous electric field
+on the point ri, which means that K
+′
+E = KE(ri, rj).
+However, for the received noise on the antenna, it can
+not directly be assumed as the point sampling of the
+noise field, because of the delta function. To solve this
+problem, we assume that K
+′
+N =
+n1
+2 Im is an identity
+matrix, and control the signal-to-noise ratio (SNR) of this
+model the same as that of the continuous model to ensure
+the fairness of the comparison. The SNR at the receiver
+of the continuous model is �∞
+i=1
+λi
+n0/2, where λi is the
+ith eigenvalue of the operator TE. From Lemma 1 we
+know that �∞
+i=1
+λi
+n0/2 =
+P
+n0/2
+� l
+0
+� l
+0 G(r, s)G∗(r, s)drds
+is finite. The SNR at the receiver of the discrete model is
+�m
+i=1
+λ
+′
+i
+n1/2, where λ
+′
+i is the ith eigenvalue of the matrix
+K
+′
+E.
+The SNR control scheme is necessary because if we
+do not control the SNR, the mutual information that can
+be obtained from the discrete antennas in the receiver
+may infinitely increase. Let us take a counter-example
+where the power of received signal and received noise on
+
+5
+each point antenna remain unchanged when the number
+of antennas in a given aperture increases. For dense
+antennas we can assume that N received signals of the
+antennas in a small aperture are nearly the same, while
+the corresponding noises are independent according to
+the model. Then, the SNR for the N antennas will
+keep near-linearity increasing with N, since when we
+perform combing of the N received signals we have
+SNR =
+E[(�N
+i=1 Ei)(�N
+i=1 E∗
+i )]
+E[(�N
+i=1 Ni)(�N
+i=1 N ∗
+i )] ≈ N E[E1E∗
+1 ]
+E[N1N ∗
+1 ]. Therefore
+the mutual information that can be obtained from the
+N antennas will keep near-logarithm increasing with N,
+which corresponds to the simulation in [28].
+According to (9), the noise power in the discrete
+receiver model can be controlled by
+n1 = n0
+�m
+i=1 KE(ri, ri)
+� l
+0 KE(r, r)dr
+.
+(12)
+We denote the determinant of matrix K ∈ Cm×m by
+det(Ki,j)m
+i,j=1. Then we can express the mutual informa-
+tion between the transceivers by
+I1 = log
+�
+det(K
+′
+N + K
+′
+E)
+det(K
+′
+N)
+�
+= logdet
+�
+1i=j + KE(ri, rj)
+n1/2
+�m
+i,j=1
+.
+(13)
+Remark 2. Here the SNR on each of the point antennas in
+the discrete model changes with the density of point anten-
+nas. Notice that L
+m
+�m
+i=1 KE(ri, ri) is the approximation
+of the integral
+� l
+0 KE(r, r)dr. When m approximates
+infinity, n1 will approximate mn0
+2l . This phenomenon has
+several annotations, including the increase of the noise
+power on each point antenna, the reduction of antenna
+efficiency, and the corollary of the discretization of EIT
+continuous models.
+From the perspective of noise power, we can explain it
+by spatial sampling. For the point antenna arrays, more
+antennas on a given aperture corresponds to a higher
+sampling rate in the spatial domain and a wider lowpass
+filter in the wavenumber domain. Since a wide lowpass
+filter can receive more noise power from the white noise
+field, the noise power should increase with the density of
+the antennas.
+From the perspective of antenna efficiency, the well-
+known Hannan’s efficiency shows that for both transmit-
+ting and receiving antennas, the antenna gain is propor-
+tional to lxly for two-dimensional surface antennas [29].
+Therefore, for the linear model we considered, the an-
+tenna gain will be inversely proportional to the sampling
+number when the antennas are dense enough.
+Besides these two annotations, another perspective is
+viewing the model of discrete point antennas as the
+discretization from the EIT continuous model. If we
+consider m linear continuous antennas instead of point
+antennas in the destination region. All the antennas are
+connected head to tail to occupy the [0, l] position in the
+space and detect the electric field by inner producting it
+with its eigenmode. This model fulfills the requirement of
+discretizing the continuous receiver to discrete receiving
+antennas. The signal received by the ith antenna is
+Yi =
+� ai+1
+ai
+Y (r)φ(r)dr, where [ai, ai+1] is the occupied
+region of the ith antenna, and φ(r) is the eigenmode of the
+antenna. If we assume φ(r) ≡ 1, the correlation matrix
+of the received electric field can be expressed by
+(KE)i,j = E
+�� ai+1
+ai
+� aj+1
+aj
+E(r)E∗(r′)drdr′
+�
+= (ai+1 − ai)(aj+1 − aj)KE(ri, rj),
+(14)
+where ri ∈ [ai, ai+1] and rj ∈ [aj, aj+1] according to
+the mean value theorem for integrals. For the noise field
+on the destination, we have
+(KN)i,j = E
+�� ai+1
+ai
+� aj+1
+aj
+N(r)N ∗(r′)drdr′
+�
+=
+�
+(ai+1 − ai) n0
+2
+i = j
+0
+i ̸= j .
+(15)
+Therefore, the SNR after the discretization will discreases
+by ai+1 − ai, which is the case when the antennas are
+dense enough.
+After explaining the rationality of the SNR control
+scheme, we will introduce the following lemma to show
+the convergence of the noise power on each discrete point
+antenna, which will be useful for the following proofs.
+Lemma 1. When the number of antennas m in a given
+aperture increases, the noise power on each antenna n1/2
+will approach
+mn0
+2l . The difference between them is at
+most inverse-proportional to m.
+Proof: From (12) and the middle point quadrature
+rule, we have
+����
+l
+mn1 − n0
+���� = n0
+���
+� l
+0 KE(r, r)dr − l/m �m
+i=1 KE(ri, ri)
+���
+���
+� l
+0 KE(r, r)dr
+���
+⩽
+n0l3 ���K
+′′
+E(r, r)
+���
+L∞(0,l)
+24m2
+���
+� l
+0 KE(r, r)dr
+���
+,
+(16)
+which completes the proof.
+D. Discrete transceivers
+The models discussed in the above subsections keep
+the transmitter continuous and only perform discretization
+on the receiver. However, the commonly used model
+to depict wireless communication is the discrete MIMO
+model, in which both the transceivers are modeled as
+discrete point antennas. Therefore, in this section, we
+will introduce a model which discretizes the transceivers
+simultaneously, which is the extension of the model with
+continuous transmitter and discrete receiver. Then, similar
+
+6
+d
+Continuous 
+Transmitter
+Continuous 
+Receiver
+d
+Discrete 
+Receiver
+l
+/ 2
+
+l
+d
+Discrete 
+Receiver
+l
+/ 2
+
+Discrete 
+Transmitter
+Continuous 
+Transmitter
+0I
+1I
+2I
+Fig. 1. Comparison between the three models in this section with continuous transceivers and the model with discrete transceivers.
+to the scheme in the above subsection, we will provide the
+corresponding SNR control scheme to ensure the fairness
+of the comparison.
+Specifically, we build a model with m point antennas
+on a length-l segment in the source region and m point
+antennas on a length-l segment in the destination region.
+Similar to the above subsection, we assume that the ith
+point antenna is placed at si in the source region and ri
+in the destination region. The correlation matrix of the
+signals in the source region is set to be an identity matrix
+K
+′′
+J = PIm, which corresponds to the power allocation
+scheme with no channel state information at the transmit-
+ter. The channel gain from the ith antenna in the source
+region and the jth antenna in the destination region can be
+expressed by Hi,j = G(si, rj). The correlation matrix of
+the received signal is denoted by K
+′′
+E = HK
+′′
+JHH. The
+noise matrix is denoted by K
+′′
+N = n2
+2 Im. Similar SNR
+control on the receiver side is used, which is expressed
+by
+n2 = n0
+�m
+i=1
+�m
+j=1 G(ri, sj)G∗(ri, sj)
+� l
+0
+� l
+0 G(r, s)G∗(r, s)drds
+.
+(17)
+The mutual information between the transceivers is ex-
+pressed as:
+I2 = log
+�
+det(K
+′′
+N + K
+′′
+E)
+det(K
+′′
+N)
+�
+= logdet
+�
+1i=j +
+�
+k G(ri, rk)G∗(rj, rk)
+n2/2
+�m
+i,j=1
+.
+(18)
+The comparison between the three models built in Sec-
+tion. II-B, Section. II-C and in this subsection is shown
+in Fig. 1. In the following two sections we will introduce
+the intermediate quantity I
+′
+0 and I
+′′
+0 to theoretically prove
+that I1 and I2 converge to I0. The flow chart of the proof
+is shown in Fig. 2
+III. PERFORMANCE COMPARISON BETWEEN DISCRETE
+AND CONTINUOUS RECEIVERS
+In the above section we have proposed the mod-
+els of continuous and discrete transceivers and derived
+the corresponding mutual information. Before compar-
+ing the performance between CAP-MIMO with contin-
+uous transceivers and traditional MIMO with discrete
+transceivers, we first utilize the simplified model with
+continuous transmitter and discrete receiver in this sec-
+tion. Under this simplified model, the transmitter is con-
+tinuous, which is the same as that in the CAP-MIMO
+system. The comparison is based on the convergence
+analysis of the mutual information when the number
+of antennas in the discrete receiver increases. Numer-
+ical analysis is provided to verify the correctness of
+the convergence analysis. The discussion about discrete
+transceivers inspired by the analysis in this part will be
+in the next section.
+A. Convergence analysis of the mutual information
+To compare the mutual information I0 and I1, we intro-
+duce an intermediate quantity I
+′
+0 = logdet
+�
+1 + mTE
+ln1/2
+�
+.
+We can bound |I0 −I1| by |I0 −I
+′
+0|+|I1 −I
+′
+0|. According
+to [30], I1 can be viewed as the approximation of I
+′
+0
+using a numerical integral scheme. In our discussion the
+point antennas in the destination region are evenly spaced,
+which means that ai = (i−1)l/m and ri = (i−0.5)l/m.
+To bound |I1 − I
+′
+0|, we introduce the following lemma
+from [30]:
+Lemma 2. We define d(z) := det(1+zT) and dQ(z) :=
+det (1i=j + wjzK(ri, rj))m
+i,j=1, where K is the kernel of
+the operator T. The difference between d(z) and dQ(z)
+
+7
+0I
+'
+0I
+1I
+2I
+''
+0I
+Discretize the receiver
+Discretize the transmitter
+Discretize the transceivers
+Lemma 1
+Lemma 2
+Lemma 3
+Lemma 4
+Lemma 5
+Theorem 1
+Theorem 2
+Fig. 2. Flow chart of the proof in this paper.
+is
+d(z) − dQ(z) =
+∞
+�
+n=1
+zn
+n!
+�
+Qn
+m(Kn)
+−
+�
+[a,b]n Kn(x1, · · · , xn)dx1 · · · dxn
+�
+,
+(19)
+where Kn(x1, · · · , xn)
+=
+det (K(xi, xj))n
+i,j=1, and
+Qn
+m(f) = �m
+j1=1,··· ,jn=1
+�n
+i=1 wjif(rj1, · · · , rjn).
+Lemma 2 provides a method to compare the difference
+between a Fredholm determinant of operator and a clas-
+sical determinant of matrix. In our model, the operator T
+corresponds to the integral operator TE, z equals 2m
+ln1 ,and
+wj = l/m according to the equally spaced antennas.
+Notice that Qn
+m(f) is the numerical approximation of
+the integral
+�
+[a,b]n Kn(x1, · · · , xn)dx1 · · · dxn, we need
+to use numerical integral theory to estiamte the approxi-
+mation error. For the model with equally spaced antennas,
+this expression corresponds to a multivariate m-point
+composite midpoint quadrature rule.
+For the error bound of a m-point composite midpoint
+quadrature [31], we have
+�����Qm(f) −
+� l
+0
+f(x)dx
+����� ⩽
+l3
+24m2 ∥f
+′′∥L∞(0,l)
+(20)
+According to [32], the numerical approximation error
+for multiple integrals in a n-dimensional unit cube can be
+bounded by
+�����
+�
+Gn
+f −
+� n
+�
+i=1
+�
+Qi(f)
+����� ⩽ E1 +
+n
+�
+i=2
+i−1
+�
+j=1
+WjEi,
+(21)
+where Qj(g) := �
+j wi,jg(xi,j), Wi = �
+j |wi,j| and
+Ei ⩾
+���Qi(f; xi) −
+� 1
+0 f(x1, · · · , xn)dxi
+���. According to
+the models in this paper, we have wi,j = l/m and Wi = l.
+By simple variation of the integral band, we can bound the
+approximation error of the multi-dimensional numerical
+integral quadrature rule by
+�����Qn
+m(Kn) −
+�
+[0,l]n Kn(x1, · · · , xn)dx1 · · · dxn
+�����
+⩽ ni−1
+n
+�
+i=1
+Ei,
+(22)
+where
+Ei =
+�����Qi(Kn; xi) −
+� l
+0
+Kn(x1, · · · , xn)dxi
+����� .
+(23)
+Therefore, we have
+�����Qn
+m(Kn) −
+�
+[0,l]n Kn(x1, · · · , xn)dx1 · · · dxn
+�����
+⩽ nln+2
+24m2 |Kn|2
+(24)
+where |Kn|2 = max
+i ∥ ∂2Kn
+∂x2
+i ∥L∞((0,l)n).
+Similar to [30, Lemma A.4], we can bound |Kn|k by
+using the Hadamard’s inequality, which leads to
+|Kn|k ⩽ 2knn/2
+�
+� max
+i+j⩽k
+�����
+∂i
+x∂j
+yK(x, y)
+∂xi∂yj
+�����
+L∞((0,l)2)
+�
+�
+n
+.
+(25)
+Next we will show that
+���
+∂i
+x∂j
+yK(x,y)
+∂xi∂yj
+��� is upper-bounded.
+Since we have K(x, y) =
+� l
+0 G(x, s)G∗(y, s)ds, we
+will first analyze the property of G(x, s). We decom-
+pose G(x, s) as G1(x, s) + jG2(x, s), where G1, G2 ∈
+C∞([0, l]2). The smoothness of G1, G2 in their domains
+
+8
+is trivial since they are compositions of polynomial func-
+tions, trigonometric functions and square root functions.
+Consider the integral kernel K(x, y) expressed in terms
+of G1, G2, i.e.,
+K(x, y) =
+� l
+0
+�
+G1(x, s)G1(y, s) + G2(x, s)G2(y, s)
+�
+ds
++ j
+� l
+0
+�
+G1(y, s)G2(x, s) − G1(x, s)G2(y, s)
+�
+ds.
+(26)
+Since G1(x, s) and G2(y, s) are smooth in [0, l]2,
+we can conclude that f1(x, y) = G1(x, s)G1(y, s) +
+G2(x, s)G2(y, s) and f2(x, y) = G1(y, s)G2(x, s) −
+G1(x, s)G2(y, s) are smooth in the same domain. Since
+compactly supported smooth functions attain their maxi-
+mum or minimum values, the partial derivatives of K(·, ·)
+are upper-bounded for any order i, j, i.e.,
+����
+∂i+jK(x, y)
+∂xi∂yj
+���� < ∞,
+∀i, j.
+(27)
+Therefore, by substituting (24) and (25) into Lemma
+2, we can bound the difference between the mutual
+information I
+′
+0 and I1 by the following lemma:
+Lemma 3. The mutual information I1 converges to the
+mutual information I
+′
+0. The difference
+���I1 − I
+′
+0
+��� is at most
+inverse-proportional to m2.
+Proof: From (6) we know that for the operator
+TE, the kernel function can be expressed by K(x, y) =
+� L
+0 g(x, s)g∗(y, s)ds. From (25) we have
+|Kn|2 ⩽ 4nn/2An,
+(28)
+where A = max
+��� ∂i+jK(x,y)
+∂xi∂yj
+���
+L∞((0,l)2) is a constant.
+Therefore we have
+|d(z) − dQ(z)| ⩽
+∞
+�
+n=1
+zn
+n!
+nln+2
+24m2 max
+i
+����
+∂2Kn
+∂x2
+����
+L∞((0,l)n)
+⩽
+∞
+�
+n=1
+zn
+n!
+nln+2
+6m2 nn/2An.
+(29)
+According to the Stirling’s approximation, we have n! ⩾
+nne−n√
+2πn, which leads to
+|d(z) − dQ(z)| ⩽
+l2
+6m2
+∞
+�
+n=1
+� n
+2π
+(Aezl)n
+nn/2
+.
+(30)
+Since it is obvious that �∞
+n=1
+� n
+2π
+(Aezl)n
+nn/2
+is convergent,
+the difference between d(z) and dQ(z) is proportional to
+m−2. For the difference between mutual information I1
+and I
+′
+0, we have
+|I1−I
+′
+0| ⩽
+|d(z) − dQ(z)|
+min(d(z), dQ(z)) <
+l2
+6m2
+∞
+�
+n=1
+� n
+2π
+(Aezl)n
+nn/2
+,
+(31)
+where z =
+m
+ln1/2. From Lemma 1 we know that
+l
+mn1 ⩾
+n0 −
+n0l3���K
+′′
+E(r,r)
+���
+L∞(0,l)
+24m2|
+� l
+0 KE(r,r)dr| . Therefore, z is upperbounded,
+which completes the proof of Lemma 3.
+According to Lemma 1 and Lemma 3, we have
+Theorem 1, which bounds the difference between I0 and
+I1.
+Theorem 1. The mutual information I1 that can be
+obtained from the discrete receiver converges to the
+mutual information I0 that can be obtained from the
+continuous receiver when the number of points increases.
+The convergence rate is at least inverse-proportional to
+the square of the sampling number m.
+Proof: Since the Fredholm determinant f(z) =
+det(1 + zTE) is an analytic function, we have
+|det(1 + zTE) − det(1 + z1TE)|
+= |z − z1|
+����
+∂det(1 + xTE)
+∂x
+����
+x∈[min(z,z1),max(z,z1)]
+.
+(32)
+In our assumption z
+=
+2
+n0
+and z1
+=
+2m
+ln1 . The
+analycity of det(1 + zTE) implies that
+∂det(1+xTE)
+∂x
+is also an anlytic function and is bounded on the
+interval
+[min(z, z1), max(z, z1)].
+We
+denote
+M
+=
+max
+x
+��� ∂det(1+xTE)
+∂x
+���, where x ∈ [min(z, z1), max(z, z1)].
+From Lemma 1 we have
+����
+l
+mn1 − n0
+���� ⩽
+n0l3 ���K
+′′
+E(r, r)
+���
+L∞(0,l)
+24m2 � l
+0 KE(r, r)dr
+.
+(33)
+Since n1/m → n0/l when m approximates infinity, we
+denote the minimum value of n1/m by c. Therefore,
+���det(1 +
+2
+n0 TE) − det(1 + 2m
+ln1 TE)
+��� can be bounded by
+����det(1 + 2
+n0
+TE) − det(1 + 2m
+ln1
+TE)
+����
+⩽
+Ml2 ���K
+′′
+E(r, r)
+���
+L∞(0,l)
+12m2c
+� l
+0 KE(r, r)dr
+.
+(34)
+Similar to the direvation of (31), we know that when m
+increases, I
+′
+0 will converge to I0. The convergence rate
+is at least inverse-proportional to m2. From Lemma 3
+we know that
+���I1 − I
+′
+0
+��� is at most inverse-proportional to
+m2. Since |I0 − I1| ⩽
+���I0 − I
+′
+0
+��� +
+���I1 − I
+′
+0
+���, Theorem 1
+is proved.
+Remark 3. Theorem 1 shows that the SNR control
+scheme between the discrete and continuous models is
+appropriate, since the limit of the mutual information of
+the discrete model is proved to be that of the continu-
+ous model. That is to say, our proposed model is self-
+consistent. Therefore, we can use the proposed model
+to compare the mutual information from the discrete
+and continuous receivers. Our analysis is based on
+
+9
+0
+50
+100
+150
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+continuous receiver
+discrete receiver
+5 10 15 20
+1.5
+2
+2.5
+3
+continuous receiver
+discrete receiver
+Fig. 3. The mutual information as a function of the sampling number.
+The transmitter is kept continuous and the receiver is discretized.
+RE(r, r′) = P
+� l
+0 G(r, s)G∗(r′, s)ds which corresponds
+to the scenario when no CSI can be obtained at the
+transmitter but not limited to this scenario. It can be eas-
+ily extended to other shapes of autocorrelation functions
+after power allocation at the transmitter, as long as the
+analyticity of RE(r, r′) is guaranteed.
+B. Numerical analysis about the mutual information
+As proven in the above subsection, the mutual infor-
+mation between the continuous transmitter and discrete
+receiver converges to the mutual information between
+continuous transceivers. Therefore, the model of the dis-
+crete receiver can be viewed as the discretization of the
+continuous receiver. In this subsection, we will use nu-
+merical analysis to show the correctness of the theoretical
+results. Moreover, we will show the near-optimality of the
+discrete receiver with half-wavelength sampling.
+We set the length l of the transceivers to 2 m. The trans-
+mitter is kept continuous, while the receiver is discretized
+to m point antennas. The wavelength of the electromag-
+netic field is fixed to 0.04 m, which correpsonds to the fre-
+quency of 7.5 GHz. The distance between the transceivers
+varies from 10 m to 0.1 m. The simulation results are
+shown in Fig. 3. From the simulation, we can observe
+the convergence of the mutual information between the
+continuous transmitter and the discrete receiver, which
+verifies the theoretical analysis. For the three distances
+between transceivers, the half-wavelength sampling al-
+most achieves the supremum mutual information between
+continuous transceivers. Therefore, half-wavelength sam-
+pling of the receiver is suboptimal. Moreover, when the
+distance between transceivers decreases, we can observe
+that the mutual information converges slower. When the
+distance equals 0.1 m, the half wavelength sampling is at
+the critical state of convergence. If the distance is less
+than 0.1 m, a performance gap between the model with
+the continuous receiver and that with the discrete receiver
+may be observed. This performance gap has theoretical
+meaning but may not be useful because the distance will
+be comparable to the wavelength in this scenario, where
+the evanescent wave components will hold a dominant
+position.
+IV. COMPARISON BETWEEN CONTINUOUS AND
+DISCRETE TRANSCEIVERS
+In the above section we have compared the mutual
+information between the models with continuous and
+discrete receivers. For both models the transmitter is kept
+continuous, which simplifies the analyzing procedure. In-
+spired by the analysis in the above section, in this section
+we will compare the mutual information between contin-
+uous transceivers and that between discrete transceivers.
+Numerical analysis is then provided to show the near-
+optimality of the half-wavelength sampling scheme.
+A. Convergence analysis of the mutual information
+The analysis in this section focuses on the difference
+between I0 and I2. It is an extension of the conver-
+gence analysis in the above section. We define I
+′′
+0
+=
+logdet
+�
+1 + m2TE
+l2n2/2
+�
+as an intermediate variable similar
+to I
+′
+0. First we will discuss the convergence of |I0 − I
+′′
+0 |
+in the following lemma:
+Lemma 4. The mutual information I
+′′
+0 converges to the
+mutual information I0. The difference
+���I0 − I
+′′
+0
+��� is at most
+inverse-proportional to m2.
+Proof: From the SNR control scheme of discrete
+transceivers (17) and the multivariate m-point composite
+midpoint quadrature rule, we have
+����n0 − l2
+m2 n2
+���� =
+n0
+� l
+0 K(r, r)dr
+�����
+� l
+0
+� l
+0
+g(r, r, z)dzdr
+− l2
+m2
+m
+�
+i=1,j=1
+g(ri, ri, rj)
+�����
+⩽
+n0l4
+24m2 � l
+0 K(r, r)dr
+� ����
+∂2g(r, r, z)
+∂r2
+����
+L∞((0,l)2)
++
+����
+∂2g(r, r, z)
+∂z2
+����
+L∞((0,l)2)
+�
+,
+(35)
+where g(x, y, z)
+:=
+G(x, z)G∗(y, z), ri
+=
+(i −
+0.5)l/m. It is obvious that n2/m2 converges to n0/l2
+when m
+→
+∞. We denote the minimum value
+of n2/m2 by c. Then, according to (32), we know
+that
+���det(1 +
+2
+n0 TE) − det(1 + 2m2
+l2n2 TE)
+��� converges to 0
+when m → ∞. Therefore, |I0 − I
+′′
+0 | converges to 0, and
+the convergence rate is at least inversely proportional to
+m2.
+
+10
+Then we will discuss the convergence of |I2 − I
+′′
+0 | in
+the following lemma:
+Lemma 5. The difference
+���I2 − I
+′′
+0
+��� approaches 0 when
+m approaches infinity. Moreover, it is at most inverse-
+proportional to m2.
+Proof: We denote the Fredholm determinant and
+its discretization by d(z) = det(1 + zT) and dV (z) =
+det (1i=j + wjz �m
+k=1 wkG(ri, rk)G∗(rj, rk))m
+i,j=1,
+where K is the kernel of the operator T. To bound
+the difference between d(z) and dV (z), we define
+gn(x1, · · · , xn, s1, · · · , sn) as
+gn(x1, · · · , xn, s1, · · · , sn)
+= det
+�
+�
+g(x1, x1, s1)
+· · ·
+g(x1, xn, s1)
+· · ·
+g(xi, xj, si)
+· · ·
+g(xn, x1, sn)
+· · ·
+g(xn, xn, sn)
+�
+� .
+(36)
+From the definition of g(x, y, z), we know that
+� l
+0 g(xi, xj, si)dsi
+=
+K(xi, xj). According
+to the
+property
+of
+determinants
+that
+det(ai,j)m
+i,j=1
+=
+�
+k1,··· ,km(−1)ka1,k1 · · · am,km,
+where
+k1 · · · km
+is
+the kth exchange of 1 · · · n, we can find that
+Kn(x1, · · · , xn) =
+�
+[0,l]n gn(x1, · · · , xn, s1, · · · , sn)
+ds1 · · · dsn.
+(37)
+If we define Cn
+m(gn) by (38) we have (39). Here
+wαi correspondes to the distance between antennas in
+the source region and s correpsonds to the location
+of the antennas in the source region. When further
+considering
+the
+discretization
+of
+the
+receiver
+as
+in (19), we should set xn
+1
+to the location of the
+antennas in the destination region, and add additional
+weights w which equals the distance between antennas
+in the destination region. Similar to the definition
+of Qn
+m in (19), we define V n
+m(gn) by V n
+m(gn)
+=
+�m
+j1,··· ,jn=1
+�
+i wjiCn
+m(gn(rj1, · · · , rjn, s1,α1, · · · , sn,αn)).
+When sj,αi = rαi, we have
+V n
+m(gn) =
+m
+�
+j1,··· ,jn=1
+m
+�
+α1,··· ,αn=1
+� n
+�
+i=1
+ji
+� � n
+�
+i=1
+αi
+�
+gn(rj1, · · · , rjn, rα1, · · · , rαn)
+=
+m
+�
+j1,··· ,j2n=1
+� 2n
+�
+i=1
+ji
+�
+gn(rj1, · · · , rj2n).
+(40)
+The difference between d(z) and dV (z) is
+d(z) − dV (z)
+=
+∞
+�
+n=1
+zn
+n!
+�
+V n
+m(gn) −
+�
+[0,l]n Kn(x1, · · · , xn)dx1 · · · dxn
+�
+=
+∞
+�
+n=1
+zn
+n!
+�
+V n
+m(gn) −
+�
+[0,l]2n gn(x1, · · · , x2n)dx1 · · · dx2n
+�
+(41)
+Note that V n
+m(gn) is the numerical discretization of the
+function gn with 2n variables, we can bound V n
+m(gn) −
+�
+[a,b]2n gn(x1, · · · , x2n)dx1 · · · dx2n by using the multi-
+variate numerical integration error bound:
+�����V n
+m(gn) −
+�
+[0,l]2n gn(x1, · · · , x2n)dx1 · · · dx2n
+�����
+⩽ l2n−1
+2n
+�
+i=1
+Ei,
+(42)
+where
+Ei =
+�����Qi(gn; xi) −
+� l
+0
+gn(x1, · · · , x2n)dxi
+����� .
+(43)
+According to the m-point composite midpoint quadrature
+rule, we have
+�����Qi(gn; xi) −
+� l
+0
+gn(x1, · · · , x2n)dxi
+�����
+⩽
+l3
+24m2
+����
+∂2gn
+∂x2
+i
+����
+L∞((0,l)2n)
+,
+(44)
+From the Hadamard’s inequality [33], we can further
+bound it by
+�����V n
+m(gn) −
+�
+[a,b]2n gn(x1, · · · , x2n)dx1 · · · dx2n
+�����
+⩽ 2nl2n+2
+24m2 max
+j
+�����
+∂2gn
+∂x2
+j
+�����
+L∞((0,l)2n)
+⩽ 2nl2n+2
+24m2 max
+�
+�nn/2
+�����
+∂2g(x, y, z)
+∂z2
+����
+L∞((0,l)3)
+�n
+,
+4nn/2
+�
+� max
+i+j⩽2
+�����
+∂i
+x∂j
+yg(x, y, z)
+∂xi∂yj
+�����
+L∞((0,l)3)
+�
+�
+n �
+�
+⩽ l2n+2
+3m2 n(n+2)/2
+�
+� max
+i+j+k⩽2
+�����
+∂i
+x∂j
+y∂k
+z g(x, y, z)
+∂xi∂yj∂zk
+�����
+L∞((0,l)3)
+�
+�
+n
+.
+(45)
+Similar to Lemma 3, we know that |I2 −I
+′
+0| converges
+to 0, and the error is at most inverse proportional to m2.
+Therefore, we have Theorem 2:
+Theorem 2. The mutual information I2 that can be
+obtained from the discrete transceivers converges to the
+mutual information I0 that can be obtained from the
+continuous transceivers when the number of antennas in
+the discrete transceivers increases. The difference |I0−I2|
+is at least inverse-proportional to the square of the
+sampling number m.
+Remark
+4. Similar to Remark
+3 in Section. III,
+the convergence analysis in this section is not lim-
+ited to the scenario with equal power allocation.
+
+11
+Cn
+m(gn) = det
+�
+�
+�
+α1 wα1g(x1, x1, s1,α1)
+· · ·
+�
+α1 wα1g(xn, xn, s1,α1)
+· · ·
+�
+αi wαig(xi, xj, si,αi)
+· · ·
+�
+αn wαng(xn, x1, sn,αn)
+· · ·
+�
+αn wαng(xn, xn, sn,αn),
+�
+� .
+(38)
+Cn
+m(gn) =
+�
+k1···kn
+(−1)k(
+m
+�
+α1=1
+wα1g(x1, xk1, s1,α1)) · · · (
+m
+�
+αn=1
+wαng(xn, xkn, sn,αn))
+=
+�
+k1···kn
+�
+(−1)k
+m
+�
+α1,··· ,αn=1
+n
+�
+i=1
+wαig(xi, xki, si,αi)
+�
+=
+m
+�
+α1,··· ,αn=1
+�� n
+�
+i=1
+wαi
+� �
+k1···kn
+(−1)k
+n
+�
+i=1
+g(xi, xki, si,αi)
+�
+=
+m
+�
+α1,··· ,αn=1
+�� n
+�
+i=1
+wαi
+�
+gn(x1, · · · , xn, s1,α1, · · · , sn,αn)
+�
+.
+(39)
+0
+50
+100
+150
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+continuous transceiver
+discrete receiver
+discrete transceiver
+10
+20
+0.1188
+0.119
+0.1192
+Fig. 4. The mutual information variation with different sampling num-
+bers. The mutual information that correpsonds to the three models with
+continuous and discrete transceivers is plotted. The distance between
+the transceivers is large.
+For arbitrary analytic function RJ(s, s′), the conver-
+gence of |I0 − I2| can be obtained. Instead of dis-
+cretizing
+�
+G(r, z)G∗(r′, z)dz to �
+i G(r, ri)G(r′, ri),
+we will discretize
+��
+G(r, z)RJ(z, z′)G∗(r′, z′)dzdz′ to
+�
+i,j G(r, ri)RJ(ri, rj)G(r′, rj) in the extended sce-
+narios
+with
+power
+allocation
+schemes.
+Then,
+in-
+stead of g(x, y, z) we need a four-variable function
+h(x, y, z, ω) := G(x, z)RJ(z, ω)G(y, ω) and the deriva-
+tion procedure of the convergence has no essential differ-
+ence with Theorem 2.
+B. Numerical analysis about the mutual information
+In this subsection, we will verify the correctness of
+the convergence analysis in the above subsection by
+simulations. The length l of the transceivers is fixed
+to 2 m. We have plotted the mutual information of the
+three models: continuous transceiver, continuous trans-
+mitter and discrete receiver, and discrete transceiver. The
+transceivers are both discretized to m point antennas.
+The wavelength of the electromagnetic field is fixed to
+0.04 m, which corresponds to the frequency of 7.5 GHz.
+First we will show the scenarios when the distance be-
+tween the transceivers is large. The distance between the
+transceivers varies from 50 m to 200 m. The simulation
+results are shown in Fig. 4.
+From the simulation results we find that the mutual
+information nearly keeps the same when the sampling
+number increases. The reason for this phenomenon is that
+the DoF of the channel is nearly inverse-proportional to
+the distance between transceivers. For example, the DoF
+when the distance equals 50 m can be approximated by
+l2/(dλ) = 2, which means that when the sampling num-
+ber is 5, the multiplexing gain is almost fully explored.
+Therefore, for large distances between transceivers, the
+dominant limitation is the channel DoF, which means that
+the suboptimal performance can be achieved by sampling
+sparser than half-wavelength.
+Moreover, we have shown the variation of the mutual
+information with the sampling number when the distance
+between transceivers is small. In Fig. 5 the distance
+between the transceiver is 0.1 m and 1 m. We can find that
+when the distance decreases, the dense sampling of the
+transceivers becomes important to fully explore the limit
+of the mutual information. However, the half-wavelength
+sampling of the transceivers still achieves suboptimal
+performance, which means that denser sampling schemes
+are not necessary.
+V. CONCLUSION
+In this paper, we proposed a comparison scheme be-
+tween continuous and discrete MIMO systems which is
+
+12
+0
+50
+100
+150
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+continuous transceiver
+discrete receiver
+discrete transceiver
+Fig. 5. The mutual information as a function of the sampling number.
+The mutual information values that correspond to the three models with
+continuous and discrete transceivers are plotted. The distance between
+the transceivers is small.
+based on a precise non-asymptotic analysis framework.
+Three information-theoretic models of the continuous
+and discrete transceivers were built, with the first model
+corresponds to the fully continuous electromagnetic infor-
+mation theory model, and the third model corresponds to
+the matrix-vector MIMO model. We proposed physically
+consistent SNR control schemes to ensure the fairness of
+the comparison, and proved that the mutual information
+between discrete MIMO transceivers converges to that
+of continuous electromagnetic transceivers. Numerical
+results verified the theoretical analysis and showed the
+near-optimality of the half-wavelength sampling scheme.
+Further works can be done by extending the lin-
+ear transceivers to rectangular or other two-dimensional
+transceivers for generality. The analysis based on the
+capacity after water-filling of the mutual information also
+remains to be explored.
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+
diff --git a/1NFAT4oBgHgl3EQfCxwU/content/tmp_files/load_file.txt b/1NFAT4oBgHgl3EQfCxwU/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1931c106211eee4f7b00a4c79cc3c7356ce8cc62
--- /dev/null
+++ b/1NFAT4oBgHgl3EQfCxwU/content/tmp_files/load_file.txt
@@ -0,0 +1,759 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf,len=758
+page_content='1  Can Continuous Aperture MIMO Achieve Much  Better Performance than Discrete MIMO?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Zhongzhichao Wan, Jieao Zhu, and Linglong Dai, Fellow, IEEE  Abstract—The concept of continuous-aperture multiple-  input multiple-output (CAP-MIMO) technology has been  proposed recently, which aims at achieving high spectrum  density by deploying extremely dense antennas or even  continuous antennas in a given aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The fundamental  question of CAP-MIMO is whether it can achieve much  better performance than the traditional discrete MIMO  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' In this paper, to model the CAP-MIMO, we use self-  adjoint operators to depict the structural characteristics of  the continuous random electromagnetic fields from physical  laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Then, we propose a non-asymptotic performance  comparison scheme between continuous and discrete MIMO  systems based on the analysis of mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We  show the consistency of the proposed scheme by proving  that the mutual information between discretized transceivers  converges to that between continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Numeri-  cal analysis verifies the theoretical results, and suggests that  the mutual information obtained from the discrete MIMO  with widely adopted half-wavelength spaced antennas al-  most achieves the mutual information obtained from CAP-  MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Index Terms—Multiple-input multiple-output (MIMO),  Continuous-aperture MIMO (CAP-MIMO), mutual infor-  mation, random fields, Fredholm determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' INTRODUCTION  The spectrum efficiency of wireless communication  systems has been greatly improved from 3G to 5G  because of the use of multiple-input multiple-output  (MIMO) technology [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The MIMO systems utilize  multiple antennas to exploit the spatial multiplexing gain  [4], where the antennas are modeled as discrete points  in the continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Along with the tendency of  increasing the number of antennas to achieve higher  spectrum efficiency, people are considering deploying  extremely dense antennas in a given aperture [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  When the number of antennas in a given aperture tends to  infinity, the traditional MIMO systems with transceivers  composed of discrete point antennas are equivalent  to the MIMO systems with continuously controllable  transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, the MIMO with extremely dense  All authors are with the Department of Electronic Engineer-  ing, Tsinghua University as well as Beijing National Research  Center  for  Information Science  and  Technology  (BNRist), Bei-  jing 100084, China (E-mails: {wzzc20, zja21}@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  daill@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  This work was supported in part by the National Key Research  and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 2020YFB1807201), in  part by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  62031019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  antennas is called continuous-aperture MIMO (CAP-  MIMO), and is also called holographic MIMO [7]–[9]  or large intelligent surface [5], [10] in the recent litera-  ture1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' It has attracted increasing interest in the research  of MIMO technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Recent works about CAP-MIMO  include pattern optimization [6], antenna design [11],  channel estimation [7], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For CAP-MIMO, the  fundamental question is whether the CAP-MIMO system  can achieve much better performance than the traditional  discrete MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Related works  The structure of CAP-MIMO has been defined in the  previous part but there are many structures for realizing  the discrete MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, we need to choose which  structure of the discrete MIMO to compare with CAP-  MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' A representative structure of discrete MIMO  uses half-wavelength spaced antennas to compose the  transceivers [12]–[14], because half-wavelength sampling  of the electromagnetic field can reconstruct the original  field according to the sampling theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  There have been several works discussing the per-  formance comparison between CAP-MIMO and discrete  MIMO with half-wavelength spaced antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The perfor-  mance comparison is from the degrees of freedom (DoF)  perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Specifically, when discarding the evanescent  wave components, the Fourier transform of the received  field, which is in the wavenumber domain, is concentrated  in a circle or a segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' This concentration phenomenon  means that the field is bandlimited in the wavenumber  domain, thus it can be perfectly recovered from the half-  wavelength sampling points in the spatial domain [15]  according to the Nyquist sampling theorem [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The  above conclusion is based on the assumption that we can  observe the received field in the infinitely large spatial  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' However, in practice, the destination where we  can observe the field is in a finitely large aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  For a rigorous analysis framework of the DoF in a  finitely large aperture, the prolate spheroidal wave func-  tion (PSWF) [17] is introduced to perform orthogonal  expansion on the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Specifically, to  1The MIMO with extremely dense antennas can be accurately de-  scribed by the name CAP-MIMO, while holographic MIMO and large  intelligent surface do not focus on the continuity of the transceiver  apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, in the rest part of the paper, we will prefer using  the name CAP-MIMO rather than using other names like holographic  MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='08411v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='IT]  20 Jan 2023  2  reconstruct the wavenumber-bandlimited electromagnetic  field observed in a length-l spatial region, the PSWFs  were used as the basis based on the Slepian’s concentra-  tion problem [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Such an electromagnetic field can be  perfectly reconstructed from infinite number of PSWFs,  and approximately reconstructed from a finite number  of PSWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' If the reconstruction error can be controlled  within a given threshold by using N0 PSWFs, the number  of DoFs of the field can be approximated by N0 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  This analyzing scheme is strict for arbitrary l, but can  only provide the asymptotic result of the DoF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=', the  quantitative result of N0 can be obtained only when  the length l or the frequency tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' However,  the practical systems are with finitely large aperture and  finite frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The asymptotic result can not provide  quantitative number of DoFs for practical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' There-  fore, a non-asymptotic performance comparison scheme  between CAP-MIMO and discrete MIMO is required for  the accurate performance comparison with finitely large  apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Our contributions  To solve this problem, in this paper, we provide a non-  asymptotic performance comparison scheme between  CAP-MIMO and discrete MIMO, and we further prove  the rationality of the scheme2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Specifically, the contribu-  tions of this paper can be summarized as follows:  We build models of CAP-MIMO and discrete MIMO  based on electromagnetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For CAP-MIMO  with continuous transceivers, we model the structural  characteristics of the continuous random electro-  magnetic fields from physical laws by using self-  adjoint operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Based on this model, we can  utilize the spectrum theory of operators to derive the  information that can be obtained from the received  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The existing models of MIMO with discrete  transceivers are spatially discretized from the contin-  uous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Moreover, signal-to-noise ratio (SNR)  control schemes are introduced to ensure the fairness  of the comparison between CAP-MIMO and discrete  MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Then, before comparing the performance between  CAP-MIMO with continuous transceivers and tra-  ditional MIMO with discrete transceivers, we first  utilize the simplified model with continuous trans-  mitter and discrete receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Under this simplified  model, the transmitter is continuous, which is the  same as that in the CAP-MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' By theo-  retically analyzing the mutual information that can  be obtained from the discrete receiver in this sim-  plified model, we can obtain some insights about  how the discretization of the receiver affects the  mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Moreover, the theoretical proof  2Simulation  codes  will  be  provided  to  reproduce  the  re-  sults in this paper: http://oa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='cn/dailinglong/publications/  publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  of the convergence of the mutual information in the  simplified model can inspire the analysis of a more  practical scenario, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=', the discrete transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Finally, we extend the convergence proof from the  model with discrete receiver to the model with  discrete transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We prove that the mutual in-  formation between the discrete transceivers con-  verges to the mutual information between continuous  transceivers when the number of antennas of the  discretized transceivers tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore,  the fairness of the performance comparison is guar-  anteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Numerical results are provided to verify the  theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Moreover, it shows the near-  optimality of the half-wavelength sampling of the  transceivers in traditional discrete MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Organization and notation  Organization: The rest of our paper is organized as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' II introduces the basic model of EIT and  proposes models with continuous or discrete transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  The mutual information between the transceivers is also  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' III proves the convergence of the mutual  information between continuous transmitter and discrete  receiver when the number of discrete antennas increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Then, the convergence of the mutual information between  discrete transceivers is illustrated in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Finally,  we conclude the paper in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Notation: bold characters denote matrices and vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  j is the imaginary unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' E [x] denotes the mean of random  variable x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' x∗ denotes the conjugation of a number or  a function x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' XH denotes the conjugate transpose of  a vector or a matrix X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' µ0 is the permeability of a  vacuum, Z0 is the free-space intrinsic impedance and  c is the speed of light in a vacuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' ∇ is the nabla  operator, and ∇× is the curl operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' |φ⟩ is the quantum  mechanical notation of a function φ, where the inner  product is denoted by ⟨ψ| φ⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' det(·) denotes the matrix  determinant or the Fredholm determinant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' tr(·) denotes  the trace of a matrix or an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Im denotes the m×m  identity matrix, 1 denotes the indentity operator, δ(x)  denotes the delta function, and 1i=j denotes the indicator  function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' |x| denotes the modulus of a complex variable,  and ∥f(x)∥L∞(a,b) is the uniform norm of the function  f(x) over the interval [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' C∞(K) denotes the set of  smooth functions supported on a compact set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' MODELS OF CONTINUOUS AND DISCRETE  SYSTEMS  In this section, we introduce the models of contin-  uous and discrete systems for performance comparison  between CAP-MIMO and discrete MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We control  the SNR at the receiver side to ensure the fairness of the  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The information obtained from these models  is derived from operators and matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  3  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Basic model of electromagnetic information theory  To model the transceivers and the channel, we follow  the approach of electromagnetic information theory (EIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  The EIT is an interdisciplinary subject that integrates the  classical electromagnetic theory and information theory to  build an analysis framework for the ultimate performance  bound of wireless communication systems [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The anal-  ysis framework of EIT is based on spatially continuous  electromagnetic fields, which provides us the tool to  model and analyze the continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Then, for  the consistency, the model of discrete transceivers are  viewed as the discretization of the continuous model from  EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  The model of EIT is built on the vector wave equa-  tion [21] without boundary conditions, which is expressed  by  ∇×∇×E (r)−κ2  0E (r) = jωµ0J (r) = jκ0Z0J (r) , (1)  where κ0 = ω√µ0ε0 is the wavenumber, and Z0 =  µ0c = 120π [Ω] is the free-space intrinsic impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  We assume that the transceivers are confined in two  regions Vs and Vr, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The current density at the  source is J(s), where s ∈ R3 is the coordinate of the  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The induced electric field at the destination is  E(r), where r ∈ R3 is the coordinate of the field observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  To solve the linear partial differential equation (1), a  general theoretical approach is to introduce the dyadic  Green’s function G(r, s) ∈ C3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' According to the  linearity of (1), the electric field E(r) can be expressed  by  E(r) =  �  Vs  G(r, s)J(s)ds,  r ∈ Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (2)  By exploiting the symmetric properties of the free space,  the Green’s function in unbounded, homogeneous medi-  ums at a fixed frequency point is [22]  G(r, s) = jκ0Z0  4π  �  I + ∇r∇H  r  κ2  0  � ejκ0∥r−s∥  ∥r − s∥  = jκ0Z0  4π  ejκ0∥r−s∥  ∥r − s∥  �  �  I − ˆpˆpH�  +  j  2π ∥r − s∥ /λ  �  I − 3ˆpˆpH�  −  1  (2π ∥r − s∥ /λ)2  �  I − 3ˆpˆpH�  �  [Ω/m2],  (3)  where ˆp =  p  ∥p∥ and p = r − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Since there are some non-ideal factors at the receiver  that corrupts the recieved field, we call them the noise  field N(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The received electric field can be expressed  by Y(r) = E(r) + N(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The above equations represent  the deterministic model in the electromagnetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' To  satisfy the demand of wireless communication, we need  to convey information through the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Specifically, the wireless communiation system encodes  the information in the current J(s), and decodes the  information from the noisy electric field Y(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Due to  the randomness of the transmitted bit source, the electro-  magnetic fields are randomly excited by the transmitter  equipments before being radiated into the propagation  media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, the electromagnetic fields should be  modeled as random fields [23], which are random func-  tions with several arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We denote the autocorre-  lation function of the current and the electric field as  matrix-valued functions RJ(s, s  ′) = E[J(s)JH(s  ′)] and  RE(r, r  ′) = E[E(r)EH(r  ′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The relationship between  RJ and RE is determined by the Green’s function, which  is  RE(r, r′) =  �  Vs  �  Vs  G(r, s)RJ(s, s′)GH(r, s)dsds′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (4)  Similar definitions of the autocorrelation functions for  the noise field and the noisy electric field are repre-  sented as RN(r, r  ′) = E[N(r)NH(r  ′)] and RY(r, r  ′) =  E[Y(r)YH(r  ′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Continuous trasceivers  In this part, we will build the model of CAP-MIMO  with continuous transceivers based on the EIT model  in the above subsection, and then derive the mutual  information between the continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For sim-  plicity, in the rest part of the paper, we assume that the  transceivers are linear along the ˆz-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Moreover,  since the current J can only exist on the linear source  and we only observe the electric field on the linear  receiver, we express all the physical quantities in a  Cartesian coordinate system that satisfies s = (0, 0, s) and  r = (d, 0, r), where d is the distance between the parallel  source and destination line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' This model corresponds to  single-polarized linear antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Through this simplifica-  tion scheme, we use J(s) and E(r) instead of J(s) and  E(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The relationship between them can be expressed by  E(r) =  � l  0 G(r, s)J(s)ds, where G(r, s) is the upper left  element of the matrix G(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We can derive G(r, s) as  G(r, s) =jZ0ej2π  √  x2+d2/λ  2λ  √  x2 + d2  �  j  2π  √  x2 + d2/λ  d2 − 2x2  x2 + d2  +  d2  x2 + d2 −  1  (2π/λ)2(x2 + d2)  d2 − 2x2  x2 + d2  �  ,  (5)  where x = r − s and λ = 2π/κ0 is the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Here we consider the scenario with no channel state  information, which means that the signals on the source  are under equal power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The second moments  (autocorrelation) of J are denoted by RJ(s, s′) = Pδ(s−  s′), s, s′ ∈ [0, l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Since the noiseless received field is uniquely deter-  mined by the source and the deterministic channel, the  autocorrelation function of the electric field is expressed  4  by the source autocorrelation RJ(s, s′) and the Green’s  function G(r, s), written as  RE(r, r′) =  � l  0  � l  0  G(r, s)RJ(s, s′)G∗(r′, s′)dsds′  = P  � l  0  G(r, s)G∗(r′, s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (6)  The received field on the destination is Y (r)  =  E(r) + N(r), where N(r) is the noise field at the  receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' In this paper, we consider thermal noise model  E [N(r)N ∗(r′)] =  n0  2 δ(r − r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' According to [24], we  can perform Mercer expansion on the electric field E(r)  to obtain a set of mutually independent random variables  ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The expansion can be written as E(r) = �  k ξkφk(r),  where E[ξkiξ∗  kj] = λki1i=j and ⟨φki(r), φkj(r)⟩ = δkikj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  This expansion scheme has split the continuous field  into independent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Since the white noise field  can be expanded under arbitrary orthogonal bases, the  continuous channel is also decomposed into independent  subchannels, which makes the mutual information of the  subchannels summable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Next we will show that for the operator TE := φ(r) →  � l  0 KE(r, r′)φ(r′)dr′, where KE(r, r′) = RE(r, r′) =  P  � l  0 G(r, s)G∗(r′, s)ds, all of its eigenvalues are real  and nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Moreover, the sum of its eigenval-  ues �∞  i=1 λi equals P  � l  0  � l  0 G(r, s)G∗(r, s)drds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Notice  that TE can be decomposed to T ∗T, where T  :=  φ(r) →  √  P  � l  0 G(r, s)φ(r)ds and T ∗  := φ(r) →  √  P  � l  0 G∗(r, s)φ(r)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' This decomposition means that  TE = T ∗  E is a self-adjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We assume that λ  is an eigenvalue of TE and φ(r) is the corresponding  eigenfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Since  λ = λ⟨φ(r), φ(r)⟩ = ⟨T ∗  Eφ(r), φ(r)⟩  = ⟨φ(r), TEφ(r)⟩ = λ∗,  (7)  we know that λ is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' From  λ = λ⟨φ(r), φ(r)⟩ = ⟨T ∗Tφ(r), φ(r)⟩  = ⟨Tφ(r), Tφ(r)⟩ ⩾ 0,  (8)  we know that λ is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  From [25] we know that an integral operator on [a, b]  is a trace class operator if its kernel K(x, y) satisfies  K(x, y) and ∂yK(x, y) are continuous on [a, b]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' There-  fore TE is a trace class operator, which means that the  sum of its eigenvalues is finite and can be expressed  by [26]  tr(TE) =  � l  0  KE(r, r)dr = P  � l  0  � l  0  G(r, s)G∗(r, s)drds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (9)  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The non-negative values  λk  n0/2 represent  the SNR of the independent subchannels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The mutual  information between the noisy received field and the  current on the source can be expressed by  I0(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Y ) =  +∞  �  k=1  log  �  1 +  λk  n0/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (10)  By introducing the Fredholm determinant which is the  determinant of operators, we can express (10) by  I0(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Y ) = log det  �  1 + TE  n0/2  �  ,  (11)  where (TEφ)(r) :=  � L  0 RE(r, r′)φ(r′)dr′ and λk are the  eigenvalues of TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Our analysis here is based on the sim-  plified model with uni-polarized linear antennas as the  transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' This simplification reduces the dimension of  the problem, where random fields degenerate to one-  dimensional random processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For the more general  scenarios, such analyzing schemes are still effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' If  the random field is defined in a region X, we can expand  E(r) by E(r) = �  k ξkΦk(r) and its autocorrelation  function RE(r, r  ′) by RE(r, r  ′) = �  k λkΦk(r)ΦH  k (r  ′)  [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The expansion satisfies that λk and Φk(r) are  eigenvalues and eigenfunctions of the integral equation  �  X RE(r, r  ′)Φ(r  ′)dr  ′ = λkΦ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Similar expressions of  the mutual information in (10) and (11) can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Continuous transmitter and discrete receiver  Before building the model with discrete transceivers,  in this subsection, we will first build a simplfied model  with continuous transmitter and discrete receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The  simplfied model analyzed here can bring some insights  about the discretization of both transceivers and the  SNR control schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For the continuous transmitter,  we still use the length-l linear transmitter along the ˆz-  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For the discrete receiver, we build a model  with m point antennas on a segment parallel to the  linear transmitter in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The ith point  antenna is placed on ri ∈ [0, l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The correlation matrices  of the received signals and received noise are denoted  by K  ′  E and K  ′  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For the received signals, we assume  that it is the sampling of the continuous electric field  on the point ri, which means that K  ′  E = KE(ri, rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  However, for the received noise on the antenna, it can  not directly be assumed as the point sampling of the  noise field, because of the delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' To solve this  problem, we assume that K  ′  N =  n1  2 Im is an identity  matrix, and control the signal-to-noise ratio (SNR) of this  model the same as that of the continuous model to ensure  the fairness of the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The SNR at the receiver  of the continuous model is �∞  i=1  λi  n0/2, where λi is the  ith eigenvalue of the operator TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' From Lemma 1 we  know that �∞  i=1  λi  n0/2 =  P  n0/2  � l  0  � l  0 G(r, s)G∗(r, s)drds  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The SNR at the receiver of the discrete model is  �m  i=1  λ  ′  i  n1/2, where λ  ′  i is the ith eigenvalue of the matrix  K  ′  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  The SNR control scheme is necessary because if we  do not control the SNR, the mutual information that can  be obtained from the discrete antennas in the receiver  may infinitely increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Let us take a counter-example  where the power of received signal and received noise on  5  each point antenna remain unchanged when the number  of antennas in a given aperture increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For dense  antennas we can assume that N received signals of the  antennas in a small aperture are nearly the same, while  the corresponding noises are independent according to  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Then, the SNR for the N antennas will  keep near-linearity increasing with N, since when we  perform combing of the N received signals we have  SNR =  E[(�N  i=1 Ei)(�N  i=1 E∗  i )]  E[(�N  i=1 Ni)(�N  i=1 N ∗  i )] ≈ N E[E1E∗  1 ]  E[N1N ∗  1 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore  the mutual information that can be obtained from the  N antennas will keep near-logarithm increasing with N,  which corresponds to the simulation in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  According to (9), the noise power in the discrete  receiver model can be controlled by  n1 = n0  �m  i=1 KE(ri, ri)  � l  0 KE(r, r)dr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (12)  We denote the determinant of matrix K ∈ Cm×m by  det(Ki,j)m  i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Then we can express the mutual informa-  tion between the transceivers by  I1 = log  �  det(K  ′  N + K  ′  E)  det(K  ′  N)  �  = logdet  �  1i=j + KE(ri, rj)  n1/2  �m  i,j=1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (13)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Here the SNR on each of the point antennas in  the discrete model changes with the density of point anten-  nas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Notice that L  m  �m  i=1 KE(ri, ri) is the approximation  of the integral  � l  0 KE(r, r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' When m approximates  infinity, n1 will approximate mn0  2l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' This phenomenon has  several annotations, including the increase of the noise  power on each point antenna, the reduction of antenna  efficiency, and the corollary of the discretization of EIT  continuous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  From the perspective of noise power, we can explain it  by spatial sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For the point antenna arrays, more  antennas on a given aperture corresponds to a higher  sampling rate in the spatial domain and a wider lowpass  filter in the wavenumber domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Since a wide lowpass  filter can receive more noise power from the white noise  field, the noise power should increase with the density of  the antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  From the perspective of antenna efficiency, the well-  known Hannan’s efficiency shows that for both transmit-  ting and receiving antennas, the antenna gain is propor-  tional to lxly for two-dimensional surface antennas [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Therefore, for the linear model we considered, the an-  tenna gain will be inversely proportional to the sampling  number when the antennas are dense enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Besides these two annotations, another perspective is  viewing the model of discrete point antennas as the  discretization from the EIT continuous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' If we  consider m linear continuous antennas instead of point  antennas in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' All the antennas are  connected head to tail to occupy the [0, l] position in the  space and detect the electric field by inner producting it  with its eigenmode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' This model fulfills the requirement of  discretizing the continuous receiver to discrete receiving  antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The signal received by the ith antenna is  Yi =  � ai+1  ai  Y (r)φ(r)dr, where [ai, ai+1] is the occupied  region of the ith antenna, and φ(r) is the eigenmode of the  antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' If we assume φ(r) ≡ 1, the correlation matrix  of the received electric field can be expressed by  (KE)i,j = E  �� ai+1  ai  � aj+1  aj  E(r)E∗(r′)drdr′  �  = (ai+1 − ai)(aj+1 − aj)KE(ri, rj),  (14)  where ri ∈ [ai, ai+1] and rj ∈ [aj, aj+1] according to  the mean value theorem for integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For the noise field  on the destination, we have  (KN)i,j = E  �� ai+1  ai  � aj+1  aj  N(r)N ∗(r′)drdr′  �  =  �  (ai+1 − ai) n0  2  i = j  0  i ̸= j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (15)  Therefore, the SNR after the discretization will discreases  by ai+1 − ai, which is the case when the antennas are  dense enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  After explaining the rationality of the SNR control  scheme, we will introduce the following lemma to show  the convergence of the noise power on each discrete point  antenna, which will be useful for the following proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' When the number of antennas m in a given  aperture increases, the noise power on each antenna n1/2  will approach  mn0  2l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The difference between them is at  most inverse-proportional to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Proof: From (12) and the middle point quadrature  rule, we have  ����  l  mn1 − n0  ���� = n0  ���  � l  0 KE(r, r)dr − l/m �m  i=1 KE(ri, ri)  ���  ���  � l  0 KE(r, r)dr  ���  ⩽  n0l3 ���K  ′′  E(r, r)  ���  L∞(0,l)  24m2  ���  � l  0 KE(r, r)dr  ���  ,  (16)  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Discrete transceivers  The models discussed in the above subsections keep  the transmitter continuous and only perform discretization  on the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' However, the commonly used model  to depict wireless communication is the discrete MIMO  model, in which both the transceivers are modeled as  discrete point antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, in this section, we  will introduce a model which discretizes the transceivers  simultaneously, which is the extension of the model with  continuous transmitter and discrete receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Then, similar  6  d  Continuous  Transmitter  Continuous  Receiver  d  Discrete  Receiver  l  / 2  \uf06c  l  d  Discrete  Receiver  l  / 2  \uf06c  Discrete  Transmitter  Continuous  Transmitter  0I  1I  2I  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Comparison between the three models in this section with continuous transceivers and the model with discrete transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  to the scheme in the above subsection, we will provide the  corresponding SNR control scheme to ensure the fairness  of the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Specifically, we build a model with m point antennas  on a length-l segment in the source region and m point  antennas on a length-l segment in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Similar to the above subsection, we assume that the ith  point antenna is placed at si in the source region and ri  in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The correlation matrix of the  signals in the source region is set to be an identity matrix  K  ′′  J = PIm, which corresponds to the power allocation  scheme with no channel state information at the transmit-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The channel gain from the ith antenna in the source  region and the jth antenna in the destination region can be  expressed by Hi,j = G(si, rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The correlation matrix of  the received signal is denoted by K  ′′  E = HK  ′′  JHH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The  noise matrix is denoted by K  ′′  N = n2  2 Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Similar SNR  control on the receiver side is used, which is expressed  by  n2 = n0  �m  i=1  �m  j=1 G(ri, sj)G∗(ri, sj)  � l  0  � l  0 G(r, s)G∗(r, s)drds  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (17)  The mutual information between the transceivers is ex-  pressed as:  I2 = log  �  det(K  ′′  N + K  ′′  E)  det(K  ′′  N)  �  = logdet  �  1i=j +  �  k G(ri, rk)G∗(rj, rk)  n2/2  �m  i,j=1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (18)  The comparison between the three models built in Sec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' II-B, Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' II-C and in this subsection is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' In the following two sections we will introduce  the intermediate quantity I  ′  0 and I  ′′  0 to theoretically prove  that I1 and I2 converge to I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The flow chart of the proof  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 2  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' PERFORMANCE COMPARISON BETWEEN DISCRETE  AND CONTINUOUS RECEIVERS  In the above section we have proposed the mod-  els of continuous and discrete transceivers and derived  the corresponding mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Before compar-  ing the performance between CAP-MIMO with contin-  uous transceivers and traditional MIMO with discrete  transceivers, we first utilize the simplified model with  continuous transmitter and discrete receiver in this sec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Under this simplified model, the transmitter is con-  tinuous, which is the same as that in the CAP-MIMO  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The comparison is based on the convergence  analysis of the mutual information when the number  of antennas in the discrete receiver increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Numer-  ical analysis is provided to verify the correctness of  the convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The discussion about discrete  transceivers inspired by the analysis in this part will be  in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Convergence analysis of the mutual information  To compare the mutual information I0 and I1, we intro-  duce an intermediate quantity I  ′  0 = logdet  �  1 + mTE  ln1/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  We can bound |I0 −I1| by |I0 −I  ′  0|+|I1 −I  ′  0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' According  to [30], I1 can be viewed as the approximation of I  ′  0  using a numerical integral scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' In our discussion the  point antennas in the destination region are evenly spaced,  which means that ai = (i−1)l/m and ri = (i−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='5)l/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  To bound |I1 − I  ′  0|, we introduce the following lemma  from [30]:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We define d(z) := det(1+zT) and dQ(z) :=  det (1i=j + wjzK(ri, rj))m  i,j=1, where K is the kernel of  the operator T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=" The difference between d(z) and dQ(z)  7  0I  '  0I  1I  2I  ''  0I  Discretize the receiver  Discretize the transmitter  Discretize the transceivers  Lemma 1  Lemma 2  Lemma 3  Lemma 4  Lemma 5  Theorem 1  Theorem 2  Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Flow chart of the proof in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  is  d(z) − dQ(z) =  ∞  �  n=1  zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  �  Qn  m(Kn)  −  �  [a,b]n Kn(x1, · · · , xn)dx1 · · · dxn  �  ,  (19)  where Kn(x1, · · · , xn)  =  det (K(xi, xj))n  i,j=1, and  Qn  m(f) = �m  j1=1,··· ,jn=1  �n  i=1 wjif(rj1, · · · , rjn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Lemma 2 provides a method to compare the difference  between a Fredholm determinant of operator and a clas-  sical determinant of matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' In our model, the operator T  corresponds to the integral operator TE, z equals 2m  ln1 ,and  wj = l/m according to the equally spaced antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Notice that Qn  m(f) is the numerical approximation of  the integral  �  [a,b]n Kn(x1, · · · , xn)dx1 · · · dxn, we need  to use numerical integral theory to estiamte the approxi-  mation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For the model with equally spaced antennas,  this expression corresponds to a multivariate m-point  composite midpoint quadrature rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  For the error bound of a m-point composite midpoint  quadrature [31], we have  �����Qm(f) −  � l  0  f(x)dx  ����� ⩽  l3  24m2 ∥f  ′′∥L∞(0,l)  (20)  According to [32], the numerical approximation error  for multiple integrals in a n-dimensional unit cube can be  bounded by  �����  �  Gn  f −  � n  �  i=1  �  Qi(f)  ����� ⩽ E1 +  n  �  i=2  i−1  �  j=1  WjEi,  (21)  where Qj(g) := �  j wi,jg(xi,j), Wi = �  j |wi,j| and  Ei ⩾  ���Qi(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' xi) −  � 1  0 f(x1, · · · , xn)dxi  ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' According to  the models in this paper, we have wi,j = l/m and Wi = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  By simple variation of the integral band, we can bound the  approximation error of the multi-dimensional numerical  integral quadrature rule by  �����Qn  m(Kn) −  �  [0,l]n Kn(x1, · · · , xn)dx1 · · · dxn  �����  ⩽ ni−1  n  �  i=1  Ei,  (22)  where  Ei =  �����Qi(Kn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' xi) −  � l  0  Kn(x1, · · · , xn)dxi  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (23)  Therefore, we have  �����Qn  m(Kn) −  �  [0,l]n Kn(x1, · · · , xn)dx1 · · · dxn  �����  ⩽ nln+2  24m2 |Kn|2  (24)  where |Kn|2 = max  i ∥ ∂2Kn  ∂x2  i ∥L∞((0,l)n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Similar to [30, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='4], we can bound |Kn|k by  using the Hadamard’s inequality, which leads to  |Kn|k ⩽ 2knn/2  �  � max  i+j⩽k  �����  ∂i  x∂j  yK(x, y)  ∂xi∂yj  �����  L∞((0,l)2)  �  �  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (25)  Next we will show that  ���  ∂i  x∂j  yK(x,y)  ∂xi∂yj  ��� is upper-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Since we have K(x, y) =  � l  0 G(x, s)G∗(y, s)ds, we  will first analyze the property of G(x, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We decom-  pose G(x, s) as G1(x, s) + jG2(x, s), where G1, G2 ∈  C∞([0, l]2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The smoothness of G1, G2 in their domains  8  is trivial since they are compositions of polynomial func-  tions, trigonometric functions and square root functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Consider the integral kernel K(x, y) expressed in terms  of G1, G2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=',  K(x, y) =  � l  0  �  G1(x, s)G1(y, s) + G2(x, s)G2(y, s)  �  ds  + j  � l  0  �  G1(y, s)G2(x, s) − G1(x, s)G2(y, s)  �  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (26)  Since G1(x, s) and G2(y, s) are smooth in [0, l]2,  we can conclude that f1(x, y) = G1(x, s)G1(y, s) +  G2(x, s)G2(y, s) and f2(x, y) = G1(y, s)G2(x, s) −  G1(x, s)G2(y, s) are smooth in the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Since  compactly supported smooth functions attain their maxi-  mum or minimum values, the partial derivatives of K(·, ·)  are upper-bounded for any order i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=',  ����  ∂i+jK(x, y)  ∂xi∂yj  ���� < ∞,  ∀i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (27)  Therefore, by substituting (24) and (25) into Lemma  2, we can bound the difference between the mutual  information I  ′  0 and I1 by the following lemma:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The mutual information I1 converges to the  mutual information I  ′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The difference  ���I1 − I  ′  0  ��� is at most  inverse-proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Proof: From (6) we know that for the operator  TE, the kernel function can be expressed by K(x, y) =  � L  0 g(x, s)g∗(y, s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' From (25) we have  |Kn|2 ⩽ 4nn/2An,  (28)  where A = max  ��� ∂i+jK(x,y)  ∂xi∂yj  ���  L∞((0,l)2) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Therefore we have  |d(z) − dQ(z)| ⩽  ∞  �  n=1  zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  nln+2  24m2 max  i  ����  ∂2Kn  ∂x2  ����  L∞((0,l)n)  ⩽  ∞  �  n=1  zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  nln+2  6m2 nn/2An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (29)  According to the Stirling’s approximation, we have n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' ⩾  nne−n√  2πn, which leads to  |d(z) − dQ(z)| ⩽  l2  6m2  ∞  �  n=1  � n  2π  (Aezl)n  nn/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (30)  Since it is obvious that �∞  n=1  � n  2π  (Aezl)n  nn/2  is convergent,  the difference between d(z) and dQ(z) is proportional to  m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For the difference between mutual information I1  and I  ′  0, we have  |I1−I  ′  0| ⩽  |d(z) − dQ(z)|  min(d(z), dQ(z)) <  l2  6m2  ∞  �  n=1  � n  2π  (Aezl)n  nn/2  ,  (31)  where z =  m  ln1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' From Lemma 1 we know that  l  mn1 ⩾  n0 −  n0l3���K  ′′  E(r,r)  ���  L∞(0,l)  24m2|  � l  0 KE(r,r)dr| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, z is upperbounded,  which completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  According to Lemma 1 and Lemma 3, we have  Theorem 1, which bounds the difference between I0 and  I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The mutual information I1 that can be  obtained from the discrete receiver converges to the  mutual information I0 that can be obtained from the  continuous receiver when the number of points increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  The convergence rate is at least inverse-proportional to  the square of the sampling number m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Proof: Since the Fredholm determinant f(z) =  det(1 + zTE) is an analytic function, we have  |det(1 + zTE) − det(1 + z1TE)|  = |z − z1|  ����  ∂det(1 + xTE)  ∂x  ����  x∈[min(z,z1),max(z,z1)]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (32)  In our assumption z  =  2  n0  and z1  =  2m  ln1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The  analycity of det(1 + zTE) implies that  ∂det(1+xTE)  ∂x  is also an anlytic function and is bounded on the  interval  [min(z, z1), max(z, z1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  We  denote  M  =  max  x  ��� ∂det(1+xTE)  ∂x  ���, where x ∈ [min(z, z1), max(z, z1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  From Lemma 1 we have  ����  l  mn1 − n0  ���� ⩽  n0l3 ���K  ′′  E(r, r)  ���  L∞(0,l)  24m2 � l  0 KE(r, r)dr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (33)  Since n1/m → n0/l when m approximates infinity, we  denote the minimum value of n1/m by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore,  ���det(1 +  2  n0 TE) − det(1 + 2m  ln1 TE)  ��� can be bounded by  ����det(1 + 2  n0  TE) − det(1 + 2m  ln1  TE)  ����  ⩽  Ml2 ���K  ′′  E(r, r)  ���  L∞(0,l)  12m2c  � l  0 KE(r, r)dr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (34)  Similar to the direvation of (31), we know that when m  increases, I  ′  0 will converge to I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The convergence rate  is at least inverse-proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' From Lemma 3  we know that  ���I1 − I  ′  0  ��� is at most inverse-proportional to  m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Since |I0 − I1| ⩽  ���I0 − I  ′  0  ��� +  ���I1 − I  ′  0  ���, Theorem 1  is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Theorem 1 shows that the SNR control  scheme between the discrete and continuous models is  appropriate, since the limit of the mutual information of  the discrete model is proved to be that of the continu-  ous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' That is to say, our proposed model is self-  consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, we can use the proposed model  to compare the mutual information from the discrete  and continuous receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Our analysis is based on  9  0  50  100  150  0  20  40  60  80  100  120  140  160  180  continuous receiver  discrete receiver  5 10 15 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='5  3  continuous receiver  discrete receiver  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The mutual information as a function of the sampling number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  The transmitter is kept continuous and the receiver is discretized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  RE(r, r′) = P  � l  0 G(r, s)G∗(r′, s)ds which corresponds  to the scenario when no CSI can be obtained at the  transmitter but not limited to this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' It can be eas-  ily extended to other shapes of autocorrelation functions  after power allocation at the transmitter, as long as the  analyticity of RE(r, r′) is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Numerical analysis about the mutual information  As proven in the above subsection, the mutual infor-  mation between the continuous transmitter and discrete  receiver converges to the mutual information between  continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, the model of the dis-  crete receiver can be viewed as the discretization of the  continuous receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' In this subsection, we will use nu-  merical analysis to show the correctness of the theoretical  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Moreover, we will show the near-optimality of the  discrete receiver with half-wavelength sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  We set the length l of the transceivers to 2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The trans-  mitter is kept continuous, while the receiver is discretized  to m point antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The wavelength of the electromag-  netic field is fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='04 m, which correpsonds to the fre-  quency of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The distance between the transceivers  varies from 10 m to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The simulation results are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' From the simulation, we can observe  the convergence of the mutual information between the  continuous transmitter and the discrete receiver, which  verifies the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For the three distances  between transceivers, the half-wavelength sampling al-  most achieves the supremum mutual information between  continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, half-wavelength sam-  pling of the receiver is suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Moreover, when the  distance between transceivers decreases, we can observe  that the mutual information converges slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' When the  distance equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='1 m, the half wavelength sampling is at  the critical state of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' If the distance is less  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='1 m, a performance gap between the model with  the continuous receiver and that with the discrete receiver  may be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' This performance gap has theoretical  meaning but may not be useful because the distance will  be comparable to the wavelength in this scenario, where  the evanescent wave components will hold a dominant  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' COMPARISON BETWEEN CONTINUOUS AND  DISCRETE TRANSCEIVERS  In the above section we have compared the mutual  information between the models with continuous and  discrete receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For both models the transmitter is kept  continuous, which simplifies the analyzing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' In-  spired by the analysis in the above section, in this section  we will compare the mutual information between contin-  uous transceivers and that between discrete transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Numerical analysis is then provided to show the near-  optimality of the half-wavelength sampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Convergence analysis of the mutual information  The analysis in this section focuses on the difference  between I0 and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' It is an extension of the conver-  gence analysis in the above section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We define I  ′′  0  =  logdet  �  1 + m2TE  l2n2/2  �  as an intermediate variable similar  to I  ′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' First we will discuss the convergence of |I0 − I  ′′  0 |  in the following lemma:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The mutual information I  ′′  0 converges to the  mutual information I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The difference  ���I0 − I  ′′  0  ��� is at most  inverse-proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Proof: From the SNR control scheme of discrete  transceivers (17) and the multivariate m-point composite  midpoint quadrature rule, we have  ����n0 − l2  m2 n2  ���� =  n0  � l  0 K(r, r)dr  �����  � l  0  � l  0  g(r, r, z)dzdr  − l2  m2  m  �  i=1,j=1  g(ri, ri, rj)  �����  ⩽  n0l4  24m2 � l  0 K(r, r)dr  � ����  ∂2g(r, r, z)  ∂r2  ����  L∞((0,l)2)  +  ����  ∂2g(r, r, z)  ∂z2  ����  L∞((0,l)2)  �  ,  (35)  where g(x, y, z)  :=  G(x, z)G∗(y, z), ri  =  (i −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='5)l/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' It is obvious that n2/m2 converges to n0/l2  when m  →  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We denote the minimum value  of n2/m2 by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Then, according to (32), we know  that  ���det(1 +  2  n0 TE) − det(1 + 2m2  l2n2 TE)  ��� converges to 0  when m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Therefore, |I0 − I  ′′  0 | converges to 0, and  the convergence rate is at least inversely proportional to  m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  10  Then we will discuss the convergence of |I2 − I  ′′  0 | in  the following lemma:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The difference  ���I2 − I  ′′  0  ��� approaches 0 when  m approaches infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Moreover, it is at most inverse-  proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Proof: We denote the Fredholm determinant and  its discretization by d(z) = det(1 + zT) and dV (z) =  det (1i=j + wjz �m  k=1 wkG(ri, rk)G∗(rj, rk))m  i,j=1,  where K is the kernel of the operator T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' To bound  the difference between d(z) and dV (z), we define  gn(x1, · · · , xn, s1, · · · , sn) as  gn(x1, · · · , xn, s1, · · · , sn)  = det  �  �  g(x1, x1, s1)  · ·  g(x1, xn, s1)  · ·  g(xi, xj, si)  · ·  g(xn, x1, sn)  · ·  g(xn, xn, sn)  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (36)  From the definition of g(x, y, z), we know that  � l  0 g(xi, xj, si)dsi  =  K(xi, xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' According  to the  property  of  determinants  that  det(ai,j)m  i,j=1  =  �  k1,··· ,km(−1)ka1,k1 · · · am,km,  where  k1 · · · km  is  the kth exchange of 1 · · · n, we can find that  Kn(x1, · · · , xn) =  �  [0,l]n gn(x1, · · · , xn, s1, · · · , sn)  ds1 · · · dsn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (37)  If we define Cn  m(gn) by (38) we have (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Here  wαi correspondes to the distance between antennas in  the source region and s correpsonds to the location  of the antennas in the source region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' When further  considering  the  discretization  of  the  receiver  as  in (19), we should set xn  1  to the location of the  antennas in the destination region, and add additional  weights w which equals the distance between antennas  in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Similar to the definition  of Qn  m in (19), we define V n  m(gn) by V n  m(gn)  =  �m  j1,··· ,jn=1  �  i wjiCn  m(gn(rj1, · · · , rjn, s1,α1, · · · , sn,αn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  When sj,αi = rαi, we have  V n  m(gn) =  m  �  j1,··· ,jn=1  m  �  α1,··· ,αn=1  � n  �  i=1  ji  � � n  �  i=1  αi  �  gn(rj1, · · · , rjn, rα1, · · · , rαn)  =  m  �  j1,··· ,j2n=1  � 2n  �  i=1  ji  �  gn(rj1, · · · , rj2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (40)  The difference between d(z) and dV (z) is  d(z) − dV (z)  =  ∞  �  n=1  zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  �  V n  m(gn) −  �  [0,l]n Kn(x1, · · · , xn)dx1 · · · dxn  �  =  ∞  �  n=1  zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  �  V n  m(gn) −  �  [0,l]2n gn(x1, · · · , x2n)dx1 · · · dx2n  �  (41)  Note that V n  m(gn) is the numerical discretization of the  function gn with 2n variables, we can bound V n  m(gn) −  �  [a,b]2n gn(x1, · · · , x2n)dx1 · · · dx2n by using the multi-  variate numerical integration error bound:  �����V n  m(gn) −  �  [0,l]2n gn(x1, · · · , x2n)dx1 · · · dx2n  �����  ⩽ l2n−1  2n  �  i=1  Ei,  (42)  where  Ei =  �����Qi(gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' xi) −  � l  0  gn(x1, · · · , x2n)dxi  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (43)  According to the m-point composite midpoint quadrature  rule, we have  �����Qi(gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' xi) −  � l  0  gn(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' x2n)dxi  �����  ⩽  l3  24m2  ����  ∂2gn  ∂x2  i  ����  L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='l)2n)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (44)  From the Hadamard’s inequality [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' we can further  bound it by  �����V n  m(gn) −  �  [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='b]2n gn(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' x2n)dx1 · · · dx2n  �����  ⩽ 2nl2n+2  24m2 max  j  �����  ∂2gn  ∂x2  j  �����  L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='l)2n)  ⩽ 2nl2n+2  24m2 max  �  �nn/2  �����  ∂2g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' z)  ∂z2  ����  L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='l)3)  �n  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  4nn/2  �  � max  i+j⩽2  �����  ∂i  x∂j  yg(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' z)  ∂xi∂yj  �����  L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='l)3)  �  �  n �  �  ⩽ l2n+2  3m2 n(n+2)/2  �  � max  i+j+k⩽2  �����  ∂i  x∂j  y∂k  z g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' z)  ∂xi∂yj∂zk  �����  L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='l)3)  �  �  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (45)  Similar to Lemma 3, we know that |I2 −I  ′  0| converges  to 0, and the error is at most inverse proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Therefore, we have Theorem 2:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The mutual information I2 that can be  obtained from the discrete transceivers converges to the  mutual information I0 that can be obtained from the  continuous transceivers when the number of antennas in  the discrete transceivers increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The difference |I0−I2|  is at least inverse-proportional to the square of the  sampling number m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Remark  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Similar to Remark  3 in Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' III,  the convergence analysis in this section is not lim-  ited to the scenario with equal power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  11  Cn  m(gn) = det  �  �  �  α1 wα1g(x1, x1, s1,α1)  · ·  �  α1 wα1g(xn, xn, s1,α1)  · ·  �  αi wαig(xi, xj, si,αi)  · ·  �  αn wαng(xn, x1, sn,αn)  · ·  �  αn wαng(xn, xn, sn,αn),  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (38)  Cn  m(gn) =  �  k1···kn  (−1)k(  m  �  α1=1  wα1g(x1, xk1, s1,α1)) · · · (  m  �  αn=1  wαng(xn, xkn, sn,αn))  =  �  k1···kn  �  (−1)k  m  �  α1,··· ,αn=1  n  �  i=1  wαig(xi, xki, si,αi)  �  =  m  �  α1,··· ,αn=1  �� n  �  i=1  wαi  � �  k1···kn  (−1)k  n  �  i=1  g(xi, xki, si,αi)  �  =  m  �  α1,··· ,αn=1  �� n  �  i=1  wαi  �  gn(x1, · · · , xn, s1,α1, · · · , sn,αn)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  (39)  0  50  100  150  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='12  continuous transceiver  discrete receiver  discrete transceiver  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='1188  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='119  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='1192  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The mutual information variation with different sampling num-  bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The mutual information that correpsonds to the three models with  continuous and discrete transceivers is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The distance between  the transceivers is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  For arbitrary analytic function RJ(s, s′), the conver-  gence of |I0 − I2| can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Instead of dis-  cretizing  �  G(r, z)G∗(r′, z)dz to �  i G(r, ri)G(r′, ri),  we will discretize  ��  G(r, z)RJ(z, z′)G∗(r′, z′)dzdz′ to  �  i,j G(r, ri)RJ(ri, rj)G(r′, rj) in the extended sce-  narios  with  power  allocation  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Then,  in-  stead of g(x, y, z) we need a four-variable function  h(x, y, z, ω) := G(x, z)RJ(z, ω)G(y, ω) and the deriva-  tion procedure of the convergence has no essential differ-  ence with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Numerical analysis about the mutual information  In this subsection, we will verify the correctness of  the convergence analysis in the above subsection by  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The length l of the transceivers is fixed  to 2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We have plotted the mutual information of the  three models: continuous transceiver, continuous trans-  mitter and discrete receiver, and discrete transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The  transceivers are both discretized to m point antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  The wavelength of the electromagnetic field is fixed to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='04 m, which corresponds to the frequency of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  First we will show the scenarios when the distance be-  tween the transceivers is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The distance between the  transceivers varies from 50 m to 200 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The simulation  results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  From the simulation results we find that the mutual  information nearly keeps the same when the sampling  number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The reason for this phenomenon is that  the DoF of the channel is nearly inverse-proportional to  the distance between transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' For example, the DoF  when the distance equals 50 m can be approximated by  l2/(dλ) = 2, which means that when the sampling num-  ber is 5, the multiplexing gain is almost fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Therefore, for large distances between transceivers, the  dominant limitation is the channel DoF, which means that  the suboptimal performance can be achieved by sampling  sparser than half-wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Moreover, we have shown the variation of the mutual  information with the sampling number when the distance  between transceivers is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 5 the distance  between the transceiver is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='1 m and 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We can find that  when the distance decreases, the dense sampling of the  transceivers becomes important to fully explore the limit  of the mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' However, the half-wavelength  sampling of the transceivers still achieves suboptimal  performance, which means that denser sampling schemes  are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' CONCLUSION  In this paper, we proposed a comparison scheme be-  tween continuous and discrete MIMO systems which is  12  0  50  100  150  0  20  40  60  80  100  120  140  160  180  continuous transceiver  discrete receiver  discrete transceiver  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The mutual information as a function of the sampling number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  The mutual information values that correspond to the three models with  continuous and discrete transceivers are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The distance between  the transceivers is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  based on a precise non-asymptotic analysis framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Three information-theoretic models of the continuous  and discrete transceivers were built, with the first model  corresponds to the fully continuous electromagnetic infor-  mation theory model, and the third model corresponds to  the matrix-vector MIMO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' We proposed physically  consistent SNR control schemes to ensure the fairness of  the comparison, and proved that the mutual information  between discrete MIMO transceivers converges to that  of continuous electromagnetic transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' Numerical  results verified the theoretical analysis and showed the  near-optimality of the half-wavelength sampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content='  Further works can be done by extending the lin-  ear transceivers to rectangular or other two-dimensional  transceivers for generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
+page_content=' The analysis based on the  capacity after water-filling of the mutual information also  remains to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}
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+Quantum interference in the resonance fluorescence of a J = 1/2 − J′ = 1/2
+atomic system: Quantum beats, nonclassicality, and non-Gaussianity
+H. M. Castro-Beltr´an,1, ∗ O. de los Santos-S´anchez,2 L. Guti´errez,3 and A. D. Alcantar-Vidal1
+1Centro de Investigaci´on en Ingenier´ıa y Ciencias Aplicadas and Instituto de Investigaci´on en Ciencias B´asicas y Aplicadas,
+Universidad Aut´onoma del Estado de Morelos, Avenida Universidad 1001, 62209 Cuernavaca, Morelos, M´exico
+2Tecnologico de Monterrey, Escuela de Ingenier´ıa y Ciencias,
+Ave.
+Carlos Lazo 100, Santa Fe, Mexico City, Mexico, 01389
+3Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico,
+62210 Cuernavaca, Morelos, M´exico
+(Dated: January 10, 2023)
+We study the resonance fluorescence of a system with angular momentum J = 1/2−J′ = 1/2 level
+structure driven by a single, linearly polarized, monochromatic laser field. Quantum interference
+among the two, antiparallel, π transitions leads to rich results. We develop the article around two
+broad overlapping themes: (i) the observation of quantum beats in the intensity and the dipole-
+dipole, intensity-intensity, and quadrature-intensity correlations, when the atom is subject to a
+strong laser and large Zeeman splittings. The mean and modulation frequencies of the beats are
+given by the average and difference, respectively, among two close generalized Rabi frequencies
+related to a Mollow-like spectrum with two pairs of sidebands.
+(ii) The nonclassical and non-
+Gaussian properties of phase-dependent fluorescence for the cases of weak to moderate excitation
+and in the regime of beats. The fluorescence in the beats regime is nonclassical, mainly from the
+third-order dipole fluctuations, which reveal them to be also strongly non-Gaussian. For weak to
+moderate driving laser and small detunings and Zeeman splittings the nonclassicality is an interplay
+of second- (squeezing) and third-order dipole noise.
+I.
+INTRODUCTION
+Recently, the properties of the resonance fluorescence
+of a single atomic system with angular momentum transi-
+tion J = 1/2−J′ = 1/2 driven by a monochromatic laser
+have been the subject of great interest due to the possi-
+bility of observing vacuum-induced coherence effects due
+to interference among the two antiparallel π transitions,
+emitting into the same frequency range of the electro-
+magnetic vacuum. Here, the π transitions are incoher-
+ently coupled, mediated by spontaneous emision in the σ
+transitions and then excited by the laser. The antiparal-
+lel dipoles of the transitions makes it realistic to observe
+interference effects, while V and Λ three-level systems re-
+quire additional preparation because the transitions are
+perpendicular [1, 2]. Particular attention has been de-
+voted to the spectrum [3–6], time-energy complementar-
+ity [4, 5], Young’s interference [7], photon correlations
+[8], frequency-resolved photon correlations [9], squeezing
+[10], phase shifts [11], and cooperative effects in photon
+correlations [12]. The case of additional laser excitation
+of one of the σ transitions on the spectrum and squeezing
+has been studied in [13–15].
+Quantum beats are among the more familiar mani-
+festations of quantum interference. They appear in the
+modulation of the decay by spontaneous emission of mul-
+tilevel systems due to the energy difference among transi-
+tions [2]. So far, few experiments of quantum interference
+experiments have been performed on the J = 1/2 − J′ =
+∗ hcastro@uaem.mx
+1/2, in this case observing Young-type fringes [7]. Hence,
+further experiments are desirable. Quantum beats in the
+intensity are the result of the inability to tell the path
+of a particular photon when observed by a broadband
+detector. The beats can also occur in two-time correla-
+tions. As a general rule, initial conditions should be a
+superposition state.
+In this paper we investigate theoretically effects of
+quantum interference on the total intensity and two-time
+correlations such as dipole-dipole (to calculate spectra),
+intensity-intensity, intensity-amplitude correlations, and
+variance of the light emitted into the π transitions of the
+J = 1/2−J′ = 1/2 atomic system driven by a linearly po-
+larized laser and a magnetic field to break the degeneracy.
+While we put emphasis on the regime of observation of
+quantum beats, the nonclassical and non-Gaussian prop-
+erties of the fluorescence are also investigated.
+After describing the main features of the model in
+Section II, we discuss the basic dynamic and stationary
+properties of the atomic expectation values in Section
+III. Here, we analyze the previously overlooked time-
+dependent behavior of the atomic populations.
+Those
+of the excited states, for instance, although equal in the
+steady state, evolve with different Rabi frequencies and
+amplitudes. This is at the root of the formation of beats
+in the intensity and the correlations. In the regime of
+strong laser and magnetic fields these beats are character-
+ized by well-defined oscillations at the average frequency
+among two generalized Rabi frequencies, modulated at
+the difference of those frequencies.
+To observe beats
+in the intensity both ground state populations must be
+nonzero initially, ideally equal [1]. Similarly, for the two-
+time correlations, the vector of initial conditions must
+arXiv:2301.03061v1  [quant-ph]  8 Jan 2023
+
+2
+have at least two nonzero terms.
+In Section IV we describe the scattered field intensity
+and quadratures. Here, beats depend only on the inter-
+ference of the two upper populations in the nondegener-
+ate case, with both lower populations initially nonzero.
+Cross terms of the oppposite π transitions represent in-
+terference in the steady state intensity. Then, In Section
+V, using the dressed states approach, we show that the
+double sideband spectrum [5] stems from a dipole-dipole
+correlation with beats, where the terms of addition of sin-
+gle π transitions dominate over those of the cross terms.
+In Section VI we study Brown-Twiss photon-photon
+correlations [16, 17], extending the work of Ref.[8] to the
+nondegenerate case. Besides the ubiquitous antibunching
+effect, for weak to moderate laser drivings the interplay
+of parameters, together with detuning and Zeeman split-
+tings, can make for somewhat involved evolutions, e.g.,
+long decays due to optical pumping in the non-degenerate
+case. Again, cross terms are minor contributors to the
+full correlation in the beats regime.
+Section VII is devoted to a study of phase-dependent
+fluctuations by conditional homodyne detection (CHD)
+[18, 19] in both the temporal and spectral domains. The
+CHD method is characterized by amplitude-intensity cor-
+relations (AIC), which are of third order in the field am-
+plitude. When the atomic operators are decomposed into
+a mean plus a noise operator the AIC is split into a
+second-order term which would be a measure of squeezing
+if the third-order one were negligible. But the latter is
+not negligible outside the weak field regime of resonance
+fluorescence, which make the fluctuations non-Gaussian
+and also nonclassical by the violation of classical inequal-
+ities [20]. We obtain the spectra of the total, second- and
+third-order terms of the AIC. Narrow peaks in the spec-
+tra reveal population trapping when detunings favour the
+long term population or optical pumping of the ground
+state of the more detuned transition, which in the time
+domain show the above mentioned long decays.
+The
+third-order terms make up most of the beats and thus
+they are non-Gaussian and nonclassical but not squeezed.
+In Section VIII we consider squeezing by means of the
+variance of fluctuations. As usual, squeezing in resonance
+fluorescence is small and restricted to weak or moderate
+Rabi frequencies.
+Finally, in Section IX we provide a
+discussion and conclusions, and two Appendices give de-
+tails on solution methods, initial conditions, and optimal
+appearance of beats.
+II.
+MODEL
+The system, illustrated in Fig. 1, consists of a two-level
+atom with transition J = 1/2 – J = 1/2 and states with
+magnetic quantum number m = ±J,
+|1⟩ = |J, −1/2⟩,
+|2⟩ = |J, 1/2⟩,
+|3⟩ = |J, −1/2⟩,
+|4⟩ = |J, 1/2⟩.
+(1)
+The matrix elements are
+FIG. 1.
+Scheme of the J = 1/2 – J = 1/2 atomic system
+interacting with a laser driving the |1⟩ − |3⟩ and |2⟩ − |4⟩
+transitions with Rabi frequency Ω and detuning ∆.
+There
+are spontaneous decay rates γ1, γ2 and γσ, vacuum-induced
+coherence γ12, and Zeeman frequency splittings Bℓ and Bu.
+d1 = ⟨1|ˆd|3⟩ = − 1
+√
+3Dez,
+d2 = ⟨2|ˆd|4⟩ = −d1,
+d3 = ⟨2|ˆd|3⟩ =
+�
+2
+3De−,
+d4 = ⟨1|ˆd|4⟩ = d∗
+3,
+(2)
+where D is the reduced dipole matrix element. We choose
+the field polarization basis {ez, e−, e+} (linear, left cir-
+cular, right circular), where e± = ∓(ex ± iey)/2.
+The π transitions, |1⟩ − |3⟩ and |2⟩ − |4⟩ (m = m′), are
+coupled to linearly polarized light and have their dipole
+moments antiparallel. On the other hand, the σ tran-
+sitions, |1⟩ − |4⟩ and |2⟩ − |3⟩ (m ̸= m′), are coupled
+to circularly polarized light. This configuration can be
+found, for example, in 198Hg+ [3], and 40Ca+ [12].
+The level degeneracy is removed by the application of
+a static magnetic field Bz along the z direction, the Zee-
+man effect.
+Note that the energy splittings gµBBz of
+the upper (u) and lower (ℓ) levels are different due to
+unequal Land´e g factors, gu and gℓ, respectively; µB is
+Bohr’s magneton. The difference Zeeman splitting is
+δ = (gu − gℓ)µBBz
+¯h
+= gu − gℓ
+gℓ
+Bℓ,
+(3)
+where Bℓ = glµBBz/¯h. For 198Hg+ gu = 2/3 and gℓ = 2,
+so ¯hδ = −(4/3)µBBz = −(2/3)¯hBℓ.
+The atom is driven by a monochromatic laser of fre-
+quency ωL, linearly polarized in the z direction, propa-
+gating in the x direction,
+EL(x, t) = E0ei(ωLt−kLx)ez + c.c.,
+(4)
+thus driving only the π transitions.
+The free atomic, H0, and interaction, V , parts of the
+Hamiltonian are, respectively:
+H0 = ¯hω13A11 + ¯h(ω24 + Bℓ)A22 + ¯hBℓA44,
+(5)
+V = ¯hΩ(A13 − A24)eiωLt + h.c.
+(6)
+
+3
+where Ajk = |j⟩⟨k| are atomic operators, ω13 and ω24 =
+ω13 + δ are the frequencies of the |1⟩ − |3⟩ and |2⟩ − |4⟩
+transitions, respectively, and Ω = E0D/
+√
+3 ¯h is the Rabi
+frequency. The frequencies of the other transitions are
+ω23 = ω13 − δ and ω14 = ω13 − Bℓ. Using the unitary
+transformation
+U = exp [(A11 + A22)iωLt],
+(7)
+the Hamiltonian in the frame rotating at the laser fre-
+quency is
+H = U †(H0 + V )U,
+= −¯h∆A11 − ¯h(∆ − δ)A22 + ¯hBℓ(A22 + A44)
++¯hΩ [(A13 − A24) + h.c.] ,
+(8)
+where ∆ = ωL −ω13 is the detuning of the laser from the
+|1⟩ − |3⟩ resonance transition, and ∆ − δ is the detuning
+on the |2⟩ − |4⟩ transition.
+The excited states decay either in the π transitions
+emitting photons with linear polarization at rates γ1 =
+γ2, or in the σ transitions emitting photons of circular
+polarization at rate γσ. There is also a cross-coupling
+of the excited states by the reservoir, responsible for the
+quantum interference we wish to study. In general, the
+decay rates are written as
+γij = di · d∗
+j
+|di||dj|
+√γiγj,
+i, j = 1, 2.
+(9)
+In particular, we have γii = γ1 = γ2 and γ13 = γ24 = γσ.
+Also, given that d1 and d2 are antiparallel, γ12 = γ21 =
+−√γ1γ2 = −γ1.
+The total decay rate is
+γ = γ1 + γσ = γ2 + γσ.
+(10)
+The decays for the π and σ transitions occur with the
+branching fractions bπ and bσ [5], respectively,
+γ1 = γ2 = bπγ,
+bπ = 1/3,
+(11a)
+γσ = bσγ,
+bσ = 2/3.
+(11b)
+III.
+MASTER EQUATION
+The dynamics of the atom-laser-reservoir system is de-
+scribed by the master equation for the reduced atomic
+density operator, ρ. In a frame rotating at the laser fre-
+quency (˜ρ = UρU †) it is given by
+˙˜ρ = − i
+¯h[H, ˜ρ] + Lγ ˜ρ,
+(12)
+where −(i/¯h)[H, ˜ρ] describes the coherent atom-laser in-
+teraction and Lγ ˜ρ describes the damping due to sponta-
+neous emission [5, 21]. Defining
+S−
+1 = A31,
+S−
+2 = A42,
+S−
+3 = A32,
+S−
+4 = A41,
+S+
+i = (S−
+i )†,
+(13)
+the dissipative part is written as
+Lγ ˜ρ = 1
+2
+2
+�
+i,j=1
+γij
+�
+2S−
+i ˜ρS+
+j − S+
+i S−
+j ˜ρ − ˜ρS+
+i S−
+j
+�
++γσ
+2
+4
+�
+i=3
+�
+2S−
+i ˜ρS+
+i − S+
+i S−
+i ˜ρ − ˜ρS+
+i S−
+i
+�
+. (14)
+We now define the Bloch vector of the system as
+Q ≡ (A11, A12, A13, A14, A21, A22, A23, A24,
+A31, A32, A33, A34, A41, A42, A43, A44)T . (15)
+The equations for the expectation values of the atomic
+operators, ⟨Ajk⟩ = ˜ρkj, are the so-called Bloch equations,
+which we write as
+d
+dt⟨Q(t)⟩ = MB⟨Q(t)⟩,
+(16)
+where MB is a matrix of coeficients of the full master
+equation, and the formal solution is
+⟨Q(t)⟩ = eMBt⟨Q(0)⟩.
+(17)
+Since we are interested only in properties of the fluores-
+cence emitted in the π transitions we use the simplifying
+fact, already noticed in [8], that these Bloch equations
+can be split into two decoupled homogeneous sets. Set 1
+contains the equations for the populations and the coher-
+ences of the coherently driven π transitions; these are
+⟨ ˙A11⟩ = −γ⟨A11⟩ + iΩ(⟨A31⟩ − ⟨A13⟩),
+⟨ ˙A13⟩ = −
+�γ
+2 + i∆
+�
+⟨A13⟩ − iΩ(⟨A11⟩ − ⟨A33⟩),
+⟨ ˙A22⟩ = −γ⟨A22⟩ − iΩ(⟨A42⟩ − ⟨A24⟩),
+⟨ ˙A24⟩ = −
+�γ
+2 + i(∆ − δ)
+�
+⟨A24⟩ + iΩ(⟨A22⟩ − ⟨A44⟩),
+⟨ ˙A31⟩ = −
+�γ
+2 − i∆
+�
+⟨A31⟩ + iΩ(⟨A11⟩ − ⟨A33⟩),
+⟨ ˙A33⟩ = γ1⟨A11⟩ + γσ⟨A22⟩ − iΩ(⟨A31⟩ − ⟨A13⟩),
+⟨ ˙A42⟩ = −
+�γ
+2 − i(∆ − δ)
+�
+⟨A42⟩ − iΩ(⟨A22⟩ − ⟨A44⟩),
+⟨ ˙A44⟩ = γσ⟨A11⟩ + γ2⟨A22⟩ + iΩ(⟨A42⟩ − ⟨A24⟩).
+(18)
+with Bloch vector
+R ≡ (A11, A13, A22, A24, A31, A33, A42, A44)T
+(19)
+and a corresponding matrix M, Eq. (A3). Equations (18)
+do not depend on γ12, the vacuum-induced coupling of
+the upper levels, but on the applied magnetic field only
+through the difference of Zeeman splittings, δ.
+The steady state solutions, for which we introduce the
+
+4
+0
+2
+4
+6
+8
+10
+12
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+4
+6
+8
+10
+12
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+4
+6
+8
+10
+12
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+4
+6
+8
+10
+12
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+γt
+γt
+(d) ∆ = −2γ, δ = −4γ
+(b) ∆ = 2γ, δ = −2γ
+(
) ∆ = −2γ, δ = −2γ
+⟨A22 (t)⟩
+⟨A44 (t)⟩
+⟨A11 (t)⟩
+⟨A33 (t)⟩
+(a) ∆ = 0, δ = 0
+FIG. 2.
+Time-dependent populations ⟨A11(t)⟩ (solid-black),
+⟨A22(t)⟩ (dashed-red), ⟨A33(t)⟩ (dots-green), and ⟨A44(t)⟩
+(dashed-dots-blue), with the atom initially in state |3⟩. The
+parameters are: Ω = γ and (a) ∆ = δ = 0; (b) ∆ = 2γ,
+δ = −2γ; (c) ∆ = δ = −2γ; (d) ∆ = −2γ, δ = −4γ.
+short notation αjk = ⟨Ajk⟩st, are
+α11 = α22 = Ω2
+2D,
+(20a)
+α33 = Ω2 + γ2/4 + ∆2
+2D
+,
+(20b)
+α44 = Ω2 + γ2/4 + (∆ − δ)2
+2D
+,
+(20c)
+α13 = Ω(∆ + iγ/2)
+2D
+,
+(20d)
+α24 = Ω(δ − ∆ − iγ/2)
+2D
+,
+(20e)
+αkj = α∗
+jk.
+where
+D = 2Ω2 + γ2 + δ2
+4
++
+�
+∆ − δ
+2
+�2
+.
+(21)
+Note also that in the degenerate system (δ = 0) α33 =
+α44 and that α31 = −α42, where the minus sign arises
+from the fact that the dipole moments d1 and d2 are
+antiparallel.
+Set 2 contains the equations for the coherences of the σ
+transitions and those among both upper and both lower
+levels,
+R2 ≡ (A12, A14, A21, A23, A32, A34, A41, A43)T . (22)
+The equations for their expected values do depend on Bℓ
+and γ12. These coherences vanish because the σ tran-
+sitions are driven incoherently (⟨{A14, A23, A32, A41}⟩),
+i.e., by spontaneous emission, or because they are medi-
+ated by those σ transitions (⟨{A12, A21, A34, A43}⟩). For
+completeness, we write the steady state results:
+α12 = α34 = α14 = α23 = 0,
+αkj = α∗
+jk.
+(23)
+0
+1
+2
+3
+4
+5
+6
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+1
+2
+3
+4
+5
+6
+0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+6
+0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+6
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+α11 = α22
+α33
+α44
+(d) ∆ = −2γ, δ = −4γ
+(b) ∆ = 2γ, δ = −2γ
+(a) ∆ = 0, δ = 0
+(
) ∆ = −2γ, δ = −2γ
+Ω/γ
+Ω/γ
+FIG. 3.
+Steady-state populations as a function of Rabi
+frequency: α11 = α22 (dashed-red), α33 (solid-black), and
+α44 (dots-blue). All other parameters as in Fig. 2.
+The properties of the fluorescence of the π transitions,
+the subject matter of this article, do not depend on the
+equations for Set 2. Only the second- and third-order
+amplitude-intensity correlations and the dipole correla-
+tion for the spectrum of the σ transitions would require
+the full set of Bloch equations.
+We gain valuable information on the nontrivial dynam-
+ics of the atomic system from single-time expectation val-
+ues, apparently ignored in the previous literature on the
+system. In Fig. 2 we show the populations for several
+particular cases, all with the atom initially in state |3⟩.
+In the degenerate case, δ = 0, the upper populations
+reach opposite phases by the end of the first Rabi cycle,
+Fig. 2(a). This is understandable since the electron oc-
+cupation of, say, state |1⟩ implies not to be in state |2⟩,
+and viceversa. The same occurs for the lower popula-
+tions. Next, we show three situations for the nondegen-
+erate case with δ < 0 (as it is for 198Hg+). In Fig. 2(b)
+the laser is slightly detuned above the |1⟩−|3⟩ transition,
+but highly detuned from the |2⟩−|4⟩ transition; the oscil-
+lations get out of phase and most of the population ends
+up in state |4⟩ by optical pumping. In Fig. 2(c) the laser
+is detuned below the |1⟩−|3⟩ transition, and the |2⟩−|4⟩
+transition is now on resonance with the laser; again, the
+oscillations are out of phase but most of the population
+ends up now in state |3⟩. In Fig. 2(d) we extend the pre-
+vious case but with stronger applied magnetic field, thus
+the non-degeneracy is more evident; the large detuning
+on both transitions makes it recover the opposite phases
+of the degenerate case.
+In Fig. 3 we show the steady state populations as a
+function of the Rabi frequency; the other parameters are
+the same as in Fig. 2. For strong fields the populations
+tend to be equal (1/4), but arrive at that limit at dif-
+ferent rates; for instance, for large detunings on both
+transitions, Fig. 3(d), it takes larger fields, as compared
+to the degenerate case, Fig. 3(a). On the other hand,
+for small detunings and weak-moderate fields, when one
+
+5
+transition is closer to resonance than the other, the lower
+state of the more detuned transition is more populated,
+as seen in Figs. 3 (b) and (c).
+IV.
+THE SCATTERED FIELD
+In this Section we present the main dynamical and
+stationary properties of the field scattered by the atom,
+with emphasis on the π transitions.
+A.
+Single-Time and Stationary Properties
+The positive-frequency part of the emitted field oper-
+ator is [5, 21]
+ˆE+(r, t) = ˆE+
+free(r, t) + ˆE+
+S (r, ˆt),
+(24)
+where ˆE+
+free(r, t) is the free-field part, which does not con-
+tribute to normally ordered correlations, hence we omit
+it in further calculations, and
+ˆE+
+S (r, t) = −η
+r
+4
+�
+i=1
+ω2
+i ˆr × (ˆr × di)S−
+i (ˆt)
+(25)
+is the dipole source field operator in the far-field zone,
+where ˆt = t−r/c is the retarded time and η = (4πϵ0c2)−1.
+Since ωi ≫ δ, we may approximate the four transition as
+a single one ω0 in Eq. (25, but cannot do so at the level
+of decay rates, Rabi frequencies, and splittings.
+Making ˆr = ey the direction of observation and using
+Eq. (2) we have
+ˆE+
+S (r, ˆt) = ˆE+
+π (r, ˆt) ez + ˆE+
+σ (r, ˆt) ex,
+(26)
+i.e., the fields scattered from the π and σ transitions are
+polarized in the ez and ex directions, respectively, where
+ˆE+
+π (r, ˆt) = fπ(r)
+�
+A31(ˆt) − A42(ˆt)
+�
+,
+(27a)
+ˆE+
+σ (r, ˆt) = fσ(r)
+�
+A32(ˆt) − A41(ˆt)
+�
+,
+(27b)
+are the positive-frequency source field operators of the π
+and σ transitions, and
+fπ(r) = −ηω2
+1D/
+√
+3r,
+fσ(r) =
+√
+2fπ(r),
+(28)
+are their geometric factors.
+The intensity in the π transitions is given by
+Iπ(r, ˆt) = ⟨ ˆE−
+π (r, ˆt) · ˆE+
+π (r, ˆt)⟩
+= f 2
+π(r)⟨A13(ˆt)A31(ˆt) + A24(ˆt)A42(ˆt)⟩
+= f 2
+π(r)⟨A11(ˆt) + A22(ˆt)⟩,
+(29a)
+while in the steady state is
+Ist
+π = f 2
+π(r) [α11 + α22] = Ω2
+D .
+(29b)
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0.20
+0
+2
+4
+6
+8
+10
+0
+0.1
+0.2
+0.3
+0
+2
+4
+6
+8
+10
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+2
+4
+6
+8
+10
+0
+0.1
+0.2
+0.3
+0.4
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0.20
+0.25
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0.20
+0.25
+Iπ (r, t) �f 2
+π (r)
+Iπ (r, t) �f 2
+π (r)
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+γt
+γt
+γt
+γt
+γt
+γt
+(d)
+(c)
+(a)
+(b)
+FIG. 4.
+Fluorescence intensity of the π transitions with equal
+initial ground state populations, ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2.
+The other parameters are as in Fig. 2: Ω = γ and (a) ∆ =
+δ = 0; (b) ∆ = 2γ, δ = −2γ; (c) ∆ = δ = −2γ; (d) ∆ = −2γ,
+δ = −4γ.
+The insets show the excited states populations:
+⟨A11(t)⟩ (solid), ⟨A22(t)⟩ (dashed).
+Adding the excited state populations with the atom
+initially in the single state |3⟩ in Eq. 29a gives simply
+Iπ(r, ˆt) = f 2
+π(r)⟨A11(ˆt)⟩, i.e., without the contribution
+of ⟨A22(ˆt)⟩. More interesting is the case where the ini-
+tial condition is ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2, shown in
+Fig. 4 (see the populations ⟨A11(t)⟩ and ⟨A22(t)⟩ in the
+insets). The modulation in the intensity is reminiscent
+of the quantum beats in the spontaneous decay in the V
+three-level system [1, 2]. These beats are basically due
+to the inability to tell from which of the π transitions
+a photon comes from. This is the standard Young-type
+interference [4, 5, 7]. The main requirement is that the
+initial condition for both ground states are nonzero (see
+Appendix B.
+More interesting, though, is the case of strong resonant
+laser and magnetic fields and the laser is detuned far from
+the |2⟩ − |4⟩ resonance frequency, shown in Fig. 5. Due
+to the laser detuning, the population ⟨A22(t)⟩ has larger
+frequency and smaller amplitude than that of ⟨A11(t)⟩, as
+seen in the insets. Remarkably well-defined wave-packets
+or beats are observed due to the interference of the flu-
+orescence of both π transitions with close Rabi frequen-
+cies, with clear average and modulation frequencies (see
+Fig. 5a). The beats get scrambled with larger frequency
+and amplitude differences, Fig. 5b.
+Save for the decay, these beats are more like the classic
+textbook ones, described by a modulation and an average
+frequency, unlike the beats from spontaneous emission or
+weak resonance fluorescence from two or more closely sep-
+arated levels. Henceforth, we reserve the moniker beats
+to those due to strong applied fields. Further analyses of
+the beats are given in the next Sections, as they show up
+
+6
+0
+2
+4
+6
+8
+10
+0
+0.2
+0.4
+0.6
+0
+2
+4
+6
+8
+10
+0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+0
+0.1
+0.2
+0.3
+0.4
+0
+1
+2
+3
+4
+5
+0
+0.1
+0.2
+0.3
+0.4
+Iπ (r, t)
+�
+f2
+π (r)
+Iπ (r, t)
+�
+f2
+π (r)
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+⟨A22 (t)⟩
+⟨A11 (t)⟩
+γt
+γt
+γt
+γt
+(b)
+(a)
+FIG. 5.
+Fluorescence intensity for Ω = 9γ, ∆ = 0, and (a)
+δ = −8γ and (b) δ = −15γ. The insets show the excited state
+populations ⟨A11⟩ (solid line) and ⟨A22⟩ (dotted line). The
+initial conditions are ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2, ⟨A11(0)⟩ =
+⟨A22(0)⟩ = 0.
+also in two-time correlations with particular features.
+Similarly, for the σ transitions we have
+Iσ(r, ˆt) = ⟨ ˆE−
+σ (r, ˆt) · ˆE+
+σ (r, ˆt)⟩
+= f 2
+σ(r)[⟨A23(ˆt)A32(ˆt) + A14(ˆt)A41(ˆt)⟩]
+= f 2
+σ(r)[⟨A11(ˆt) + A22(ˆt)⟩],
+(30a)
+Ist
+σ = f 2
+σ(r) [α11 + α22] ,
+(30b)
+also showing beats with intensity twice that of the π tran-
+sitions given that f 2
+σ(r) = 2f 2
+π(r).
+The field quadrature operator at any time is
+ˆEπ,φ(r, ˆt) = 1
+2
+�
+E−
+π (r, ˆt)e−iφ + E+
+π (r, ˆt)eiφ�
+= fπ(r)(S1,φ − S2,φ),
+(31)
+where φ = 0, π/2 are the quadrature phases we consider,
+and
+S1,φ = 1
+2
+�
+A13e−iφ + A31eiφ�
+,
+(32a)
+S2,φ = 1
+2
+�
+A24e−iφ + A42eiφ�
+.
+(32b)
+The mean quadrature field is given by
+⟨ ˆEπ,φ⟩st = fπ(r)
+2
+�
+(α13 − α24) e−iφ + (α31 − α42) eiφ�
+= fπ(r)Re
+�
+(α13 − α24) e−iφ�
+(33)
+= fπ(r)Re
+�Ω (∆ + (iγ − δ)/2)
+D
+e−iφ
+�
+,
+B.
+Intensity and Quadrature Fluctuations
+Here we introduce the intensity and quadratures of the
+field in terms of atomic fluctuation operators ∆Ajk =
+Ajk − ⟨Ajk⟩st, such that
+⟨AklAmn⟩ = αklαmn + ⟨∆Akl∆Amn⟩.
+(34)
+Only the π transitions have nonvanishing coherence
+terms (α13, α24 ̸= 0). The fluorescence in the σ transi-
+tions is fully incoherent (α14 = α23 = 0), so its intensity
+is given by Eq. (30b). In the remainder of this section
+we deal only with the π transition. The quadrature op-
+erators are then written as
+ˆEπ,φ(r, ˆt) = fπ(r)[απ,φ + ∆Sπ,φ(ˆt)],
+(35a)
+where
+απ,φ = 1
+2(α31 − α42)eiφ + 1
+2(α13 − α24)e−iφ,
+(35b)
+= Re
+�Ω (∆ + (iγ − δ)/2)
+D
+e−iφ
+�
+,
+∆Sπ,φ = 1
+2(∆A31 − ∆A42)eiφ + 1
+2(∆A13 − ∆24)e−iφ.
+(35c)
+From Eqs. (29b) and (34) we write the steady state
+intensity in terms of products of dipole and dipole fluc-
+tuation operator expectation values,
+Ist
+π (r) = f 2
+π(r)
+�
+Icoh
+π,0 + Iinc
+π,0 + Icoh
+π,cross + Iinc
+π,cross
+�
+,(36)
+where
+Icoh
+π,0 = |⟨A13⟩st|2 + |⟨A24⟩st|2,
+(37a)
+Iinc
+π,0 = ⟨∆A13∆A31⟩ + ⟨∆A24∆A42⟩,
+(37b)
+Icoh
+π,cross = −⟨A13⟩st⟨A42⟩st − ⟨A24⟩st⟨A31⟩st
+= −2Re (⟨A13⟩st⟨A42⟩st) ,
+(37c)
+Iinc
+π,cross = −⟨∆A13∆A42⟩ − ⟨∆A24∆A31⟩
+= −2Re (⟨∆A13∆A42⟩) .
+(37d)
+Superindices coh and inc stand, respectively, for the co-
+herent (depending on mean dipoles) and incoherent (de-
+pending on noise terms) parts of the emission. Subindex
+0 stands for terms with the addition of single transition
+products, giving the total intensity, while subindex cross
+stands for terms with products of the two π transitions,
+the steady state interference part of the intensity.
+In
+
+7
+terms of atomic expectation values these intensities are:
+Icoh
+π,0 = |α13|2 + |α24|2
+(38a)
+= Ω2
+4D2
+�γ2
+2 + ∆2 + (δ − ∆)2
+�
+,
+Iinc
+π,0 = α11 + α22 − |α13|2 − |α24|2
+(38b)
+= Ω2
+D2
+�
+2Ω2 − γ2
+4 − ∆2 − δ2
+�
+,
+Icoh
+π,cross = −2Re (α13α42)
+(38c)
+= Ω2
+2D2
+�γ2
+4 + ∆(∆ − δ)
+�
+,
+Iinc
+π,cross = 2Re (α13α42) = −Icoh
+π,cross,
+(38d)
+The sum of these terms is, of course, the total intensity,
+Eq. (29a). As usual in resonance fluorescence, the coher-
+ent and incoherent intensities are similar only in the weak
+field regime, Ω ≤ γ. Here, in particular, the term Iinc
+π,0
+(no interference) becomes much larger than the others
+for strong driving.
+C.
+Degree of Interference - Coherent Part
+In Ref. [5], a measure of the effect of interference in
+the coherent part of the intensity was as
+Icoh
+π,0 + Icoh
+π,cross = Icoh
+π,0 (1 + C(δ)),
+C(δ) = Icoh
+π,cross
+Icoh
+π,0
+=
+γ2/4 + ∆(∆ − δ)
+γ2/4 + δ2/4 + (∆ − δ/2)2 , (39)
+independent
+of
+the
+Rabi
+frequency
+and
+shown
+in
+Fig. 6(a).
+Some special cases are found analytically:
+C(0) = 1,
+δ = 0,
+(40a)
+C(δ0) = 0,
+δ0 = ∆[1 + (γ/2∆)2],
+(40b)
+C(δmin) =
+−1
+1 + γ2/2∆2 ,
+δmin = 2∆[1 + (γ/2∆)2],
+(40c)
+C(δ±
+1/2) = 1/2,
+δ±
+1/2 = −∆ ±
+�
+3∆2 + (γ2/2).
+(40d)
+In the degenerate case, C(δ = 0) = 1 means perfect
+constructive interference. That is because at δ = 0 both
+π transitions (and both σ transitions) share the same
+reservoir environment. Increasing δ the reservoir overlap
+decreases, so is the interference. Negative values of C
+indicate destructive interference; its minimum is given
+by δmin. For large detunings, ∆2 ≫ γ2 we have
+δ0 = ∆,
+δmin = 2∆,
+δ±
+1/2 = −∆ ±
+√
+3 |∆|.
+(40e)
+We have used the special cases δ = {0, δ0, δmin} as a
+guide to obtain many of the figures in this paper.
+-1.0
+-0.5
+0
+0.5
+1.0
+-20
+-10
+0
+10
+20
+0
+0.5
+1.0
+δ/γ
+∆ = −5γ
+K(δ)
+C(δ)
+∆ = −2γ
+∆ = 0
+(a)
+(b)
+FIG. 6.
+Relative weight of the interference terms C(δ) (a)
+and K(δ) (b). In (b) Ω = γ/4. For 198Hg+, δ ≤ 0.
+D.
+Degree of Interference - Incoherent Part
+Likewise, we define a measure, K(δ), of the effect of
+interference in the intensity’s incoherent part,
+Iinc
+π,0 + Iinc
+π,cross = Iinc
+π,0(1 + K(δ)),
+K(δ) = Iinc
+π,cross
+Iinc
+π,0
+=
+γ2/4 + ∆(∆ − δ)
+2 [γ2/4 + δ2 + ∆2 − 2Ω2]. (41)
+Unlike C(δ), K(δ) also depends on the Rabi frequency
+as Ω−2, since fluctuations increase with laser intensity.
+Special cases are:
+K(0) =
+γ2/4 + ∆2
+2 [γ2/4 + ∆2 − 2Ω2],
+δ = 0,
+(42a)
+K(δ) = 0,
+δ = ∆ + γ2
+4∆
+or
+Ω ≫ γ, ∆, δ.
+(42b)
+The behavior of K(δ) with ∆ is more subtle. It is ba-
+sically required that ∆ ∼ Ω in order to preserve the
+shape seen in Fig. 6(b), in which case the minima for
+C(δ) and K(δ) are very similar. On-resonance, for ex-
+ample, Ω should be no larger than 0.35γ. Also, we can
+infer that the beats are little affected by the interference
+term unless ∆ >∼ Ω ≫ γ.
+V.
+TWO-TIME DIPOLE CORRELATIONS AND
+POWER SPECTRUM
+The resonance fluorescence spectrum of the J = 1/2 →
+J = 1/2 atomic system was first considered in [3] and
+then very thoroughly in [4, 5]. Thus, here we only con-
+sider basic definitions and issues related to the observa-
+tion of beats.
+The stationary Wiener-Khintchine power spectrum is
+given by the Fourier transform of the field autocorrelation
+function
+Sπ(ω) = Re
+� ∞
+0
+dτe−iωτ⟨ ˆE−
+π (0) ˆE+
+π (τ)⟩,
+(43)
+
+8
+0
+2
+4
+6
+8
+10
+0
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+-20
+-10
+0
+10
+20
+ω/γ
+Sinc
+π (ω) (arb. units)
+γτ
+�
+∆E−
+π (0) ∆E+
+π (τ)
+� �
+f2
+π (r)
+FIG. 7.
+Dipole correlation function ⟨∆ ˆE−
+π (0)∆ ˆE+
+π (τ)⟩ for
+Ω = 9γ, δ = −8γ, and ∆ = 0. The inset shows the corre-
+sponding incoherent spectrum Sinc
+π
+(ω).
+such that
+� ∞
+−∞ Sπ(ω)dω = Ist
+π . By writing the atomic
+operators in Eq. (27a) as Ajk(t) = αjk + ∆Ajk(t), we
+separate the spectrum in two parts: a coherent one,
+Scoh
+π
+(ω) = Re
+� ∞
+0
+e−iωτdτ
+�
+Icoh
+π,0 + Icoh
+π,cross
+�
+= π
+�
+Icoh
+π,0 + Icoh
+π,cross
+�
+δ(ω)
+= πΩ2
+D2
+�
+γ2
+4 +
+�
+∆ − δ
+2
+�2�
+δ(ω),
+(44)
+due to elastic scattering, where Icoh
+π,0 and Icoh
+π,cross are given
+by Eqs. (38) (a) and (c), respectively; and an incoherent
+part,
+Sinc
+π (ω) = Re
+� ∞
+0
+dτe−iωτ⟨∆ ˆE−
+π (0)∆ ˆE+
+π (τ)⟩,
+specifically,
+Sinc
+π (ω) = Re
+� ∞
+0
+dτe−iωτ [⟨∆A13(0)∆A31(τ)⟩
++⟨∆A24(0)∆A42(τ)⟩ − ⟨∆A13(0)∆A42(τ)⟩
+−⟨∆A24(0)∆A31(τ)⟩] ,
+(45)
+due to atomic fluctuations. An outline of the numerical
+calculation is given in Appendix A.
+The dipole correlation ⟨ ˆE−
+π (0) ˆE+
+π (τ)⟩ and the incoher-
+ent spectrum in the strong driving regime and strong
+nondegeneracy (large δ) are shown in Fig. 7. The spec-
+trum (inset) displays a central peak and two pairs of
+Mollow-like-sidebands [22] with peaks at the Rabi side-
+bands ±Ω1 and ±Ω2, while the correlation features de-
+caying quantum beats due to the closeness of the Rabi
+peaks.
+As usual in the strong-field regime, the dressed system
+approach allows to discern the origin of the peaks from
+the transitions among the dressed states, to find their
+positions [5], and thus find the frequencies of the beats.
+The generalized Rabi frequencies are
+Ω1 = E+
+1 − E−
+1 =
+�
+4Ω2 + ∆2,
+(46a)
+Ω2 = E+
+2 − E−
+2 =
+�
+4Ω2 + (δ − ∆)2,
+(46b)
+TABLE I. Eigenvalues of matrix M/γ and initial conditions
+of the correlations in Eq. (45) for Ω = 9γ and ∆ = 0.
+Eigenvalues
+δ = −8γ
+δ = −15γ
+λ1
+−0.749386 + 0i
+−0.836531 + 0i
+λ2
+−0.583099 − 18.0094i
+−0.583308 − 17.9981i
+λ3
+−0.583099 + 18.0094i
+−0.583308 + 17.9981i
+λ4
+−0.569785 − 19.6808i
+−0.5492 − 23.4257i
+λ5
+−0.569785 + 19.6808i
+−0.5492 + 23.4257i
+λ6
+−0.5 + 0i
+−0.5 + 0i
+λ7
+−0.444846 + 0i
+−0.398452 + 0i
+λ8
+0 + 0i
+0 + 0i
+Init. cond.
+⟨∆A13∆A31⟩
+0.20836 + 0i
+0.14734 + 0i
+⟨∆A24∆A42⟩
+0.174014 + 0i
+0.086982 + 0i
+⟨∆A13∆A42⟩
+0.000134 + 0.002146i
+0.000067 + 0.002011i
+⟨∆A24∆A31⟩
+0.000134 − 0.002146i
+0.000067 − 0.002011i
+where
+E±
+1 = −∆
+2 ± 1
+2
+�
+4Ω2 + ∆2,
+(47a)
+E±
+2 = Bℓ + δ − ∆
+2
+± 1
+2
+�
+4Ω2 + (δ − ∆)2,
+(47b)
+are the eigenvalues of the Hamiltonian (8). Due to the
+spontaneous decays these frequencies would have to be
+corrected, but they are very good in the relevant strong
+field limit. Indeed, we notice that Ω1 and Ω2 are very
+close to the imaginary parts of the eigenvalues λ2,3 and
+λ4,5, respectively, of matrix M, shown in Table I.
+The beats are the result of the superposition of waves
+at the frequencies Ω1 and Ω2 of the spectral sidebands,
+with average frequency
+Ωav = Ω2 + Ω1
+2
+=
+�
+4Ω2 + (δ − ∆)2 +
+√
+4Ω2 + ∆2
+2
+,
+(48)
+and beat or modulation frequency
+Ωbeat = Ω2 − Ω1
+2
+=
+�
+4Ω2 + (δ − ∆)2 −
+√
+4Ω2 + ∆2
+2
+.
+(49)
+Now, we can identify the origin and modulation fre-
+quency of the beats in the time-dependent intensity,
+Eq. (29a), since the excited state populations ⟨A11(t)⟩
+and ⟨A22(t)⟩ oscillate at the generalized Rabi frequen-
+cies Ω1 and Ω2, respectively, with initial conditions
+given by a nonzero superposition of ground state pop-
+ulations at t = 0. In the case of the dipole correlation
+⟨ ˆE−
+π (0) ˆE+
+π (τ)⟩, however, the initial conditions are given
+by products of stationary atomic expectation values,
+most of them the coherences α13, α24, which become very
+small in the regime of beats. Thus, as seen in Table I,
+the terms ⟨∆A13(0)∆A31(τ)⟩ and ⟨∆A24(0)∆A42(τ)⟩ are
+
+9
+0
+3
+6
+9
+12
+15
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0
+3
+6
+9
+12
+15
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+0
+3
+6
+9
+12
+15
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+γτ
+∆ = −2γ, δ = −4γ
+∆ = −2γ, δ = −2γ
+∆ = 2γ,
+δ = −2γ
+∆ = 0,
+δ = 0
+(
) Ω = γ
+(a) Ω = γ/4
+(b) Ω = γ/2
+g(2)
+π (τ)
+g(2)
+π (τ)
+g(2)
+π (τ)
+FIG. 8.
+Photon correlations for (a) Ω = 0.25γ, (b) Ω = 0.5γ
+and (c) Ω = γ. The pairs of values (∆, δ) are the same as
+those in Fig. 2.
+much larger than the cross terms ⟨∆A13(0)∆A42(τ)⟩ and
+⟨∆A24(0)∆A31(τ)⟩, so the beats are basically due to the
+interference of those dominant terms.
+VI.
+PHOTON-PHOTON CORRELATIONS
+The standard method to investigate intensity fluctua-
+tions of a light source uses Brown-Twiss photon-photon
+correlations [16, 17]. The conditional character of this
+type of measurement makes it nearly free of detector in-
+efficiencies, unlike a single-detector measurement of the
+photoelectron probability distribution.
+In Ref. [8] the
+correlations of two photons from the π transitions were
+studied, albeit only for the degenerate case. In this paper
+we extend it to the case of nondegenerate states. These
+correlations are defined as
+g(2)
+π (τ) =
+G(2)
+π (τ)
+G(2)
+π (τ → ∞)
+(50)
+where, using Eq. (27a) for the field operators,
+G(2)
+π (τ) = ⟨ ˆE−
+π (0) ˆE−
+π (τ) ˆE+
+π (τ) ˆE+
+π (0)⟩
+= f 4
+π(r)⟨[A13(0) − A24(0)][A11(τ) + A22(τ)]
+×[A31(0) − A42(0)]⟩,
+(51a)
+and
+G(2)
+π (τ → ∞) =
+�
+Ist
+π
+�2 = f 4
+π(r) (α11 + α22)2 (51b)
+is the normalization factor. G(2)
+π (τ) can be further re-
+duced, since ⟨A13Ajk(τ)A42(0)⟩ = ⟨A24Ajk(τ)A31(0)⟩ =
+0, due to having vanishing initial conditions.
+Figure 8 shows g(2)
+π (τ) for moderate values of the Rabi
+frequency (near saturation) and the same sets of detun-
+ings ∆ and δ of Fig. 2. As usual in resonance fluores-
+cence, the correlation shows antibunching, g(2)
+π (0) = 0,
+0
+2
+4
+6
+8
+10
+0
+0.5
+1.0
+1.5
+2.0
+0
+2
+4
+6
+8
+10
+0
+0.5
+1.0
+1.5
+2.0
+0
+2
+4
+6
+8
+10
+0
+0.5
+1.0
+1.5
+2.0
+0
+2
+4
+6
+8
+10
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+g(2)
+π (τ)
+γτ
+γτ
+g(2)
+π (τ)
+(b) δ = −10γ
+(d) δ = −15γ
+(
) δ = −12γ
+(a) δ = −8γ
+FIG. 9.
+Photon-photon correlations showing beats in the
+strong field limit, Ω = 9γ, ∆ = 0, and large Zeeman splittings.
+The horizontal line helps to see that the wave packet is slightly
+rised.
+that is, a single atom cannot emit two photons simultane-
+ously. Unlike the two-level atom resonance fluorescence,
+the correlation is not simply the normalized population
+of the excited state, nor it is only the sum of the cor-
+relations of each single π transition. Besides the terms
+⟨A13(0)A11(τ)A31(0)⟩ and ⟨A24(0)A22(τ)A42(0)⟩, which
+are also out of phase, as seen from the time-dependent
+populations of their excited states (Fig. 2), there are six
+cross terms in the full correlation. In the nondegenerate
+case the multiple contributions cause in some cases quite
+irregular evolution. For instance, as we will see in the
+next Section, the slow decay of the correlation when the
+laser drives the atom near saturation, but below the ω13
+resonance transition, is related to a very narrow peak in
+the spectrum.
+The case of strong driving and large nondegeneracy
+is shown in Fig. 9, featuring quantum beats. There are
+several effects resulting from the increase of the nonde-
+generacy factor δ: (i) the larger number of visible wave
+packets; (ii) both average and beat frequencies approach
+one another, so the wave packets get shorter for larger
+photon-pair intervals τ, containing very few of the fast
+oscillations, as seen in Fig. 9(d); (iii) the wavepackets are
+initially slightly lifted above the g(2)(τ) = 1 value.
+VII.
+QUADRATURE FLUCTUATIONS
+Squeezing, the reduction of noise in one quadrature
+below that of a coherent state at the expense of the
+other, is the hallmark of phase-dependent fluctuations
+of the electromagnetic field [cite]. It is usually measured
+by balanced homodyne detection (BHD), but low quan-
+tum detector efficiency degrade the weak squeezing pro-
+duced in resonance fluorescence and cavity QED systems.
+One alternative our group has used is conditional homo-
+dyne detection (CHD) [18, 19], which correlates a quadra-
+ture amplitude on the cue of an intensity measurement.
+CHD measures a third-order amplitude-intensity corre-
+
+10
+lation (AIC) which, in the weak driving limit is reduced
+to the second-order one and that allows for measuring
+squeezing. Being a conditional measurement it is nearly
+free of detector inefficiencies.
+While the original goal of CHD was to measure the
+weak squeezing in cavity QED [18, 19], it was soon re-
+alized that nonzero third-order fluctuations of the am-
+plitude provide clear evidence of non-Gaussian fluctua-
+tions and higher-order field nonclassicality. In the present
+work the fluctuations are mainly third-order ones, due to
+near and above saturation excitation, and violate classi-
+cal bounds. We thus explore the phase-dependent fluctu-
+ations under conditions of quantum interference following
+our recent work [20, 23, 24].
+A.
+Amplitude-Intensity Correlations
+In CHD a quadrature’s field Eφ is measured by BHD
+on the cue of photon countings in a separate detector,
+where φ = 0, π/2 is the phase of the local oscillator. This
+is characterized by a correlation among the amplitude
+and the intensity of the field,
+hπ,φ(τ) =
+Hπ,φ(τ)
+Hπ,φ(τ → ∞),
+(52)
+where
+Hπ,φ(τ) = ⟨: ˆE−
+π (0) ˆE+
+π (0) ˆEπ,φ(τ) :⟩,
+(53a)
+the dots :: indicating time and normal operator orderings,
+and
+Hπ,φ(τ → ∞) = Ist
+π ⟨Eπ,φ⟩st
+(53b)
+= f 3
+π(r) [α11 + α22] Re
+�
+(α13 − α24) e−iφ�
+= f 3
+π(r) Ω3
+D2 Re
+�
+(∆ + (iγ − δ)/2) e−iφ�
+is the normalization factor.
+For the sake of concreteness, in this Section we limit
+our discussion to the out-of-phase quadrature, φ = π/2,
+which is the one that features squeezing when ωL = ω13,
+that is ∆ = 0. We do consider, however, squeezing in the
+in-phase quadrature φ = 0 in Sect. VIII on the variance.
+In several atom-laser systems hπ,φ(τ) has been proven
+to be time-asymmetric [20, 24]. This is not the case with
+the J = 1/2 → J = 1/2 system so we limit the analysis
+to positive intervals τ ≥ 0.
+Omitting the geometrical
+factor f 3
+π(r), which is later cancelled by normalization,
+we have
+Hπ,φ(τ) = ⟨ ˆE−
+π (0) ˆEπ,φ(τ) ˆE+
+π (0)⟩
+= Re
+�
+e−iφ⟨A13(0)[A13(τ) − A24(τ)]A31(0)
++A24(0)[A13(τ) − A24(τ)]A42(0)⟩} .
+(54)
+Note that Hπ,φ(0) = 0 meaning that, like antibunching
+in g(2), the atom has to build a new photon wavepacket
+after one has been emitted.
+The AIC suggests nontrivial behavior when we take
+dipole fluctuations into account, that is, when the atomic
+operators are split into their mean plus noise, Ajk =
+αjk + ∆Ajk; upon substitution in Eq. (54) we get
+Hπ,φ(τ) = Ist
+π ⟨Eπ,φ⟩st + H(2)
+π,φ(τ) + H(3)
+π,φ(τ),
+(55)
+or in normalized form as
+hπ,φ(τ) = 1 +
+H(2)
+π,φ(τ)
+Ist
+π ⟨Eπ,φ⟩st
++
+H(3)
+π,φ(τ)
+Ist
+π ⟨Eπ,φ⟩st
+,
+(56)
+where
+H(2)
+π,φ(τ) = 2Re
+�
+⟨ ˆE+
+π ⟩st⟨∆ ˆE−
+π (0)∆ ˆEπ,φ(τ)⟩
+�
+= Re
+�
+(α31 − α42) [⟨(∆A13(0) − ∆A24(0))
+�
+∆A13(τ) − ∆A24(τ))⟩e−iφ
++⟨(∆A13(0) − ∆A24(0))
+�
+∆A31(τ)⟩ − ∆A42(τ))⟩eiφ��
+,
+(57)
+H(3)
+π,φ(τ) = ⟨∆ ˆE−
+π (0)∆ ˆEπ,φ(τ)∆ ˆE+
+π (0)⟩
+= Re
+�
+eiφ⟨[∆A13(0) − ∆A24(0)] [∆A31(τ) − ∆A42(τ)] [∆A31(0) − ∆A42(0)]⟩
+�
+.
+(58)
+The initial conditions of the correlations are given in Ap-
+pendix A.
+From hπ,π/2(0) = 0 we can obtain analytically the ini-
+tial values of the second- and third-order terms,
+h(2)
+π,π/2(0) = 1 − (2∆ − δ)2 + γ2
+2D
+,
+(59)
+h(3)
+π,π/2(0) = (2∆ − δ)2 + γ2
+2D
+− 2,
+(60)
+where D is given by Eq. (21).
+
+11
+Being the AIC a function of odd-order in the field am-
+plitude we rightly expect a richer landscape than that
+of the intensity correlations, more so when one considers
+quantum interference and the complex parameter space.
+For instance, the correlation can take on not only nega-
+tive values but break classical bounds [18, 19]:
+0 ≤ hφ(τ) − 1 ≤ 1 ,
+(61a)
+|h(2)
+φ (τ) − 1| ≤ |h(2)
+φ (0) − 1| ≤ 1 ,
+(61b)
+where the second line is valid only for weak fields such
+that h(3)
+φ (τ) ∼ 0. These classical bounds are stronger cri-
+teria for nonclassicality of the emitted field than squeezed
+light measurements, the more familiar probing of phase-
+dependent fluctuations. A detailed hierarchy of nonclas-
+sicality measures for higher-order correlation functions is
+presented in Refs. [25, 26]. In Ref. [20] an inequality was
+obtained that considers the full hφ(τ) by calculating the
+AIC for a field in a coherent state,
+−1 ≤ hφ(τ) ≤ 1 .
+(62)
+For a meaningful violation of Poisson statistics, hφ(τ)
+must be outside these bounds.
+Also, hφ(τ) is a measure of non-Gaussian fluctuations,
+here of third-order in the field fluctuations. Resonance
+fluorescence is a particularly strong case of non-Gaussian
+noise by being a highly nonlinear stationary nonequilib-
+rium process [20, 23, 24, 27, 28], thanks also to its small
+Hilbert space. This makes resonance fluorescence unsuit-
+able to a quasiprobability distribution approach.
+B.
+Fluctuations Spectra
+Since quadrature fluctuations, such as squeezing, are
+often studied in the frequency domain we now define the
+spectrum of the amplitude-intensity correlations:
+Sπ,φ(ω) = 8γ1
+� ∞
+0
+dτ cos (ωτ) [hπ,φ(τ) − 1]
+(63)
+which, following Eqs. (52) and (55), can be decomposed
+into terms of second- and third-order in the dipole fluc-
+tuations
+S(q)
+π,φ(ω) = 8γ1
+� ∞
+0
+dτ cos (ωτ)h(q)
+π,φ(τ),
+(64)
+where q = 2, 3, so that Sπ,φ(ω) = S(2)
+π,φ(ω) + S(3)
+π,φ(ω).
+As mentioned above, the AIC was devised initially to
+measure squeezing without the issue of imperfect detec-
+tion efficiencies. Obviously, hπ,φ(τ) and Sπ,φ(ω) are not
+measures of squeezing. They measure a third-order mo-
+ment in the field’s amplitude, while squeezing is a second-
+order one in its fluctuations.
+The so-called spectrum
+of squeezing is the one for q = 2, with the advantage
+of the AIC of not depending on the efficiency of detec-
+tion. Squeezing is signaled by frequency intervals where
+0
+2
+4
+6
+8
+10
+12
+0
+2
+4
+6
+-10
+-5
+0
+5
+10
+0
+1
+2
+0
+2
+4
+6
+8
+10
+12
+0
+2
+4
+6
+-10
+-5
+0
+5
+10
+-1
+0
+1
+2
+0
+2
+4
+6
+8
+10
+12
+0
+2
+4
+6
+-10
+-5
+0
+5
+10
+-1
+0
+1
+2
+Sπ,π/2 (ω)
+hπ,π/2 (τ)
+(b) Ω = γ/2
+γτ
+ω/γ
+(a) Ω = γ/4
+(d) Ω = γ/4
+(e) Ω = γ/2
+(f
+) Ω = γ
+(
) Ω = γ
+FIG. 10.
+Amplitude-intensity correlations (left panel) and
+spectra (right panel) for the φ = π/2 quadrature in the weak-
+moderate field limit. Parameters and line styles are the same
+as in Fig. 8: ∆ = δ = 0 (solid-black); ∆ = 2γ and δ = −2γ
+(dots-red); ∆ = −2γ and δ = −2γ (dashed-green); ∆ = −2γ
+and δ = −4γ (dot-dashed-blue).
+S(2)
+π,φ(ω) < 0. As a further note, the full incoherent spec-
+trum, Eq. (45), can be obtained by adding the squeezing
+spectra of both quadratures [29],
+Sinc
+π (ω) =
+1
+8γ1
+�
+S(2)
+π,0(ω) + S(2)
+π,π/2(ω)
+�
+.
+(65)
+C.
+Results
+We now show plots of the AICs and their spectra in
+Figs. 10-12 for the φ = π/2 quadrature and the same sets
+of detunings ∆, δ of Fig. 2, and weak to moderate Rabi
+frequencies, γ/4 < Ω < γ. With the three parameters Ω,
+∆, and δ, the landscape of effects is vast.
+We first notice a few general features seen in hπ,π/2(τ),
+Fig. 10.
+With increasing Rabi frequencies, detunings,
+and Zeeman splittings we observe the clear breakdown of
+the classical inequalities besides the one at τ = 0. Cor-
+respondingly, in the spectra, the extrema get displaced
+and broadened.
+Now, we want to single out the case
+of nondegeneracy with small detuning on the |1⟩ − |3⟩
+transition but large on the |2⟩ − |4⟩ one, ∆ = −δ = 2γ
+(green-dashed line). For weak field, Ω = γ/4, the AIC
+does not have a regular evolution for short times but it
+does decay very slowly, with a correponding very narrow
+spectral peak. The slow decay is also clearly visible in the
+photon correlation, Fig. 8a. As we mentioned in Sect. III
+regarding Fig. 2b, state |4⟩ ends up with a large portion
+of the steady state population due to optical pumping;
+not quite a trapping state, so there is no electron shelv-
+ing per se, as argued in [5]. This effect is washed out for
+larger Rabi frequencies, which allow for faster recycling
+of the populations. To a lesser degree, slow decay and
+sharp peak occur for opposite signs of ∆ and δ.
+
+12
+0
+2
+4
+6
+8
+10
+-2
+-1
+0
+1
+2
+3
+4
+-10 -8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+0
+2
+4
+6
+8
+10
+-2
+-1
+0
+1
+2
+3
+4
+-10 -8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+3
+-10 -8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+-0.5
+0
+0.5
+1.0
+(f
+) Ω = γ
+(
) Ω = γ
+(b) Ω = γ/2
+(e) Ω = γ/2
+(a) Ω = γ/4
+(d) Ω = γ/4
+h(2)
+π,π/2 (τ)
+ω/γ
+γτ
+S(2)
+π,π/2 (ω)
+FIG. 11.
+Second-order component of the AIC and spectra of
+Fig. 10.
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+-10 -8 -6 -4 -2
+0
+2
+4
+6
+8
+10
+0
+1
+2
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+-10 -8 -6 -4 -2
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+0
+2
+4
+6
+8
+10
+-1
+0
+1
+2
+3
+-10 -8 -6 -4 -2
+0
+2
+4
+6
+8
+10
+-2
+-1
+0
+1
+(f
+) Ω = γ
+(
) Ω = γ
+(b) Ω = γ/2
+(e) Ω = γ/2
+(a) Ω = γ/4
+(d) Ω = γ/4
+h(3)
+π,π/2 (τ)
+ω/γ
+γτ
+S(3)
+π,π/2 (ω)
+FIG. 12.
+Third-order component of the AIC and spectra of
+Fig. 10.
+The splitting of the AIC and spectra into components
+of second and third order in the fluctuations, Figs. 11, 12,
+helps to understand better the quadrature fluctuations.
+For the second-order ones we have the squeezing spectra:
+around ω = 0 for ∆ = 0 and small Rabi frequencies,
+Ω < γ/4; and in sidebands for larger detunings, Rabi
+frequencies and Zeeman splittings. In h(2)
+π,π/2(τ) there is a
+reduction in amplitudes and nonclassicality for increasing
+Rabi frequencies except for the case of oppposite signs of
+detuning and difference Zeeman splitting. Note that the
+sharp spectral peak in the latter case takes up most of
+the corresponding peak in Fig. 10. This is because both
+π transitions are largely detuned from the laser, keeping
+Ω small.
+Increasing the laser strength the third-order effects
+overcome the second-order ones. For instance, regarding
+the size of the features. Also, a comparison of Figs. 11
+0
+2
+4
+6
+8
+10
+-10
+0
+10
+20
+-40
+-20
+0
+20
+40
+-10
+-5
+0
+5
+10
+0
+2
+4
+6
+8
+10
+-10
+0
+10
+20
+-40
+-20
+0
+20
+40
+-10
+-5
+0
+5
+10
+0
+2
+4
+6
+8
+10
+-10
+0
+10
+20
+-40
+-20
+0
+20
+40
+-10
+-5
+0
+5
+10
+0
+2
+4
+6
+8
+10
+-10
+0
+10
+20
+-40
+-20
+0
+20
+40
+-5
+0
+5
+hπ,π/2 (τ)
+h
+(2)
+π,π/2 (τ)
+h
+(3)
+π,π/2 (τ)
+S
+(3)
+π,π/2 (τ)
+S
+(2)
+π,π/2 (τ)
+Sπ,π/2 (τ)
+γτ
+ω/γ
+(b)
+(f
+)
+(
)
+(g)
+(d)
+(h)
+(a)
+(e)
+FIG. 13.
+AIC and spectra for Ω = 9γ, ∆ = 0, (a,e) δ = −8γ,
+(b,f) δ = −10γ, (c,g) δ = −12γ, (d,h) δ = −15γ. Lines are:
+full AIC and spectra (solid-black), second-order (dots-red),
+and third-order (dashed-blue).
+and 12 shows that h(3)
+φ (τ) is mainly responsible for the
+breakdown of the classical bounds when the driving field
+is on or above saturation.
+Moreover, we see that the
+slow-decay–sharp-peak is mainly a third-order effect.
+To close this Section, the AIC and spectra for very
+strong fields and large Zeeman splittings, Ω, |δ| ≫ γ are
+shown in Fig. 13. The AIC shows beats as in the photon
+correlations.
+Unlike those in g(2)(τ), these wavepack-
+ets oscillate around h(τ) = 1.
+Because the regime
+is that of strong excitation the third-order component
+clearly dominates, making the fluorescence notably non-
+Gaussian, and clearly violates the classical inequalities.
+The spectral peaks are localized around the Rabi frequen-
+cies ±Ω1, ±Ω2. Studies of the spectrum of squeezing for
+the J = 1/2 − J = 1/2 system were reported in [10].
+Those authors choose ±Ω1, ±Ω2 with a less strong laser
+but large detuning and large Zeeman splittings, observ-
+ing the double sidebands, but no mention or hint of beats
+was made.
+VIII.
+VARIANCE
+The variance is a measure of the total noise in a
+quadrature; it is defined as
+Vφ = ⟨: (∆Eφ)2 :⟩ = Re
+�
+e−iφ⟨∆ ˆE−∆ ˆEφ⟩st
+�
+, (66)
+
+13
+and is related to the spectrum of squeezing as
+Vφ =
+1
+4πγη
+� ∞
+−∞
+dωS(2)
+φ (ω).
+(67)
+where η is the detector efficiency. The maximum value
+of Vφ is 1/4, obtained when there is very strong driving,
+when almost all the emitted light is incoherent. Negative
+values of the variance are a signature of squeezing but,
+unlike the quadrature spectra, the squeezing is the total
+one in the field, independent of frequency.
+For the π transitions we have
+Vπ,φ = f 2
+π(r)
+2
+Re
+�
+−(α13 − α24)2e−2iφ
++(α11 + α22 − |α13 − α24|2)
+�
+,
+(68)
+= f 2
+π(r)
+2
+Ω2
+D
+�
+1 − [(2∆ − δ) cos φ + γ sin φ]2
+2D
+�
+.
+(69)
+For φ = π/2 and φ = 0 we have, respectively,
+Vπ,π/2 = f 2
+π(r)
+2
+Ω2
+D
+�
+1 − γ2
+2D
+�
+,
+(70a)
+Vπ,0 = f 2
+π(r)
+2
+Ω2
+D
+�
+1 − (2∆ − δ)2
+2D
+�
+,
+(70b)
+where D is given by Eq. (21).
+In Fig. 14 we plot the variances of the out-of-phase
+φ = π/2 (left panel) and in-phase φ = 0 (right panel)
+quadratures. The interplay of parameters is a complex
+one, but we mostly use the ones of previous figures. For
+φ = π/2 and ∆ = 0, as usual in resonance fluorescence
+systems, squeezing is restricted to a small range of Rabi
+frequencies, detunings, and Zeeman splittings. For φ = 0
+nonzero laser or Zeeman detunings are necessary to pro-
+duce squeezing, with a strong dependence on their sign:
+on-resonance (not shown) there is no squeezing, as for
+a two-level atom; in Fig. 14(d) the laser is tuned be-
+low that transition, ∆ = −2γ, and there is no squeezing
+(positive variance) but the variance is reduced for large
+δ; in Fig. 14(e) the laser is tuned above the transition,
+∆ = −2γ, and there is squeezing for larger Rabi frequen-
+cies. Large values of δ tend to reduce the variance, be it
+positive or negative.
+A.
+Out-of-phase quadrature
+We now discuss a complementary view of the variance.
+For φ = π/2 we can identify the Rabi frequency interval
+within which squeezing takes place,
+0 < Ω < 1
+2
+�
+γ2/2 − δ2/2 − 2(∆ − δ/2)2,
+(71)
+and the Rabi frequency for maximum squeezing is
+˜Ωπ/2 = 1
+2
+�
+γ4/2 − 2[(δ − ∆)2 + ∆2]2
+3γ2 + 2[(δ − ∆)2 + ∆2]2 .
+(72)
+0
+0.5
+1.0
+1.5
+2.0
+-0.05
+0
+0.05
+0.10
+0.15
+0.20
+0
+0.5
+1.0
+1.5
+2.0
+0
+0.04
+0.08
+0.12
+0.16
+0.20
+0.24
+0
+0.5
+1.0
+1.5
+2.0
+0
+0.04
+0.08
+0.12
+0.16
+0
+0.5
+1.0
+1.5
+2.0
+-0.02
+0
+0.02
+0.04
+-3
+-2
+-1
+0
+1
+2
+3
+-0.03
+-0.02
+-0.01
+0
+0.01
+-3
+-2
+-1
+0
+1
+2
+3
+-0.04
+0
+0.04
+0.08
+0.12
+0.16
+0.2
+Vπ,π/2/f2
+π(r)
+Vπ,0/f2
+π(r)
+Ω/γ
+Ω/γ
+Ω/γ
+∆/γ
+∆/γ
+Ω/γ
+(
)
+(a)
+(b)
+(d)
+(e)
+(f
+)
+FIG. 14.
+Variance of the quadratures of the fluorescence of
+the π transitions: left panel for φ = π/2 and right panel for
+φ = 0. (a,b,d,e) as a function of Rabi frequency and (c,f)
+as a function of detuning.
+In all cases δ = 0 is given by
+a solid-black line, and δ = −0.5γ by a dashed-red line; the
+dotted-blue line is δ = −2γ in (a,b,d,e) and δ = −γ in (c,f).
+Additionally, (a) ∆ = 0, (b) ∆ = −2γ, (c) Ω = 0.2γ, (d)
+∆ = 0, (e) ∆ = 2γ, (f) Ω = 0.8γ.
+Thus, the variance at ˜Ωπ/2 is
+V
+(˜Ωπ/2)
+π,π/2 (∆ = 0, δ) = f 2
+π(r)
+16
+(γ4/2 − 2δ4)(δ2 − γ2)
+γ2(γ2 + 2δ2)(δ2 + γ2),
+(73a)
+for ∆ = 0 and |δ/γ| < 1/
+√
+2;
+V
+(˜Ωπ/2)
+π,π/2 (∆, δ = 0) = f 2
+π(r)
+16
+(γ4/2 − 8∆4)(4∆2 − γ2)
+γ2(γ2 + 4∆2)2
+,
+(73b)
+for δ = 0 and |∆/γ| < 1/
+√
+2; and the maximum total
+squeezing is obtained at ∆ = δ = 0,
+V
+(˜Ωπ/2)
+π,π/2 (0, 0) = −f 2
+π(r)
+32 ,
+˜Ωπ/2 =
+γ
+2
+√
+6.
+(73c)
+For φ = π/2 squeezing is limited to elliptical regions
+of weak driving and small detunings ∆ and δ:
+2δ2 + 8Ω2 < γ2,
+∆ = 0,
+(74a)
+4∆2 + 8Ω2 < γ2,
+δ = 0.
+(74b)
+B.
+In-phase quadrature
+For φ = 0, squeezing is obtained in the Rabi frequency
+interval, for δ = 0,
+0 < Ω <
+1
+√
+2
+�
+∆2 − γ2/4,
+|∆| > γ/2,
+(75)
+
+14
+with maximum squeezing at the Rabi frequency
+˜Ω0 =
+1
+2
+√
+2
+�
+16∆2 − γ2
+12∆2 + γ2 ,
+(76)
+requiring finite detuning from both π transitions (∆ ̸= 0)
+and stronger driving, Ω ∼ γ [see Fig. 14(d)-(f)].
+Thus, the variance at ˜Ω0 is
+V (˜Ω0)
+π,0 (δ) = −f 2
+π(r)
+128
+4∆2 − γ2
+∆2(4∆2 + γ2),
+|∆| ≥ γ/2. (77)
+This expression gets the asymptotic value
+lim
+∆→∞ V (˜Ω0)
+π,0
+= −f 2
+π(r)
+32 ,
+(78)
+which is the same as that for the π/2 quadrature. The
+region for squeezing obeys the relation
+4∆2 − 8Ω2 < γ2.
+(79)
+So, to obtain squeezing in this quadrature it is necessary
+to have detunings ∆ > γ/4 for any Rabi frequency.
+IX.
+DISCUSSION AND CONCLUSIONS
+We have studied several properties of the resonance
+fluorescence of the π transitions in a J = 1/2 − J = 1/2
+angular momentum atomic system driven by a linearly
+polarized laser field and a magnetic field along the π tran-
+sition to lift the level degeneracies. Interference among
+the various transition amplitudes create a rich landscape
+of effects. Most notable among our results is the observa-
+tion of quantum beats when the atom is subject to large
+laser and magnetic fields. In this regime, two close Rabi
+frequencies interfere, giving rise to a well-defined modu-
+lation of the fast oscillations. These Rabi frequencies are
+the source of the two pairs of sidebands in the incoherent
+part of the power spectrum [5] and in the squeezing spec-
+trum [10]. We studied beats in the total intensity and
+two-time functions such as the dipole-dipole, intensity-
+intensity and intensity-amplitude correlations.
+In the
+beats’ regime the role of vacuum-induced coherence is
+small because the upper levels are very separated due to
+very large difference Zeeman splitting.
+Before the beats we considered the previously over-
+looked time-dependent populations and reviewed aspects
+of the known stationary ones. The fact that the upper
+state populations evolve out of phase should not be a
+surprise.
+This, and nonzero initial population of both
+ground states (in contrast to nonzero populations of ex-
+cited states for spontaneous emission), are major factors
+in the interference among the terms in the intensity. Ex-
+cept for very strong laser fields, the steady state popula-
+tions depend strongly on the difference Zeeman splitting.
+The AIC also permits to quantify the degree of non-
+Gaussianity; the fluctuations of third-order in the field
+quadrature amplitude due to strong atom-laser nonlin-
+earity dominate over the second-order ones with strong
+driving.
+The beats are in the strongly non-Gaussian
+regime.
+The correlations show nonclassical features of the fluo-
+rescence light such as antibunching, g(2)(0) = 0, and vio-
+lation of classical inequalities in the amplitude-intensity
+correlations, Eqs. (61 -62). We studied squeezing using
+the variance, i. e., the total noise in a quadrature, as
+well as using the second-order part of the spectrum. In
+the regime of beats there is squeezing, near the effective
+Rabi frequencies, but none in the total noise.
+For a system with many parameters the interplay
+among them is a complex one, making the interpretation
+of results nontrivial. Thus, for most of our plots we chose
+parameters in two groups: i) where they are relatively
+small, Ω, ∆, δ ∼ γ, chosen to illustrate several degrees of
+vacuum-induced coherence; and ii) where they are large,
+Ω, ∆, δ ≫ γ, and quantum beats are revealed. Overall,
+particular care must be taken regarding detunings. On
+the one hand, large difference Zeeman splitting means
+that the excited levels would be very separated and in-
+teract with different frequency portions of the reservoir,
+hence diminishing the vacuum-induced coherence.
+On
+the other, large laser-atom detunings, which might in-
+crease the VIC, mean reduced fluorescence rates, which
+may also be detrimental in measurements. The beats,
+then, would be better observed if ∆ ≤ γ and δ of just
+several γ in the strong field regime.
+X.
+ACKNOWLEDGMENTS.
+The authors thank Dr. Ricardo Rom´an-Ancheyta and
+Dr. Ir´an Ramos-Prieto for useful comments at an early
+stage of the project. ADAV thanks CONACYT, Mexico,
+for scholarship No. 804318.
+ORCID
+numbers:
+H´ector
+M.
+Castro-Beltr´an
+https://orcid.org/0000-0002-3400-7652, Octavio de los
+Santos-S´anchez https://orcid.org/0000-0002-4316-0114,
+Luis Guti´errez https://orcid.org/0000-0002-5144-4782,
+Appendix A: Time-Dependent Matrix Solutions and
+Spectra
+The two-time photon correlations under study have
+the general form ⟨W(τ)⟩ = ⟨O1(0)R(τ)O2(0)⟩, where
+R is the Bloch vector and O1,2 are system operators.
+The same applies to correlations of fluctuation operators
+∆R, ∆O1,2. Using the quantum regression formula [30],
+the correlations obey the equation
+⟨ ˙W(τ)⟩ = M⟨W(τ)⟩,
+(A1)
+which has the formal solution
+⟨W(τ)⟩ = eMτ⟨W(0)⟩,
+(A2)
+where M is given by
+
+15
+M =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−γ
+−iΩ
+0
+0
+iΩ
+0
+0
+0
+−iΩ −
+� γ
+2 + i∆
+�
+0
+0
+0
+iΩ
+0
+0
+0
+0
+−γ
+iΩ
+0
+0
+−iΩ
+0
+0
+0
+iΩ
+−
+� γ
+2 + i(∆ − δ)
+�
+0
+0
+0
+−iΩ
+iΩ
+0
+0
+0
+−
+� γ
+2 − i∆
+�
+−iΩ
+0
+0
+γ1
+iΩ
+γσ
+0
+−iΩ
+0
+0
+0
+0
+0
+−iΩ
+0
+0
+0
+−
+� γ
+2 − i(∆ − δ)
+�
+iΩ
+γσ
+0
+γ2
+−iΩ
+0
+0
+iΩ
+0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(A3)
+Also, spectra of stationary systems can be evaluated
+more effectively using the above formal approach.
+Be
+g(τ) = ⟨W(τ)⟩. Then, a spectrum is calculated as
+S(ω) ∝
+� ∞
+0
+cos ωτ g(τ) dτ =
+� ∞
+0
+cos ωτ eMτg(0) dτ
+= Re
+� ∞
+0
+e−(iω1−M)τg(0) dτ
+= Re
+�
+(iω1 − M)−1g(0)
+�
+,
+(A4)
+where 1 is the identity matrix. For example, the inco-
+herent spectrum requires calculations of the type
+Sinc(ω) = Re
+� ∞
+0
+dτe−iωτeMτ⟨∆Aij(0)∆Akl(0)⟩st
+= Re
+�
+(M − iω1)−1⟨∆Aij(0)∆Akl(0)⟩st
+�
+. (A5)
+For the initial conditions of the correlations we use the
+following operator products and correlations in compact
+form:
+AklAmn = Aknδlm ,
+(A6a)
+⟨AklAmn⟩ = αknδlm,
+(A6b)
+AijAklAmn = Ainδjkδlm,
+(A6c)
+⟨AijAklAmn⟩ = αinδjkδlm.
+(A6d)
+Hence, the relevant initial conditions are:
+⟨A13R⟩ = (0, 0, 0, 0, α11, α13, 0, 0)T ,
+(A7a)
+⟨A24R⟩ = (0, 0, 0, 0, 0, 0, α22, α24)T ,
+(A7b)
+⟨A13RA31⟩ = (0, 0, 0, 0, 0, α11, 0, 0)T ,
+(A7c)
+⟨A24RA42⟩ = (0, 0, 0, 0, 0, 0, 0, α22)T ,
+(A7d)
+⟨A13RA42⟩ = ⟨A24RA31⟩ = 0,
+(A7e)
+where R = (A11, A13, A22, A24, A31, A33, A42, A44)T is
+the Bloch vector. For correlations with fluctuation oper-
+ator products, ∆Aij = Aij − αij, we have
+⟨∆Akl∆Amn⟩ = αknδlm − αklαmn,
+(A8)
+⟨∆Aij∆Akl∆Amn⟩ = αinδlmδjk − αilαmnδjk
+−αinαklδjm − αijαknδlm
++2αijαklαmn.
+(A9)
+Now, recalling that α12 = α14 = α23 = α34 = 0, we
+write the detailed initial conditions of the correlations
+(Set 1 of Bloch equations and quantum regression for-
+mula):
+⟨∆A13∆R⟩ =
+�
+−α13α11, −α2
+13, −α13α22, −α13α24, α11 − |α13|2, α13 − α13α33, −α13α42, −α13α44
+�T ,
+(A10a)
+⟨∆A24∆R⟩ =
+�
+−α24α11, −α24α13, −α24α22, −α2
+24, −α24α31, −α24α33, α22 − |α24|2, α24 − α24α44
+�T ,
+(A10b)
+⟨∆A13∆R∆A31⟩ =
+�
+2|α13|2α11 − α2
+11, 2|α13|2α13 − 2α11α13,
+2|α13|2α22 − α11α22, 2|α13|2α24 − α11α24,
+2|α13|2α31 − 2α11α31, 2|α13|2α33 + α11 − 2|α13|2 − α11α33,
+2|α13|2α42 − 2α11α42, 2|α13|2α44 − α11α44
+�T .
+(A10c)
+⟨∆A24∆R∆A42⟩ =
+�
+2|α24|2α11 − α11α22, 2|α24|2α13 − α22α13,
+2|α24|2α22 − α2
+22, 2|α24|2α24 − 2α22α24,
+2|α24|2α31 − α22α31, 2|α24|2α33 − α22α33,
+2|α24|2α42 − 2α22α42, 2|α24|2α44 + α22 − 2|α24|2 − α22α44
+�T .
+(A10d)
+
+16
+⟨∆A13∆R∆A42⟩ =
+�
+2α13α11α42, 2α2
+13α42, 2α13α22α42, (2|α24|2 − α22)α13,
+(2|α13|2 − α11)α42, (2α13α33 − α13)α42, 2α13α2
+42, (2α13α44 − α13)α42
+�T ,
+(A10e)
+⟨∆A24∆R∆A31⟩ =
+�
+2α24α11α31, (2|α13|2 − α11)α24, 2α24α22α31, 2α2
+24α31,
+2α24α2
+31, (2α24α33 − α24)α31, (2|α24|2 − α22)α31, (2α24α44 − α24)α31
+�T .
+(A10f)
+Appendix B: Condition for Optimal Appearance of
+Beats in the Intensity
+We consider a simplified, unitary, model to estimate
+the optimal initial population of the ground states to
+make well-formed beats. First, we diagonalize the Hamil-
+tonian Eq. (8). The eigenvalues and eigenstates are
+E±
+1 = −∆
+2 ± 1
+2
+�
+4Ω2 + ∆2,
+(B1a)
+E±
+2 = Bℓ + δ − ∆
+2
+± 1
+2
+�
+4Ω2 + (δ − ∆)2,
+(B1b)
+and
+|u1⟩ = sin Θ1|1⟩ + cos Θ1|3⟩,
+|u2⟩ = − cos Θ1|1⟩ + sin Θ1|3⟩,
+|u3⟩ = sin Θ2|2⟩ + cos Θ2|4⟩,
+|u4⟩ = − cos Θ2|2⟩ + sin Θ2|4⟩,
+(B2)
+respectively, where
+sin Θ1 =
+2Ω
+��
+∆ +
+√
+∆2 + 4Ω2�2 + 4Ω2
+,
+cos Θ1 =
+∆ +
+√
+∆2 + 4Ω2
+��
+∆ +
+√
+∆2 + 4Ω2�2 + 4Ω2
+,
+sin Θ2 =
+2Ω
+��
+(δ − ∆) +
+�
+(δ − ∆)2 + 4Ω2
+�2
++ 4Ω2
+,
+cos Θ2 =
+(δ − ∆) +
+�
+(δ − ∆)2 + 4Ω2
+��
+(δ − ∆) +
+�
+(δ − ∆)2 + 4Ω2
+�2
++ 4Ω2
+.
+(B3)
+It is now straightforward to obtain the excited-state
+populations. If the initial state of the system is ρ(0) =
+⟨A33(0)⟩|3⟩⟨3| + ⟨A44(0)⟩|4⟩⟨4| we get
+⟨A33(t)⟩ = 1
+2⟨A33(0)⟩ sin2 (2Θ1)(1 − cos (Ω1t)), (B4a)
+⟨A44(t)⟩ = 1
+2⟨A44(0)⟩ sin2 (2Θ2)(1 − cos (Ω2t)), (B4b)
+and the intensity of the field is
+Iπ(r, t)
+f 2π(r) = ⟨A33(0)⟩ sin2 (2Θ1) + A44(0)⟩ sin2 (2Θ2)
+−⟨A33(0)⟩ sin2 (2Θ1) cos (Ω1t)
+−⟨A44(0)⟩ sin2 (2Θ2) cos (Ω2t).
+(B5)
+A necessary condition for the beating behavior to oc-
+cur is that the initial ground-state populations are both
+nonvanishing in the nondegenerate case. Now, assuming
+the relation
+⟨A33(0)⟩
+⟨A44(0)⟩ = sin2 (2Θ2)
+sin2 (2Θ1)
+(B6)
+is satisfied by chossing appropriate parameter values
+(Ω, δ, ∆) for given values of initial ground state popu-
+lations we would get
+Iπ(r, t) = f 2
+π(r)⟨A33(0)⟩ sin2 (2Θ1)
+× [1 − cos (Ωbeatt) cos (Ωavt)] ,
+(B7)
+where Ωbeat = (Ω2 − Ω1)/2 and Ωav = (Ω2 + Ω1)/2.
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diff --git a/2tE1T4oBgHgl3EQfSAOy/content/tmp_files/load_file.txt b/2tE1T4oBgHgl3EQfSAOy/content/tmp_files/load_file.txt
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index 0000000000000000000000000000000000000000..56716d872c8798d6effe2b36027ce7e36b4654f6
--- /dev/null
+++ b/2tE1T4oBgHgl3EQfSAOy/content/tmp_files/load_file.txt
@@ -0,0 +1,1037 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf,len=1036
+page_content='Quantum interference in the resonance fluorescence of a J = 1/2 − J′ = 1/2  atomic system: Quantum beats, nonclassicality, and non-Gaussianity  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Castro-Beltr´an,1, ∗ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' de los Santos-S´anchez,2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Guti´errez,3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Alcantar-Vidal1  1Centro de Investigaci´on en Ingenier´ıa y Ciencias Aplicadas and Instituto de Investigaci´on en Ciencias B´asicas y Aplicadas,  Universidad Aut´onoma del Estado de Morelos, Avenida Universidad 1001, 62209 Cuernavaca, Morelos, M´exico  2Tecnologico de Monterrey, Escuela de Ingenier´ıa y Ciencias,  Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Carlos Lazo 100, Santa Fe, Mexico City, Mexico, 01389  3Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico,  62210 Cuernavaca, Morelos, M´exico  (Dated: January 10, 2023)  We study the resonance fluorescence of a system with angular momentum J = 1/2−J′ = 1/2 level  structure driven by a single, linearly polarized, monochromatic laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Quantum interference  among the two, antiparallel, π transitions leads to rich results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' We develop the article around two  broad overlapping themes: (i) the observation of quantum beats in the intensity and the dipole-  dipole, intensity-intensity, and quadrature-intensity correlations, when the atom is subject to a  strong laser and large Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The mean and modulation frequencies of the beats are  given by the average and difference, respectively, among two close generalized Rabi frequencies  related to a Mollow-like spectrum with two pairs of sidebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (ii) The nonclassical and non-  Gaussian properties of phase-dependent fluorescence for the cases of weak to moderate excitation  and in the regime of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The fluorescence in the beats regime is nonclassical, mainly from the  third-order dipole fluctuations, which reveal them to be also strongly non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For weak to  moderate driving laser and small detunings and Zeeman splittings the nonclassicality is an interplay  of second- (squeezing) and third-order dipole noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  INTRODUCTION  Recently, the properties of the resonance fluorescence  of a single atomic system with angular momentum transi-  tion J = 1/2−J′ = 1/2 driven by a monochromatic laser  have been the subject of great interest due to the possi-  bility of observing vacuum-induced coherence effects due  to interference among the two antiparallel π transitions,  emitting into the same frequency range of the electro-  magnetic vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Here, the π transitions are incoher-  ently coupled, mediated by spontaneous emision in the σ  transitions and then excited by the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The antiparal-  lel dipoles of the transitions makes it realistic to observe  interference effects, while V and Λ three-level systems re-  quire additional preparation because the transitions are  perpendicular [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Particular attention has been de-  voted to the spectrum [3–6], time-energy complementar-  ity [4, 5], Young’s interference [7], photon correlations  [8], frequency-resolved photon correlations [9], squeezing  [10], phase shifts [11], and cooperative effects in photon  correlations [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The case of additional laser excitation  of one of the σ transitions on the spectrum and squeezing  has been studied in [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Quantum beats are among the more familiar mani-  festations of quantum interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' They appear in the  modulation of the decay by spontaneous emission of mul-  tilevel systems due to the energy difference among transi-  tions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' So far, few experiments of quantum interference  experiments have been performed on the J = 1/2 − J′ =  ∗ hcastro@uaem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='mx  1/2, in this case observing Young-type fringes [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Hence,  further experiments are desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Quantum beats in the  intensity are the result of the inability to tell the path  of a particular photon when observed by a broadband  detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The beats can also occur in two-time correla-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' As a general rule, initial conditions should be a  superposition state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In this paper we investigate theoretically effects of  quantum interference on the total intensity and two-time  correlations such as dipole-dipole (to calculate spectra),  intensity-intensity, intensity-amplitude correlations, and  variance of the light emitted into the π transitions of the  J = 1/2−J′ = 1/2 atomic system driven by a linearly po-  larized laser and a magnetic field to break the degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  While we put emphasis on the regime of observation of  quantum beats, the nonclassical and non-Gaussian prop-  erties of the fluorescence are also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  After describing the main features of the model in  Section II, we discuss the basic dynamic and stationary  properties of the atomic expectation values in Section  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Here, we analyze the previously overlooked time-  dependent behavior of the atomic populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Those  of the excited states, for instance, although equal in the  steady state, evolve with different Rabi frequencies and  amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' This is at the root of the formation of beats  in the intensity and the correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In the regime of  strong laser and magnetic fields these beats are character-  ized by well-defined oscillations at the average frequency  among two generalized Rabi frequencies, modulated at  the difference of those frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  To observe beats  in the intensity both ground state populations must be  nonzero initially, ideally equal [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Similarly, for the two-  time correlations, the vector of initial conditions must  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='03061v1  [quant-ph]  8 Jan 2023  2  have at least two nonzero terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In Section IV we describe the scattered field intensity  and quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Here, beats depend only on the inter-  ference of the two upper populations in the nondegener-  ate case, with both lower populations initially nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Cross terms of the oppposite π transitions represent in-  terference in the steady state intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Then, In Section  V, using the dressed states approach, we show that the  double sideband spectrum [5] stems from a dipole-dipole  correlation with beats, where the terms of addition of sin-  gle π transitions dominate over those of the cross terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In Section VI we study Brown-Twiss photon-photon  correlations [16, 17], extending the work of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' [8] to the  nondegenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Besides the ubiquitous antibunching  effect, for weak to moderate laser drivings the interplay  of parameters, together with detuning and Zeeman split-  tings, can make for somewhat involved evolutions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=',  long decays due to optical pumping in the non-degenerate  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Again, cross terms are minor contributors to the  full correlation in the beats regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Section VII is devoted to a study of phase-dependent  fluctuations by conditional homodyne detection (CHD)  [18, 19] in both the temporal and spectral domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The  CHD method is characterized by amplitude-intensity cor-  relations (AIC), which are of third order in the field am-  plitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' When the atomic operators are decomposed into  a mean plus a noise operator the AIC is split into a  second-order term which would be a measure of squeezing  if the third-order one were negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' But the latter is  not negligible outside the weak field regime of resonance  fluorescence, which make the fluctuations non-Gaussian  and also nonclassical by the violation of classical inequal-  ities [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' We obtain the spectra of the total, second- and  third-order terms of the AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Narrow peaks in the spec-  tra reveal population trapping when detunings favour the  long term population or optical pumping of the ground  state of the more detuned transition, which in the time  domain show the above mentioned long decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The  third-order terms make up most of the beats and thus  they are non-Gaussian and nonclassical but not squeezed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In Section VIII we consider squeezing by means of the  variance of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' As usual, squeezing in resonance  fluorescence is small and restricted to weak or moderate  Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Finally, in Section IX we provide a  discussion and conclusions, and two Appendices give de-  tails on solution methods, initial conditions, and optimal  appearance of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  MODEL  The system, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 1, consists of a two-level  atom with transition J = 1/2 – J = 1/2 and states with  magnetic quantum number m = ±J,  |1⟩ = |J, −1/2⟩,  |2⟩ = |J, 1/2⟩,  |3⟩ = |J, −1/2⟩,  |4⟩ = |J, 1/2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (1)  The matrix elements are  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Scheme of the J = 1/2 – J = 1/2 atomic system  interacting with a laser driving the |1⟩ − |3⟩ and |2⟩ − |4⟩  transitions with Rabi frequency Ω and detuning ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  There  are spontaneous decay rates γ1, γ2 and γσ, vacuum-induced  coherence γ12, and Zeeman frequency splittings Bℓ and Bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  d1 = ⟨1|ˆd|3⟩ = − 1  √  3Dez,  d2 = ⟨2|ˆd|4⟩ = −d1,  d3 = ⟨2|ˆd|3⟩ =  �  2  3De−,  d4 = ⟨1|ˆd|4⟩ = d∗  3,  (2)  where D is the reduced dipole matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' We choose  the field polarization basis {ez, e−, e+} (linear, left cir-  cular, right circular), where e± = ∓(ex ± iey)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The π transitions, |1⟩ − |3⟩ and |2⟩ − |4⟩ (m = m′), are  coupled to linearly polarized light and have their dipole  moments antiparallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' On the other hand, the σ tran-  sitions, |1⟩ − |4⟩ and |2⟩ − |3⟩ (m ̸= m′), are coupled  to circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' This configuration can be  found, for example, in 198Hg+ [3], and 40Ca+ [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The level degeneracy is removed by the application of  a static magnetic field Bz along the z direction, the Zee-  man effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Note that the energy splittings gµBBz of  the upper (u) and lower (ℓ) levels are different due to  unequal Land´e g factors, gu and gℓ, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' µB is  Bohr’s magneton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The difference Zeeman splitting is  δ = (gu − gℓ)µBBz  ¯h  = gu − gℓ  gℓ  Bℓ,  (3)  where Bℓ = glµBBz/¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For 198Hg+ gu = 2/3 and gℓ = 2,  so ¯hδ = −(4/3)µBBz = −(2/3)¯hBℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The atom is driven by a monochromatic laser of fre-  quency ωL, linearly polarized in the z direction, propa-  gating in the x direction,  EL(x, t) = E0ei(ωLt−kLx)ez + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=',  (4)  thus driving only the π transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The free atomic, H0, and interaction, V , parts of the  Hamiltonian are, respectively:  H0 = ¯hω13A11 + ¯h(ω24 + Bℓ)A22 + ¯hBℓA44,  (5)  V = ¯hΩ(A13 − A24)eiωLt + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (6)  3  where Ajk = |j⟩⟨k| are atomic operators, ω13 and ω24 =  ω13 + δ are the frequencies of the |1⟩ − |3⟩ and |2⟩ − |4⟩  transitions, respectively, and Ω = E0D/  √  3 ¯h is the Rabi  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The frequencies of the other transitions are  ω23 = ω13 − δ and ω14 = ω13 − Bℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Using the unitary  transformation  U = exp [(A11 + A22)iωLt],  (7)  the Hamiltonian in the frame rotating at the laser fre-  quency is  H = U †(H0 + V )U,  = −¯h∆A11 − ¯h(∆ − δ)A22 + ¯hBℓ(A22 + A44)  +¯hΩ [(A13 − A24) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='] ,  (8)  where ∆ = ωL −ω13 is the detuning of the laser from the  |1⟩ − |3⟩ resonance transition, and ∆ − δ is the detuning  on the |2⟩ − |4⟩ transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The excited states decay either in the π transitions  emitting photons with linear polarization at rates γ1 =  γ2, or in the σ transitions emitting photons of circular  polarization at rate γσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' There is also a cross-coupling  of the excited states by the reservoir, responsible for the  quantum interference we wish to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In general, the  decay rates are written as  γij = di · d∗  j  |di||dj|  √γiγj,  i, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (9)  In particular, we have γii = γ1 = γ2 and γ13 = γ24 = γσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Also, given that d1 and d2 are antiparallel, γ12 = γ21 =  −√γ1γ2 = −γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The total decay rate is  γ = γ1 + γσ = γ2 + γσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (10)  The decays for the π and σ transitions occur with the  branching fractions bπ and bσ [5], respectively,  γ1 = γ2 = bπγ,  bπ = 1/3,  (11a)  γσ = bσγ,  bσ = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (11b)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  MASTER EQUATION  The dynamics of the atom-laser-reservoir system is de-  scribed by the master equation for the reduced atomic  density operator, ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In a frame rotating at the laser fre-  quency (˜ρ = UρU †) it is given by  ˙˜ρ = − i  ¯h[H, ˜ρ] + Lγ ˜ρ,  (12)  where −(i/¯h)[H, ˜ρ] describes the coherent atom-laser in-  teraction and Lγ ˜ρ describes the damping due to sponta-  neous emission [5, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Defining  S−  1 = A31,  S−  2 = A42,  S−  3 = A32,  S−  4 = A41,  S+  i = (S−  i )†,  (13)  the dissipative part is written as  Lγ ˜ρ = 1  2  2  �  i,j=1  γij  �  2S−  i ˜ρS+  j − S+  i S−  j ˜ρ − ˜ρS+  i S−  j  �  +γσ  2  4  �  i=3  �  2S−  i ˜ρS+  i − S+  i S−  i ˜ρ − ˜ρS+  i S−  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (14)  We now define the Bloch vector of the system as  Q ≡ (A11, A12, A13, A14, A21, A22, A23, A24,  A31, A32, A33, A34, A41, A42, A43, A44)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (15)  The equations for the expectation values of the atomic  operators, ⟨Ajk⟩ = ˜ρkj, are the so-called Bloch equations,  which we write as  d  dt⟨Q(t)⟩ = MB⟨Q(t)⟩,  (16)  where MB is a matrix of coeficients of the full master  equation, and the formal solution is  ⟨Q(t)⟩ = eMBt⟨Q(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (17)  Since we are interested only in properties of the fluores-  cence emitted in the π transitions we use the simplifying  fact, already noticed in [8], that these Bloch equations  can be split into two decoupled homogeneous sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Set 1  contains the equations for the populations and the coher-  ences of the coherently driven π transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' these are  ⟨ ˙A11⟩ = −γ⟨A11⟩ + iΩ(⟨A31⟩ − ⟨A13⟩),  ⟨ ˙A13⟩ = −  �γ  2 + i∆  �  ⟨A13⟩ − iΩ(⟨A11⟩ − ⟨A33⟩),  ⟨ ˙A22⟩ = −γ⟨A22⟩ − iΩ(⟨A42⟩ − ⟨A24⟩),  ⟨ ˙A24⟩ = −  �γ  2 + i(∆ − δ)  �  ⟨A24⟩ + iΩ(⟨A22⟩ − ⟨A44⟩),  ⟨ ˙A31⟩ = −  �γ  2 − i∆  �  ⟨A31⟩ + iΩ(⟨A11⟩ − ⟨A33⟩),  ⟨ ˙A33⟩ = γ1⟨A11⟩ + γσ⟨A22⟩ − iΩ(⟨A31⟩ − ⟨A13⟩),  ⟨ ˙A42⟩ = −  �γ  2 − i(∆ − δ)  �  ⟨A42⟩ − iΩ(⟨A22⟩ − ⟨A44⟩),  ⟨ ˙A44⟩ = γσ⟨A11⟩ + γ2⟨A22⟩ + iΩ(⟨A42⟩ − ⟨A24⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (18)  with Bloch vector  R ≡ (A11, A13, A22, A24, A31, A33, A42, A44)T  (19)  and a corresponding matrix M, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Equations (18)  do not depend on γ12, the vacuum-induced coupling of  the upper levels, but on the applied magnetic field only  through the difference of Zeeman splittings, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The steady state solutions, for which we introduce the  4  0  2  4  6  8  10  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  0  2  4  6  8  10  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  0  2  4  6  8  10  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  0  2  4  6  8  10  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  γt  γt  (d) ∆ = −2γ, δ = −4γ  (b) ∆ = 2γ, δ = −2γ  ( ) ∆ = −2γ, δ = −2γ  ⟨A22 (t)⟩  ⟨A44 (t)⟩  ⟨A11 (t)⟩  ⟨A33 (t)⟩  (a) ∆ = 0, δ = 0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Time-dependent populations ⟨A11(t)⟩ (solid-black),  ⟨A22(t)⟩ (dashed-red), ⟨A33(t)⟩ (dots-green), and ⟨A44(t)⟩  (dashed-dots-blue), with the atom initially in state |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The  parameters are: Ω = γ and (a) ∆ = δ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (b) ∆ = 2γ,  δ = −2γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (c) ∆ = δ = −2γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (d) ∆ = −2γ, δ = −4γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  short notation αjk = ⟨Ajk⟩st, are  α11 = α22 = Ω2  2D,  (20a)  α33 = Ω2 + γ2/4 + ∆2  2D  ,  (20b)  α44 = Ω2 + γ2/4 + (∆ − δ)2  2D  ,  (20c)  α13 = Ω(∆ + iγ/2)  2D  ,  (20d)  α24 = Ω(δ − ∆ − iγ/2)  2D  ,  (20e)  αkj = α∗  jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  where  D = 2Ω2 + γ2 + δ2  4  +  �  ∆ − δ  2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (21)  Note also that in the degenerate system (δ = 0) α33 =  α44 and that α31 = −α42, where the minus sign arises  from the fact that the dipole moments d1 and d2 are  antiparallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Set 2 contains the equations for the coherences of the σ  transitions and those among both upper and both lower  levels,  R2 ≡ (A12, A14, A21, A23, A32, A34, A41, A43)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (22)  The equations for their expected values do depend on Bℓ  and γ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' These coherences vanish because the σ tran-  sitions are driven incoherently (⟨{A14, A23, A32, A41}⟩),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=', by spontaneous emission, or because they are medi-  ated by those σ transitions (⟨{A12, A21, A34, A43}⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For  completeness, we write the steady state results:  α12 = α34 = α14 = α23 = 0,  αkj = α∗  jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (23)  0  1  2  3  4  5  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  0  1  2  3  4  5  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8  0  1  2  3  4  5  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8  0  1  2  3  4  5  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  α11 = α22  α33  α44  (d) ∆ = −2γ, δ = −4γ  (b) ∆ = 2γ, δ = −2γ  (a) ∆ = 0, δ = 0  ( ) ∆ = −2γ, δ = −2γ  Ω/γ  Ω/γ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Steady-state populations as a function of Rabi  frequency: α11 = α22 (dashed-red), α33 (solid-black), and  α44 (dots-blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' All other parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The properties of the fluorescence of the π transitions,  the subject matter of this article, do not depend on the  equations for Set 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Only the second- and third-order  amplitude-intensity correlations and the dipole correla-  tion for the spectrum of the σ transitions would require  the full set of Bloch equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  We gain valuable information on the nontrivial dynam-  ics of the atomic system from single-time expectation val-  ues, apparently ignored in the previous literature on the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2 we show the populations for several  particular cases, all with the atom initially in state |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In the degenerate case, δ = 0, the upper populations  reach opposite phases by the end of the first Rabi cycle,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' This is understandable since the electron oc-  cupation of, say, state |1⟩ implies not to be in state |2⟩,  and viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The same occurs for the lower popula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Next, we show three situations for the nondegen-  erate case with δ < 0 (as it is for 198Hg+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2(b)  the laser is slightly detuned above the |1⟩−|3⟩ transition,  but highly detuned from the |2⟩−|4⟩ transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' the oscil-  lations get out of phase and most of the population ends  up in state |4⟩ by optical pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2(c) the laser  is detuned below the |1⟩−|3⟩ transition, and the |2⟩−|4⟩  transition is now on resonance with the laser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' again, the  oscillations are out of phase but most of the population  ends up now in state |3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2(d) we extend the pre-  vious case but with stronger applied magnetic field, thus  the non-degeneracy is more evident;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' the large detuning  on both transitions makes it recover the opposite phases  of the degenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 3 we show the steady state populations as a  function of the Rabi frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' the other parameters are  the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For strong fields the populations  tend to be equal (1/4), but arrive at that limit at dif-  ferent rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' for instance, for large detunings on both  transitions, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 3(d), it takes larger fields, as compared  to the degenerate case, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' On the other hand,  for small detunings and weak-moderate fields, when one  5  transition is closer to resonance than the other, the lower  state of the more detuned transition is more populated,  as seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 3 (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  THE SCATTERED FIELD  In this Section we present the main dynamical and  stationary properties of the field scattered by the atom,  with emphasis on the π transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Single-Time and Stationary Properties  The positive-frequency part of the emitted field oper-  ator is [5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 21]  ˆE+(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' t) = ˆE+  free(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' t) + ˆE+  S (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' ˆt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (24)  where ˆE+  free(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' t) is the free-field part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' which does not con-  tribute to normally ordered correlations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' hence we omit  it in further calculations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' and  ˆE+  S (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' t) = −η  r  4  �  i=1  ω2  i ˆr × (ˆr × di)S−  i (ˆt)  (25)  is the dipole source field operator in the far-field zone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  where ˆt = t−r/c is the retarded time and η = (4πϵ0c2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Since ωi ≫ δ, we may approximate the four transition as  a single one ω0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (25, but cannot do so at the level  of decay rates, Rabi frequencies, and splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Making ˆr = ey the direction of observation and using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (2) we have  ˆE+  S (r, ˆt) = ˆE+  π (r, ˆt) ez + ˆE+  σ (r, ˆt) ex,  (26)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=', the fields scattered from the π and σ transitions are  polarized in the ez and ex directions, respectively, where  ˆE+  π (r, ˆt) = fπ(r)  �  A31(ˆt) − A42(ˆt)  �  ,  (27a)  ˆE+  σ (r, ˆt) = fσ(r)  �  A32(ˆt) − A41(ˆt)  �  ,  (27b)  are the positive-frequency source field operators of the π  and σ transitions, and  fπ(r) = −ηω2  1D/  √  3r,  fσ(r) =  √  2fπ(r),  (28)  are their geometric factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The intensity in the π transitions is given by  Iπ(r, ˆt) = ⟨ ˆE−  π (r, ˆt) · ˆE+  π (r, ˆt)⟩  = f 2  π(r)⟨A13(ˆt)A31(ˆt) + A24(ˆt)A42(ˆt)⟩  = f 2  π(r)⟨A11(ˆt) + A22(ˆt)⟩,  (29a)  while in the steady state is  Ist  π = f 2  π(r) [α11 + α22] = Ω2  D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (29b)  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='20  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='3  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='25  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='15  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='15  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='25  Iπ (r, t) �f 2  π (r)  Iπ (r, t) �f 2  π (r)  ⟨A11 (t)⟩  ⟨A22 (t)⟩  ⟨A11 (t)⟩  ⟨A22 (t)⟩  ⟨A11 (t)⟩  ⟨A22 (t)⟩  ⟨A11 (t)⟩  ⟨A22 (t)⟩  γt  γt  γt  γt  γt  γt  (d)  (c)  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Fluorescence intensity of the π transitions with equal  initial ground state populations, ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The other parameters are as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2: Ω = γ and (a) ∆ =  δ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (b) ∆ = 2γ, δ = −2γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (c) ∆ = δ = −2γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (d) ∆ = −2γ,  δ = −4γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The insets show the excited states populations:  ⟨A11(t)⟩ (solid), ⟨A22(t)⟩ (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Adding the excited state populations with the atom  initially in the single state |3⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 29a gives simply  Iπ(r, ˆt) = f 2  π(r)⟨A11(ˆt)⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=', without the contribution  of ⟨A22(ˆt)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' More interesting is the case where the ini-  tial condition is ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2, shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 4 (see the populations ⟨A11(t)⟩ and ⟨A22(t)⟩ in the  insets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The modulation in the intensity is reminiscent  of the quantum beats in the spontaneous decay in the V  three-level system [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' These beats are basically due  to the inability to tell from which of the π transitions  a photon comes from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' This is the standard Young-type  interference [4, 5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The main requirement is that the  initial condition for both ground states are nonzero (see  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  More interesting, though, is the case of strong resonant  laser and magnetic fields and the laser is detuned far from  the |2⟩ − |4⟩ resonance frequency, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Due  to the laser detuning, the population ⟨A22(t)⟩ has larger  frequency and smaller amplitude than that of ⟨A11(t)⟩, as  seen in the insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Remarkably well-defined wave-packets  or beats are observed due to the interference of the flu-  orescence of both π transitions with close Rabi frequen-  cies, with clear average and modulation frequencies (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The beats get scrambled with larger frequency  and amplitude differences, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Save for the decay, these beats are more like the classic  textbook ones, described by a modulation and an average  frequency, unlike the beats from spontaneous emission or  weak resonance fluorescence from two or more closely sep-  arated levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Henceforth, we reserve the moniker beats  to those due to strong applied fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Further analyses of  the beats are given in the next Sections, as they show up  6  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8  0  1  2  3  4  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0  1  2  3  4  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  Iπ (r, t)  �  f2  π (r)  Iπ (r, t)  �  f2  π (r)  ⟨A22 (t)⟩  ⟨A11 (t)⟩  ⟨A22 (t)⟩  ⟨A11 (t)⟩  γt  γt  γt  γt  (b)  (a)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Fluorescence intensity for Ω = 9γ, ∆ = 0, and (a)  δ = −8γ and (b) δ = −15γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The insets show the excited state  populations ⟨A11⟩ (solid line) and ⟨A22⟩ (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The  initial conditions are ⟨A33(0)⟩ = ⟨A44(0)⟩ = 1/2, ⟨A11(0)⟩ =  ⟨A22(0)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  also in two-time correlations with particular features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Similarly, for the σ transitions we have  Iσ(r, ˆt) = ⟨ ˆE−  σ (r, ˆt) · ˆE+  σ (r, ˆt)⟩  = f 2  σ(r)[⟨A23(ˆt)A32(ˆt) + A14(ˆt)A41(ˆt)⟩]  = f 2  σ(r)[⟨A11(ˆt) + A22(ˆt)⟩],  (30a)  Ist  σ = f 2  σ(r) [α11 + α22] ,  (30b)  also showing beats with intensity twice that of the π tran-  sitions given that f 2  σ(r) = 2f 2  π(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The field quadrature operator at any time is  ˆEπ,φ(r, ˆt) = 1  2  �  E−  π (r, ˆt)e−iφ + E+  π (r, ˆt)eiφ�  = fπ(r)(S1,φ − S2,φ),  (31)  where φ = 0, π/2 are the quadrature phases we consider,  and  S1,φ = 1  2  �  A13e−iφ + A31eiφ�  ,  (32a)  S2,φ = 1  2  �  A24e−iφ + A42eiφ�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (32b)  The mean quadrature field is given by  ⟨ ˆEπ,φ⟩st = fπ(r)  2  �  (α13 − α24) e−iφ + (α31 − α42) eiφ�  = fπ(r)Re  �  (α13 − α24) e−iφ�  (33)  = fπ(r)Re  �Ω (∆ + (iγ − δ)/2)  D  e−iφ  �  ,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Intensity and Quadrature Fluctuations  Here we introduce the intensity and quadratures of the  field in terms of atomic fluctuation operators ∆Ajk =  Ajk − ⟨Ajk⟩st, such that  ⟨AklAmn⟩ = αklαmn + ⟨∆Akl∆Amn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (34)  Only the π transitions have nonvanishing coherence  terms (α13, α24 ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The fluorescence in the σ transi-  tions is fully incoherent (α14 = α23 = 0), so its intensity  is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (30b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In the remainder of this section  we deal only with the π transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The quadrature op-  erators are then written as  ˆEπ,φ(r, ˆt) = fπ(r)[απ,φ + ∆Sπ,φ(ˆt)],  (35a)  where  απ,φ = 1  2(α31 − α42)eiφ + 1  2(α13 − α24)e−iφ,  (35b)  = Re  �Ω (∆ + (iγ − δ)/2)  D  e−iφ  �  ,  ∆Sπ,φ = 1  2(∆A31 − ∆A42)eiφ + 1  2(∆A13 − ∆24)e−iφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (35c)  From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (29b) and (34) we write the steady state  intensity in terms of products of dipole and dipole fluc-  tuation operator expectation values,  Ist  π (r) = f 2  π(r)  �  Icoh  π,0 + Iinc  π,0 + Icoh  π,cross + Iinc  π,cross  �  ,(36)  where  Icoh  π,0 = |⟨A13⟩st|2 + |⟨A24⟩st|2,  (37a)  Iinc  π,0 = ⟨∆A13∆A31⟩ + ⟨∆A24∆A42⟩,  (37b)  Icoh  π,cross = −⟨A13⟩st⟨A42⟩st − ⟨A24⟩st⟨A31⟩st  = −2Re (⟨A13⟩st⟨A42⟩st) ,  (37c)  Iinc  π,cross = −⟨∆A13∆A42⟩ − ⟨∆A24∆A31⟩  = −2Re (⟨∆A13∆A42⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (37d)  Superindices coh and inc stand, respectively, for the co-  herent (depending on mean dipoles) and incoherent (de-  pending on noise terms) parts of the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Subindex  0 stands for terms with the addition of single transition  products, giving the total intensity, while subindex cross  stands for terms with products of the two π transitions,  the steady state interference part of the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In  7  terms of atomic expectation values these intensities are:  Icoh  π,0 = |α13|2 + |α24|2  (38a)  = Ω2  4D2  �γ2  2 + ∆2 + (δ − ∆)2  �  ,  Iinc  π,0 = α11 + α22 − |α13|2 − |α24|2  (38b)  = Ω2  D2  �  2Ω2 − γ2  4 − ∆2 − δ2  �  ,  Icoh  π,cross = −2Re (α13α42)  (38c)  = Ω2  2D2  �γ2  4 + ∆(∆ − δ)  �  ,  Iinc  π,cross = 2Re (α13α42) = −Icoh  π,cross,  (38d)  The sum of these terms is, of course, the total intensity,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (29a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' As usual in resonance fluorescence, the coher-  ent and incoherent intensities are similar only in the weak  field regime, Ω ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Here, in particular, the term Iinc  π,0  (no interference) becomes much larger than the others  for strong driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Degree of Interference - Coherent Part  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' [5], a measure of the effect of interference in  the coherent part of the intensity was as  Icoh  π,0 + Icoh  π,cross = Icoh  π,0 (1 + C(δ)),  C(δ) = Icoh  π,cross  Icoh  π,0  =  γ2/4 + ∆(∆ − δ)  γ2/4 + δ2/4 + (∆ − δ/2)2 , (39)  independent  of  the  Rabi  frequency  and  shown  in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Some special cases are found analytically:  C(0) = 1,  δ = 0,  (40a)  C(δ0) = 0,  δ0 = ∆[1 + (γ/2∆)2],  (40b)  C(δmin) =  −1  1 + γ2/2∆2 ,  δmin = 2∆[1 + (γ/2∆)2],  (40c)  C(δ±  1/2) = 1/2,  δ±  1/2 = −∆ ±  �  3∆2 + (γ2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (40d)  In the degenerate case, C(δ = 0) = 1 means perfect  constructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' That is because at δ = 0 both  π transitions (and both σ transitions) share the same  reservoir environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Increasing δ the reservoir overlap  decreases, so is the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Negative values of C  indicate destructive interference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' its minimum is given  by δmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For large detunings, ∆2 ≫ γ2 we have  δ0 = ∆,  δmin = 2∆,  δ±  1/2 = −∆ ±  √  3 |∆|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (40e)  We have used the special cases δ = {0, δ0, δmin} as a  guide to obtain many of the figures in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  20  10  0  10  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  δ/γ  ∆ = −5γ  K(δ)  C(δ)  ∆ = −2γ  ∆ = 0  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Relative weight of the interference terms C(δ) (a)  and K(δ) (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In (b) Ω = γ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For 198Hg+, δ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Degree of Interference - Incoherent Part  Likewise, we define a measure, K(δ), of the effect of  interference in the intensity’s incoherent part,  Iinc  π,0 + Iinc  π,cross = Iinc  π,0(1 + K(δ)),  K(δ) = Iinc  π,cross  Iinc  π,0  =  γ2/4 + ∆(∆ − δ)  2 [γ2/4 + δ2 + ∆2 − 2Ω2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (41)  Unlike C(δ), K(δ) also depends on the Rabi frequency  as Ω−2, since fluctuations increase with laser intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Special cases are:  K(0) =  γ2/4 + ∆2  2 [γ2/4 + ∆2 − 2Ω2],  δ = 0,  (42a)  K(δ) = 0,  δ = ∆ + γ2  4∆  or  Ω ≫ γ, ∆, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (42b)  The behavior of K(δ) with ∆ is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' It is ba-  sically required that ∆ ∼ Ω in order to preserve the  shape seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 6(b), in which case the minima for  C(δ) and K(δ) are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' On-resonance, for ex-  ample, Ω should be no larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='35γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Also, we can  infer that the beats are little affected by the interference  term unless ∆ >∼ Ω ≫ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  TWO-TIME DIPOLE CORRELATIONS AND  POWER SPECTRUM  The resonance fluorescence spectrum of the J = 1/2 →  J = 1/2 atomic system was first considered in [3] and  then very thoroughly in [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Thus, here we only con-  sider basic definitions and issues related to the observa-  tion of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The stationary Wiener-Khintchine power spectrum is  given by the Fourier transform of the field autocorrelation  function  Sπ(ω) = Re  � ∞  0  dτe−iωτ⟨ ˆE−  π (0) ˆE+  π (τ)⟩,  (43)  8  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='35  20  10  0  10  20  ω/γ  Sinc  π (ω) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' units)  γτ  �  ∆E−  π (0) ∆E+  π (τ)  � �  f2  π (r)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Dipole correlation function ⟨∆ ˆE−  π (0)∆ ˆE+  π (τ)⟩ for  Ω = 9γ, δ = −8γ, and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The inset shows the corre-  sponding incoherent spectrum Sinc  π  (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  such that  � ∞  −∞ Sπ(ω)dω = Ist  π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' By writing the atomic  operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (27a) as Ajk(t) = αjk + ∆Ajk(t), we  separate the spectrum in two parts: a coherent one,  Scoh  π  (ω) = Re  � ∞  0  e−iωτdτ  �  Icoh  π,0 + Icoh  π,cross  �  = π  �  Icoh  π,0 + Icoh  π,cross  �  δ(ω)  = πΩ2  D2  �  γ2  4 +  �  ∆ − δ  2  �2�  δ(ω),  (44)  due to elastic scattering, where Icoh  π,0 and Icoh  π,cross are given  by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (38) (a) and (c), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' and an incoherent  part,  Sinc  π (ω) = Re  � ∞  0  dτe−iωτ⟨∆ ˆE−  π (0)∆ ˆE+  π (τ)⟩,  specifically,  Sinc  π (ω) = Re  � ∞  0  dτe−iωτ [⟨∆A13(0)∆A31(τ)⟩  +⟨∆A24(0)∆A42(τ)⟩ − ⟨∆A13(0)∆A42(τ)⟩  −⟨∆A24(0)∆A31(τ)⟩] ,  (45)  due to atomic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' An outline of the numerical  calculation is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The dipole correlation ⟨ ˆE−  π (0) ˆE+  π (τ)⟩ and the incoher-  ent spectrum in the strong driving regime and strong  nondegeneracy (large δ) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The spec-  trum (inset) displays a central peak and two pairs of  Mollow-like-sidebands [22] with peaks at the Rabi side-  bands ±Ω1 and ±Ω2, while the correlation features de-  caying quantum beats due to the closeness of the Rabi  peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  As usual in the strong-field regime, the dressed system  approach allows to discern the origin of the peaks from  the transitions among the dressed states, to find their  positions [5], and thus find the frequencies of the beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The generalized Rabi frequencies are  Ω1 = E+  1 − E−  1 =  �  4Ω2 + ∆2,  (46a)  Ω2 = E+  2 − E−  2 =  �  4Ω2 + (δ − ∆)2,  (46b)  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Eigenvalues of matrix M/γ and initial conditions  of the correlations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (45) for Ω = 9γ and ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Eigenvalues  δ = −8γ  δ = −15γ  λ1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='749386 + 0i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='836531 + 0i  λ2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='583099 − 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0094i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='583308 − 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='9981i  λ3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='583099 + 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0094i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='583308 + 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='9981i  λ4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='569785 − 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6808i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5492 − 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4257i  λ5  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='569785 + 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6808i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5492 + 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4257i  λ6  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5 + 0i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5 + 0i  λ7  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='444846 + 0i  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='398452 + 0i  λ8  0 + 0i  0 + 0i  Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  ⟨∆A13∆A31⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='20836 + 0i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='14734 + 0i  ⟨∆A24∆A42⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='174014 + 0i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='086982 + 0i  ⟨∆A13∆A42⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='000134 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='002146i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='000067 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='002011i  ⟨∆A24∆A31⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='000134 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='002146i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='000067 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='002011i  where  E±  1 = −∆  2 ± 1  2  �  4Ω2 + ∆2,  (47a)  E±  2 = Bℓ + δ − ∆  2  ± 1  2  �  4Ω2 + (δ − ∆)2,  (47b)  are the eigenvalues of the Hamiltonian (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Due to the  spontaneous decays these frequencies would have to be  corrected, but they are very good in the relevant strong  field limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Indeed, we notice that Ω1 and Ω2 are very  close to the imaginary parts of the eigenvalues λ2,3 and  λ4,5, respectively, of matrix M, shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The beats are the result of the superposition of waves  at the frequencies Ω1 and Ω2 of the spectral sidebands,  with average frequency  Ωav = Ω2 + Ω1  2  =  �  4Ω2 + (δ − ∆)2 +  √  4Ω2 + ∆2  2  ,  (48)  and beat or modulation frequency  Ωbeat = Ω2 − Ω1  2  =  �  4Ω2 + (δ − ∆)2 −  √  4Ω2 + ∆2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (49)  Now, we can identify the origin and modulation fre-  quency of the beats in the time-dependent intensity,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (29a), since the excited state populations ⟨A11(t)⟩  and ⟨A22(t)⟩ oscillate at the generalized Rabi frequen-  cies Ω1 and Ω2, respectively, with initial conditions  given by a nonzero superposition of ground state pop-  ulations at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In the case of the dipole correlation  ⟨ ˆE−  π (0) ˆE+  π (τ)⟩, however, the initial conditions are given  by products of stationary atomic expectation values,  most of them the coherences α13, α24, which become very  small in the regime of beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Thus, as seen in Table I,  the terms ⟨∆A13(0)∆A31(τ)⟩ and ⟨∆A24(0)∆A42(τ)⟩ are  9  0  3  6  9  12  15  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  0  3  6  9  12  15  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  0  3  6  9  12  15  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  γτ  ∆ = −2γ, δ = −4γ  ∆ = −2γ, δ = −2γ  ∆ = 2γ,  δ = −2γ  ∆ = 0,  δ = 0  ( ) Ω = γ  (a) Ω = γ/4  (b) Ω = γ/2  g(2)  π (τ)  g(2)  π (τ)  g(2)  π (τ)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Photon correlations for (a) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='25γ, (b) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5γ  and (c) Ω = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The pairs of values (∆, δ) are the same as  those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  much larger than the cross terms ⟨∆A13(0)∆A42(τ)⟩ and  ⟨∆A24(0)∆A31(τ)⟩, so the beats are basically due to the  interference of those dominant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  PHOTON-PHOTON CORRELATIONS  The standard method to investigate intensity fluctua-  tions of a light source uses Brown-Twiss photon-photon  correlations [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The conditional character of this  type of measurement makes it nearly free of detector in-  efficiencies, unlike a single-detector measurement of the  photoelectron probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' [8] the  correlations of two photons from the π transitions were  studied, albeit only for the degenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In this paper  we extend it to the case of nondegenerate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' These  correlations are defined as  g(2)  π (τ) =  G(2)  π (τ)  G(2)  π (τ → ∞)  (50)  where, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (27a) for the field operators,  G(2)  π (τ) = ⟨ ˆE−  π (0) ˆE−  π (τ) ˆE+  π (τ) ˆE+  π (0)⟩  = f 4  π(r)⟨[A13(0) − A24(0)][A11(τ) + A22(τ)]  ×[A31(0) − A42(0)]⟩,  (51a)  and  G(2)  π (τ → ∞) =  �  Ist  π  �2 = f 4  π(r) (α11 + α22)2 (51b)  is the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' G(2)  π (τ) can be further re-  duced, since ⟨A13Ajk(τ)A42(0)⟩ = ⟨A24Ajk(τ)A31(0)⟩ =  0, due to having vanishing initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Figure 8 shows g(2)  π (τ) for moderate values of the Rabi  frequency (near saturation) and the same sets of detun-  ings ∆ and δ of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' As usual in resonance fluores-  cence, the correlation shows antibunching, g(2)  π (0) = 0,  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  0  2  4  6  8  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  g(2)  π (τ)  γτ  γτ  g(2)  π (τ)  (b) δ = −10γ  (d) δ = −15γ  ( ) δ = −12γ  (a) δ = −8γ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Photon-photon correlations showing beats in the  strong field limit, Ω = 9γ, ∆ = 0, and large Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The horizontal line helps to see that the wave packet is slightly  rised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  that is, a single atom cannot emit two photons simultane-  ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Unlike the two-level atom resonance fluorescence,  the correlation is not simply the normalized population  of the excited state, nor it is only the sum of the cor-  relations of each single π transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Besides the terms  ⟨A13(0)A11(τ)A31(0)⟩ and ⟨A24(0)A22(τ)A42(0)⟩, which  are also out of phase, as seen from the time-dependent  populations of their excited states (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2), there are six  cross terms in the full correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In the nondegenerate  case the multiple contributions cause in some cases quite  irregular evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For instance, as we will see in the  next Section, the slow decay of the correlation when the  laser drives the atom near saturation, but below the ω13  resonance transition, is related to a very narrow peak in  the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The case of strong driving and large nondegeneracy  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 9, featuring quantum beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' There are  several effects resulting from the increase of the nonde-  generacy factor δ: (i) the larger number of visible wave  packets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (ii) both average and beat frequencies approach  one another, so the wave packets get shorter for larger  photon-pair intervals τ, containing very few of the fast  oscillations, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 9(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (iii) the wavepackets are  initially slightly lifted above the g(2)(τ) = 1 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  QUADRATURE FLUCTUATIONS  Squeezing, the reduction of noise in one quadrature  below that of a coherent state at the expense of the  other, is the hallmark of phase-dependent fluctuations  of the electromagnetic field [cite].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' It is usually measured  by balanced homodyne detection (BHD), but low quan-  tum detector efficiency degrade the weak squeezing pro-  duced in resonance fluorescence and cavity QED systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  One alternative our group has used is conditional homo-  dyne detection (CHD) [18, 19], which correlates a quadra-  ture amplitude on the cue of an intensity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  CHD measures a third-order amplitude-intensity corre-  10  lation (AIC) which, in the weak driving limit is reduced  to the second-order one and that allows for measuring  squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Being a conditional measurement it is nearly  free of detector inefficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  While the original goal of CHD was to measure the  weak squeezing in cavity QED [18, 19], it was soon re-  alized that nonzero third-order fluctuations of the am-  plitude provide clear evidence of non-Gaussian fluctua-  tions and higher-order field nonclassicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In the present  work the fluctuations are mainly third-order ones, due to  near and above saturation excitation, and violate classi-  cal bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' We thus explore the phase-dependent fluctu-  ations under conditions of quantum interference following  our recent work [20, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Amplitude-Intensity Correlations  In CHD a quadrature’s field Eφ is measured by BHD  on the cue of photon countings in a separate detector,  where φ = 0, π/2 is the phase of the local oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' This  is characterized by a correlation among the amplitude  and the intensity of the field,  hπ,φ(τ) =  Hπ,φ(τ)  Hπ,φ(τ → ∞),  (52)  where  Hπ,φ(τ) = ⟨: ˆE−  π (0) ˆE+  π (0) ˆEπ,φ(τ) :⟩,  (53a)  the dots :: indicating time and normal operator orderings,  and  Hπ,φ(τ → ∞) = Ist  π ⟨Eπ,φ⟩st  (53b)  = f 3  π(r) [α11 + α22] Re  �  (α13 − α24) e−iφ�  = f 3  π(r) Ω3  D2 Re  �  (∆ + (iγ − δ)/2) e−iφ�  is the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  For the sake of concreteness, in this Section we limit  our discussion to the out-of-phase quadrature, φ = π/2,  which is the one that features squeezing when ωL = ω13,  that is ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' We do consider, however, squeezing in the  in-phase quadrature φ = 0 in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' VIII on the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In several atom-laser systems hπ,φ(τ) has been proven  to be time-asymmetric [20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' This is not the case with  the J = 1/2 → J = 1/2 system so we limit the analysis  to positive intervals τ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Omitting the geometrical  factor f 3  π(r), which is later cancelled by normalization,  we have  Hπ,φ(τ) = ⟨ ˆE−  π (0) ˆEπ,φ(τ) ˆE+  π (0)⟩  = Re  �  e−iφ⟨A13(0)[A13(τ) − A24(τ)]A31(0)  +A24(0)[A13(τ) − A24(τ)]A42(0)⟩} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (54)  Note that Hπ,φ(0) = 0 meaning that, like antibunching  in g(2), the atom has to build a new photon wavepacket  after one has been emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The AIC suggests nontrivial behavior when we take  dipole fluctuations into account, that is, when the atomic  operators are split into their mean plus noise, Ajk =  αjk + ∆Ajk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' upon substitution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (54) we get  Hπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ) = Ist  π ⟨Eπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ⟩st + H(2)  π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ) + H(3)  π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (55)  or in normalized form as  hπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ) = 1 +  H(2)  π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ)  Ist  π ⟨Eπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ⟩st  +  H(3)  π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ)  Ist  π ⟨Eπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ⟩st  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (56)  where  H(2)  π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ) = 2Re  �  ⟨ ˆE+  π ⟩st⟨∆ ˆE−  π (0)∆ ˆEπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ)⟩  �  = Re  �  (α31 − α42) [⟨(∆A13(0) − ∆A24(0))  �  ∆A13(τ) − ∆A24(τ))⟩e−iφ  +⟨(∆A13(0) − ∆A24(0))  �  ∆A31(τ)⟩ − ∆A42(τ))⟩eiφ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (57)  H(3)  π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ) = ⟨∆ ˆE−  π (0)∆ ˆEπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='φ(τ)∆ ˆE+  π (0)⟩  = Re  �  eiφ⟨[∆A13(0) − ∆A24(0)] [∆A31(τ) − ∆A42(τ)] [∆A31(0) − ∆A42(0)]⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (58)  The initial conditions of the correlations are given in Ap-  pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  From hπ,π/2(0) = 0 we can obtain analytically the ini-  tial values of the second- and third-order terms,  h(2)  π,π/2(0) = 1 − (2∆ − δ)2 + γ2  2D  ,  (59)  h(3)  π,π/2(0) = (2∆ − δ)2 + γ2  2D  − 2,  (60)  where D is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  11  Being the AIC a function of odd-order in the field am-  plitude we rightly expect a richer landscape than that  of the intensity correlations, more so when one considers  quantum interference and the complex parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  For instance, the correlation can take on not only nega-  tive values but break classical bounds [18, 19]:  0 ≤ hφ(τ) − 1 ≤ 1 ,  (61a)  |h(2)  φ (τ) − 1| ≤ |h(2)  φ (0) − 1| ≤ 1 ,  (61b)  where the second line is valid only for weak fields such  that h(3)  φ (τ) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' These classical bounds are stronger cri-  teria for nonclassicality of the emitted field than squeezed  light measurements, the more familiar probing of phase-  dependent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' A detailed hierarchy of nonclas-  sicality measures for higher-order correlation functions is  presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' [20] an inequality was  obtained that considers the full hφ(τ) by calculating the  AIC for a field in a coherent state,  −1 ≤ hφ(τ) ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (62)  For a meaningful violation of Poisson statistics, hφ(τ)  must be outside these bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Also, hφ(τ) is a measure of non-Gaussian fluctuations,  here of third-order in the field fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Resonance  fluorescence is a particularly strong case of non-Gaussian  noise by being a highly nonlinear stationary nonequilib-  rium process [20, 23, 24, 27, 28], thanks also to its small  Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' This makes resonance fluorescence unsuit-  able to a quasiprobability distribution approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Fluctuations Spectra  Since quadrature fluctuations, such as squeezing, are  often studied in the frequency domain we now define the  spectrum of the amplitude-intensity correlations:  Sπ,φ(ω) = 8γ1  � ∞  0  dτ cos (ωτ) [hπ,φ(τ) − 1]  (63)  which, following Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (52) and (55), can be decomposed  into terms of second- and third-order in the dipole fluc-  tuations  S(q)  π,φ(ω) = 8γ1  � ∞  0  dτ cos (ωτ)h(q)  π,φ(τ),  (64)  where q = 2, 3, so that Sπ,φ(ω) = S(2)  π,φ(ω) + S(3)  π,φ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  As mentioned above, the AIC was devised initially to  measure squeezing without the issue of imperfect detec-  tion efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Obviously, hπ,φ(τ) and Sπ,φ(ω) are not  measures of squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' They measure a third-order mo-  ment in the field’s amplitude, while squeezing is a second-  order one in its fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The so-called spectrum  of squeezing is the one for q = 2, with the advantage  of the AIC of not depending on the efficiency of detec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Squeezing is signaled by frequency intervals where  0  2  4  6  8  10  12  0  2  4  6  10  5  0  5  10  0  1  2  0  2  4  6  8  10  12  0  2  4  6  10  5  0  5  10  1  0  1  2  0  2  4  6  8  10  12  0  2  4  6  10  5  0  5  10  1  0  1  2  Sπ,π/2 (ω)  hπ,π/2 (τ)  (b) Ω = γ/2  γτ  ω/γ  (a) Ω = γ/4  (d) Ω = γ/4  (e) Ω = γ/2  (f  ) Ω = γ  ( ) Ω = γ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Amplitude-intensity correlations (left panel) and  spectra (right panel) for the φ = π/2 quadrature in the weak-  moderate field limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Parameters and line styles are the same  as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 8: ∆ = δ = 0 (solid-black);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' ∆ = 2γ and δ = −2γ  (dots-red);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' ∆ = −2γ and δ = −2γ (dashed-green);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' ∆ = −2γ  and δ = −4γ (dot-dashed-blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  S(2)  π,φ(ω) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' As a further note, the full incoherent spec-  trum, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (45), can be obtained by adding the squeezing  spectra of both quadratures [29],  Sinc  π (ω) =  1  8γ1  �  S(2)  π,0(ω) + S(2)  π,π/2(ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (65)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Results  We now show plots of the AICs and their spectra in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 10-12 for the φ = π/2 quadrature and the same sets  of detunings ∆, δ of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2, and weak to moderate Rabi  frequencies, γ/4 < Ω < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' With the three parameters Ω,  ∆, and δ, the landscape of effects is vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  We first notice a few general features seen in hπ,π/2(τ),  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  With increasing Rabi frequencies, detunings,  and Zeeman splittings we observe the clear breakdown of  the classical inequalities besides the one at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Cor-  respondingly, in the spectra, the extrema get displaced  and broadened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Now, we want to single out the case  of nondegeneracy with small detuning on the |1⟩ − |3⟩  transition but large on the |2⟩ − |4⟩ one, ∆ = −δ = 2γ  (green-dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For weak field, Ω = γ/4, the AIC  does not have a regular evolution for short times but it  does decay very slowly, with a correponding very narrow  spectral peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The slow decay is also clearly visible in the  photon correlation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' As we mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' III  regarding Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2b, state |4⟩ ends up with a large portion  of the steady state population due to optical pumping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  not quite a trapping state, so there is no electron shelv-  ing per se, as argued in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' This effect is washed out for  larger Rabi frequencies, which allow for faster recycling  of the populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' To a lesser degree, slow decay and  sharp peak occur for opposite signs of ∆ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  12  0  2  4  6  8  10  2  1  0  1  2  3  4  10 -8  6  4  2  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8  0  2  4  6  8  10  2  1  0  1  2  3  4  10 -8  6  4  2  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8  0  2  4  6  8  10  1  0  1  2  3  10 -8  6  4  2  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='0  (f  ) Ω = γ  ( ) Ω = γ  (b) Ω = γ/2  (e) Ω = γ/2  (a) Ω = γ/4  (d) Ω = γ/4  h(2)  π,π/2 (τ)  ω/γ  γτ  S(2)  π,π/2 (ω)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Second-order component of the AIC and spectra of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  0  2  4  6  8  10  1  0  1  2  10 -8 -6 -4 -2  0  2  4  6  8  10  0  1  2  0  2  4  6  8  10  1  0  1  2  10 -8 -6 -4 -2  0  2  4  6  8  10  1  0  1  2  0  2  4  6  8  10  1  0  1  2  3  10 -8 -6 -4 -2  0  2  4  6  8  10  2  1  0  1  (f  ) Ω = γ  ( ) Ω = γ  (b) Ω = γ/2  (e) Ω = γ/2  (a) Ω = γ/4  (d) Ω = γ/4  h(3)  π,π/2 (τ)  ω/γ  γτ  S(3)  π,π/2 (ω)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Third-order component of the AIC and spectra of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The splitting of the AIC and spectra into components  of second and third order in the fluctuations, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 11, 12,  helps to understand better the quadrature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  For the second-order ones we have the squeezing spectra:  around ω = 0 for ∆ = 0 and small Rabi frequencies,  Ω < γ/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' and in sidebands for larger detunings, Rabi  frequencies and Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In h(2)  π,π/2(τ) there is a  reduction in amplitudes and nonclassicality for increasing  Rabi frequencies except for the case of oppposite signs of  detuning and difference Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Note that the  sharp spectral peak in the latter case takes up most of  the corresponding peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' This is because both  π transitions are largely detuned from the laser, keeping  Ω small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Increasing the laser strength the third-order effects  overcome the second-order ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For instance, regarding  the size of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Also, a comparison of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 11  0  2  4  6  8  10  10  0  10  20  40  20  0  20  40  10  5  0  5  10  0  2  4  6  8  10  10  0  10  20  40  20  0  20  40  10  5  0  5  10  0  2  4  6  8  10  10  0  10  20  40  20  0  20  40  10  5  0  5  10  0  2  4  6  8  10  10  0  10  20  40  20  0  20  40  5  0  5  hπ,π/2 (τ)  h  (2)  π,π/2 (τ)  h  (3)  π,π/2 (τ)  S  (3)  π,π/2 (τ)  S  (2)  π,π/2 (τ)  Sπ,π/2 (τ)  γτ  ω/γ  (b)  (f  )  ( )  (g)  (d)  (h)  (a)  (e)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  AIC and spectra for Ω = 9γ, ∆ = 0, (a,e) δ = −8γ,  (b,f) δ = −10γ, (c,g) δ = −12γ, (d,h) δ = −15γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Lines are:  full AIC and spectra (solid-black), second-order (dots-red),  and third-order (dashed-blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  and 12 shows that h(3)  φ (τ) is mainly responsible for the  breakdown of the classical bounds when the driving field  is on or above saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Moreover, we see that the  slow-decay–sharp-peak is mainly a third-order effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  To close this Section, the AIC and spectra for very  strong fields and large Zeeman splittings, Ω, |δ| ≫ γ are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The AIC shows beats as in the photon  correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Unlike those in g(2)(τ), these wavepack-  ets oscillate around h(τ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Because the regime  is that of strong excitation the third-order component  clearly dominates, making the fluorescence notably non-  Gaussian, and clearly violates the classical inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The spectral peaks are localized around the Rabi frequen-  cies ±Ω1, ±Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Studies of the spectrum of squeezing for  the J = 1/2 − J = 1/2 system were reported in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Those authors choose ±Ω1, ±Ω2 with a less strong laser  but large detuning and large Zeeman splittings, observ-  ing the double sidebands, but no mention or hint of beats  was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  VARIANCE  The variance is a measure of the total noise in a  quadrature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' it is defined as  Vφ = ⟨: (∆Eφ)2 :⟩ = Re  �  e−iφ⟨∆ ˆE−∆ ˆEφ⟩st  �  , (66)  13  and is related to the spectrum of squeezing as  Vφ =  1  4πγη  � ∞  −∞  dωS(2)  φ (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (67)  where η is the detector efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The maximum value  of Vφ is 1/4, obtained when there is very strong driving,  when almost all the emitted light is incoherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Negative  values of the variance are a signature of squeezing but,  unlike the quadrature spectra, the squeezing is the total  one in the field, independent of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  For the π transitions we have  Vπ,φ = f 2  π(r)  2  Re  �  −(α13 − α24)2e−2iφ  +(α11 + α22 − |α13 − α24|2)  �  ,  (68)  = f 2  π(r)  2  Ω2  D  �  1 − [(2∆ − δ) cos φ + γ sin φ]2  2D  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (69)  For φ = π/2 and φ = 0 we have, respectively,  Vπ,π/2 = f 2  π(r)  2  Ω2  D  �  1 − γ2  2D  �  ,  (70a)  Vπ,0 = f 2  π(r)  2  Ω2  D  �  1 − (2∆ − δ)2  2D  �  ,  (70b)  where D is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 14 we plot the variances of the out-of-phase  φ = π/2 (left panel) and in-phase φ = 0 (right panel)  quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The interplay of parameters is a complex  one, but we mostly use the ones of previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For  φ = π/2 and ∆ = 0, as usual in resonance fluorescence  systems, squeezing is restricted to a small range of Rabi  frequencies, detunings, and Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For φ = 0  nonzero laser or Zeeman detunings are necessary to pro-  duce squeezing, with a strong dependence on their sign:  on-resonance (not shown) there is no squeezing, as for  a two-level atom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 14(d) the laser is tuned be-  low that transition, ∆ = −2γ, and there is no squeezing  (positive variance) but the variance is reduced for large  δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 14(e) the laser is tuned above the transition,  ∆ = −2γ, and there is squeezing for larger Rabi frequen-  cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Large values of δ tend to reduce the variance, be it  positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Out-of-phase quadrature  We now discuss a complementary view of the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  For φ = π/2 we can identify the Rabi frequency interval  within which squeezing takes place,  0 < Ω < 1  2  �  γ2/2 − δ2/2 − 2(∆ − δ/2)2,  (71)  and the Rabi frequency for maximum squeezing is  ˜Ωπ/2 = 1  2  �  γ4/2 − 2[(δ − ∆)2 + ∆2]2  3γ2 + 2[(δ − ∆)2 + ∆2]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (72)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content='04  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content='01  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='01  3  2  1  0  1  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='04  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2  Vπ,π/2/f2  π(r)  Vπ,0/f2  π(r)  Ω/γ  Ω/γ  Ω/γ  ∆/γ  ∆/γ  Ω/γ  ( )  (a)  (b)  (d)  (e)  (f  )  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Variance of the quadratures of the fluorescence of  the π transitions: left panel for φ = π/2 and right panel for  φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (a,b,d,e) as a function of Rabi frequency and (c,f)  as a function of detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In all cases δ = 0 is given by  a solid-black line, and δ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='5γ by a dashed-red line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' the  dotted-blue line is δ = −2γ in (a,b,d,e) and δ = −γ in (c,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Additionally, (a) ∆ = 0, (b) ∆ = −2γ, (c) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='2γ, (d)  ∆ = 0, (e) ∆ = 2γ, (f) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='8γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Thus, the variance at ˜Ωπ/2 is  V  (˜Ωπ/2)  π,π/2 (∆ = 0, δ) = f 2  π(r)  16  (γ4/2 − 2δ4)(δ2 − γ2)  γ2(γ2 + 2δ2)(δ2 + γ2),  (73a)  for ∆ = 0 and |δ/γ| < 1/  √  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  V  (˜Ωπ/2)  π,π/2 (∆, δ = 0) = f 2  π(r)  16  (γ4/2 − 8∆4)(4∆2 − γ2)  γ2(γ2 + 4∆2)2  ,  (73b)  for δ = 0 and |∆/γ| < 1/  √  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' and the maximum total  squeezing is obtained at ∆ = δ = 0,  V  (˜Ωπ/2)  π,π/2 (0, 0) = −f 2  π(r)  32 ,  ˜Ωπ/2 =  γ  2  √  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (73c)  For φ = π/2 squeezing is limited to elliptical regions  of weak driving and small detunings ∆ and δ:  2δ2 + 8Ω2 < γ2,  ∆ = 0,  (74a)  4∆2 + 8Ω2 < γ2,  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (74b)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In-phase quadrature  For φ = 0, squeezing is obtained in the Rabi frequency  interval, for δ = 0,  0 < Ω <  1  √  2  �  ∆2 − γ2/4,  |∆| > γ/2,  (75)  14  with maximum squeezing at the Rabi frequency  ˜Ω0 =  1  2  √  2  �  16∆2 − γ2  12∆2 + γ2 ,  (76)  requiring finite detuning from both π transitions (∆ ̸= 0)  and stronger driving, Ω ∼ γ [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 14(d)-(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Thus, the variance at ˜Ω0 is  V (˜Ω0)  π,0 (δ) = −f 2  π(r)  128  4∆2 − γ2  ∆2(4∆2 + γ2),  |∆| ≥ γ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (77)  This expression gets the asymptotic value  lim  ∆→∞ V (˜Ω0)  π,0  = −f 2  π(r)  32 ,  (78)  which is the same as that for the π/2 quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The  region for squeezing obeys the relation  4∆2 − 8Ω2 < γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (79)  So, to obtain squeezing in this quadrature it is necessary  to have detunings ∆ > γ/4 for any Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  DISCUSSION AND CONCLUSIONS  We have studied several properties of the resonance  fluorescence of the π transitions in a J = 1/2 − J = 1/2  angular momentum atomic system driven by a linearly  polarized laser field and a magnetic field along the π tran-  sition to lift the level degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Interference among  the various transition amplitudes create a rich landscape  of effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Most notable among our results is the observa-  tion of quantum beats when the atom is subject to large  laser and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In this regime, two close Rabi  frequencies interfere, giving rise to a well-defined modu-  lation of the fast oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' These Rabi frequencies are  the source of the two pairs of sidebands in the incoherent  part of the power spectrum [5] and in the squeezing spec-  trum [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' We studied beats in the total intensity and  two-time functions such as the dipole-dipole, intensity-  intensity and intensity-amplitude correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  In the  beats’ regime the role of vacuum-induced coherence is  small because the upper levels are very separated due to  very large difference Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Before the beats we considered the previously over-  looked time-dependent populations and reviewed aspects  of the known stationary ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The fact that the upper  state populations evolve out of phase should not be a  surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  This, and nonzero initial population of both  ground states (in contrast to nonzero populations of ex-  cited states for spontaneous emission), are major factors  in the interference among the terms in the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Ex-  cept for very strong laser fields, the steady state popula-  tions depend strongly on the difference Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The AIC also permits to quantify the degree of non-  Gaussianity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' the fluctuations of third-order in the field  quadrature amplitude due to strong atom-laser nonlin-  earity dominate over the second-order ones with strong  driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The beats are in the strongly non-Gaussian  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The correlations show nonclassical features of the fluo-  rescence light such as antibunching, g(2)(0) = 0, and vio-  lation of classical inequalities in the amplitude-intensity  correlations, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (61 -62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' We studied squeezing using  the variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=', the total noise in a quadrature, as  well as using the second-order part of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' In  the regime of beats there is squeezing, near the effective  Rabi frequencies, but none in the total noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  For a system with many parameters the interplay  among them is a complex one, making the interpretation  of results nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Thus, for most of our plots we chose  parameters in two groups: i) where they are relatively  small, Ω, ∆, δ ∼ γ, chosen to illustrate several degrees of  vacuum-induced coherence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' and ii) where they are large,  Ω, ∆, δ ≫ γ, and quantum beats are revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Overall,  particular care must be taken regarding detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' On  the one hand, large difference Zeeman splitting means  that the excited levels would be very separated and in-  teract with different frequency portions of the reservoir,  hence diminishing the vacuum-induced coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  On  the other, large laser-atom detunings, which might in-  crease the VIC, mean reduced fluorescence rates, which  may also be detrimental in measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The beats,  then, would be better observed if ∆ ≤ γ and δ of just  several γ in the strong field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  ACKNOWLEDGMENTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Ricardo Rom´an-Ancheyta and  Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Ir´an Ramos-Prieto for useful comments at an early  stage of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' ADAV thanks CONACYT, Mexico,  for scholarship No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 804318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  ORCID  numbers:  H´ector  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Castro-Beltr´an  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='org/0000-0002-3400-7652, Octavio de los  Santos-S´anchez https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='org/0000-0002-4316-0114,  Luis Guti´errez https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='org/0000-0002-5144-4782,  Appendix A: Time-Dependent Matrix Solutions and  Spectra  The two-time photon correlations under study have  the general form ⟨W(τ)⟩ = ⟨O1(0)R(τ)O2(0)⟩, where  R is the Bloch vector and O1,2 are system operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  The same applies to correlations of fluctuation operators  ∆R, ∆O1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Using the quantum regression formula [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  the correlations obey the equation  ⟨ ˙W(τ)⟩ = M⟨W(τ)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A1)  which has the formal solution  ⟨W(τ)⟩ = eMτ⟨W(0)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A2)  where M is given by  15  M =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  −γ  −iΩ  0  0  iΩ  0  0  0  −iΩ −  � γ  2 + i∆  �  0  0  0  iΩ  0  0  0  0  −γ  iΩ  0  0  −iΩ  0  0  0  iΩ  −  � γ  2 + i(∆ − δ)  �  0  0  0  −iΩ  iΩ  0  0  0  −  � γ  2 − i∆  �  −iΩ  0  0  γ1  iΩ  γσ  0  −iΩ  0  0  0  0  0  −iΩ  0  0  0  −  � γ  2 − i(∆ − δ)  �  iΩ  γσ  0  γ2  −iΩ  0  0  iΩ  0  �  �  �  �  �  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A3)  Also, spectra of stationary systems can be evaluated  more effectively using the above formal approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  Be  g(τ) = ⟨W(τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Then, a spectrum is calculated as  S(ω) ∝  � ∞  0  cos ωτ g(τ) dτ =  � ∞  0  cos ωτ eMτg(0) dτ  = Re  � ∞  0  e−(iω1−M)τg(0) dτ  = Re  �  (iω1 − M)−1g(0)  �  ,  (A4)  where 1 is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For example, the inco-  herent spectrum requires calculations of the type  Sinc(ω) = Re  � ∞  0  dτe−iωτeMτ⟨∆Aij(0)∆Akl(0)⟩st  = Re  �  (M − iω1)−1⟨∆Aij(0)∆Akl(0)⟩st  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (A5)  For the initial conditions of the correlations we use the  following operator products and correlations in compact  form:  AklAmn = Aknδlm ,  (A6a)  ⟨AklAmn⟩ = αknδlm,  (A6b)  AijAklAmn = Ainδjkδlm,  (A6c)  ⟨AijAklAmn⟩ = αinδjkδlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A6d)  Hence, the relevant initial conditions are:  ⟨A13R⟩ = (0, 0, 0, 0, α11, α13, 0, 0)T ,  (A7a)  ⟨A24R⟩ = (0, 0, 0, 0, 0, 0, α22, α24)T ,  (A7b)  ⟨A13RA31⟩ = (0, 0, 0, 0, 0, α11, 0, 0)T ,  (A7c)  ⟨A24RA42⟩ = (0, 0, 0, 0, 0, 0, 0, α22)T ,  (A7d)  ⟨A13RA42⟩ = ⟨A24RA31⟩ = 0,  (A7e)  where R = (A11, A13, A22, A24, A31, A33, A42, A44)T is  the Bloch vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' For correlations with fluctuation oper-  ator products, ∆Aij = Aij − αij, we have  ⟨∆Akl∆Amn⟩ = αknδlm − αklαmn,  (A8)  ⟨∆Aij∆Akl∆Amn⟩ = αinδlmδjk − αilαmnδjk  −αinαklδjm − αijαknδlm  +2αijαklαmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A9)  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' recalling that α12 = α14 = α23 = α34 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' we  write the detailed initial conditions of the correlations  (Set 1 of Bloch equations and quantum regression for-  mula):  ⟨∆A13∆R⟩ =  �  −α13α11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α2  13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α13α22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α13α24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' α11 − |α13|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' α13 − α13α33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α13α42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α13α44  �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A10a)  ⟨∆A24∆R⟩ =  �  −α24α11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α24α13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α24α22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α2  24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α24α31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' −α24α33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' α22 − |α24|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' α24 − α24α44  �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A10b)  ⟨∆A13∆R∆A31⟩ =  �  2|α13|2α11 − α2  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2|α13|2α13 − 2α11α13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  2|α13|2α22 − α11α22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2|α13|2α24 − α11α24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  2|α13|2α31 − 2α11α31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2|α13|2α33 + α11 − 2|α13|2 − α11α33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  2|α13|2α42 − 2α11α42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 2|α13|2α44 − α11α44  �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A10c)  ⟨∆A24∆R∆A42⟩ =  �  2|α24|2α11 − α11α22, 2|α24|2α13 − α22α13,  2|α24|2α22 − α2  22, 2|α24|2α24 − 2α22α24,  2|α24|2α31 − α22α31, 2|α24|2α33 − α22α33,  2|α24|2α42 − 2α22α42, 2|α24|2α44 + α22 − 2|α24|2 − α22α44  �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A10d)  16  ⟨∆A13∆R∆A42⟩ =  �  2α13α11α42, 2α2  13α42, 2α13α22α42, (2|α24|2 − α22)α13,  (2|α13|2 − α11)α42, (2α13α33 − α13)α42, 2α13α2  42, (2α13α44 − α13)α42  �T ,  (A10e)  ⟨∆A24∆R∆A31⟩ =  �  2α24α11α31, (2|α13|2 − α11)α24, 2α24α22α31, 2α2  24α31,  2α24α2  31, (2α24α33 − α24)α31, (2|α24|2 − α22)α31, (2α24α44 − α24)α31  �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (A10f)  Appendix B: Condition for Optimal Appearance of  Beats in the Intensity  We consider a simplified, unitary, model to estimate  the optimal initial population of the ground states to  make well-formed beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' First, we diagonalize the Hamil-  tonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' The eigenvalues and eigenstates are  E±  1 = −∆  2 ± 1  2  �  4Ω2 + ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (B1a)  E±  2 = Bℓ + δ − ∆  2  ± 1  2  �  4Ω2 + (δ − ∆)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (B1b)  and  |u1⟩ = sin Θ1|1⟩ + cos Θ1|3⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  |u2⟩ = − cos Θ1|1⟩ + sin Θ1|3⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  |u3⟩ = sin Θ2|2⟩ + cos Θ2|4⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  |u4⟩ = − cos Θ2|2⟩ + sin Θ2|4⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (B2)  respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' where  sin Θ1 =  2Ω  ��  ∆ +  √  ∆2 + 4Ω2�2 + 4Ω2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  cos Θ1 =  ∆ +  √  ∆2 + 4Ω2  ��  ∆ +  √  ∆2 + 4Ω2�2 + 4Ω2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  sin Θ2 =  2Ω  ��  (δ − ∆) +  �  (δ − ∆)2 + 4Ω2  �2  + 4Ω2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  cos Θ2 =  (δ − ∆) +  �  (δ − ∆)2 + 4Ω2  ��  (δ − ∆) +  �  (δ − ∆)2 + 4Ω2  �2  + 4Ω2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (B3)  It is now straightforward to obtain the excited-state  populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' If the initial state of the system is ρ(0) =  ⟨A33(0)⟩|3⟩⟨3| + ⟨A44(0)⟩|4⟩⟨4| we get  ⟨A33(t)⟩ = 1  2⟨A33(0)⟩ sin2 (2Θ1)(1 − cos (Ω1t)), (B4a)  ⟨A44(t)⟩ = 1  2⟨A44(0)⟩ sin2 (2Θ2)(1 − cos (Ω2t)), (B4b)  and the intensity of the field is  Iπ(r, t)  f 2π(r) = ⟨A33(0)⟩ sin2 (2Θ1) + A44(0)⟩ sin2 (2Θ2)  −⟨A33(0)⟩ sin2 (2Θ1) cos (Ω1t)  −⟨A44(0)⟩ sin2 (2Θ2) cos (Ω2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  (B5)  A necessary condition for the beating behavior to oc-  cur is that the initial ground-state populations are both  nonvanishing in the nondegenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Now, assuming  the relation  ⟨A33(0)⟩  ⟨A44(0)⟩ = sin2 (2Θ2)  sin2 (2Θ1)  (B6)  is satisfied by chossing appropriate parameter values  (Ω, δ, ∆) for given values of initial ground state popu-  lations we would get  Iπ(r, t) = f 2  π(r)⟨A33(0)⟩ sin2 (2Θ1)  × [1 − cos (Ωbeatt) cos (Ωavt)] ,  (B7)  where Ωbeat = (Ω2 − Ω1)/2 and Ωav = (Ω2 + Ω1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content='  [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Ficek and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' 10, 84 (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+page_content=' Castro-Beltr´an, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQfSAOy/content/2301.03061v1.pdf'}
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+arXiv:2301.05304v1  [math.RT]  12 Jan 2023
+A characterization of the L2-range of the Poisson transforms on a class of
+vector bundles over the quaternionic hyperbolic spaces
+Abdelhamid Boussejra ∗Achraf Ouald Chaib†
+Department of Mathematics, Faculty of Sciences
+University Ibn Tofail, Kénitra, Morocco
+Abstract
+We study the L2-boundedness of the Poisson transforms associated to the homogeneous vector bundles
+Sp(n, 1)×Sp(n)×Sp(1) Vτ over the quaternionic hyperbolic spaces Sp(n, 1)/Sp(n)× Sp(1) associated with irreducible
+representations τ of Sp(n)×Sp(1) which are trivial on Sp(n). As a consequence, we describe the image of the section
+space L2(Sp(n, 1)×Sp(n)×Sp(1) Vτ) under the generalized spectral projections associated to a family of eigensections
+of the Casimir operator.
+Keywords: Vector Poisson transform, Fourier restriction estimate, Strichartz conjecture.
+1
+Introduction
+Let G be a connected real semisimple noncompact Lie group with finite center, and K a maximal compact subgroup.
+Then X = G/K is a Riemannian symmetric space of noncompact type. Let G = KAN be an Iwasawa decomposition
+of G, and let M be the centralizer of A in K. We write g = κ(g)eH(g)n(g), for each g ∈ G according to G = KAN.
+A central result in harmonic analysis (see [17]) asserts that all joint eigenfunctions F of the algebra D(X) of invariant
+differential operators, are Poisson integrals
+F(g) = Pλf(g) :=
+�
+K
+e(iλ+ρ)H(g−1k)f(k) dk,
+of a hyperfunction f on K/M, for a generic λ ∈ a∗
+c (the complexification of a∗ the real dual of a).
+Since then a characterization of the Lp-range of the Poisson transform was developed in several articles such as [3],
+[5], [6], [7], [15], [20], [21], [22], [24], [25].
+The problem of characterizing the image of the Poisson transform Pλ of L2(K/M) with real and regular spectral
+parameter λ is intimately related to Strichartz conjecture [[25], Conjecture 4.5] on the uniform L2-boundedness of
+the generalized spectral projections associated with D(X).
+To be more specific, consider the generalized spectral
+projections Qλ defined initially for F ∈ C∞
+c (X) by
+QλF(x) =| c(λ) |−2 Pλ(FF(λ, .)(x),
+λ ∈ a∗,
+(1.1)
+where FF is the Helgason Fourier transform of F and c(λ) is the Harish-Chandra c-function.
+Conjecture (Strichartz [[25], Conjecture 4.5]). There exists a positive constant C such that for any Fλ = QλF with
+∗e-mail: boussejra.abdelhamid@uit.ac.ma
+†e-mail:achraf.oualdchaib@uit.ac.ma
+1
+
+F ∈ L2(X) we have
+C−1 ∥ F ∥2
+L2(X)≤
+sup
+R>0,y∈X
+�
+a∗
++
+1
+Rr
+�
+B(y,R)
+| Fλ(x) |2 dx dλ ≤ C ∥ F ∥2
+L2(X),
+(1.2)
+and
+∥ F ∥2
+L2(X)= γr lim
+R→∞
+�
+a∗
++
+1
+Rr
+�
+B(y,R)
+| Fλ(x) |2 dx dλ.
+(1.3)
+Conversely, if Fλ is any family of joint eigenfunctions for which the right hand side of (1.2) or (1.3) is finite, then there
+exists F ∈ L2(X) such that Fλ = QλF for a.e. λ ∈ a∗
++.
+Here r = rank X, and B(y, R) denotes the open ball in X of radius R about y. The constant γr depends on the
+normalizations of the measures dx and dλ.
+The strichartz conjecture has been recently settled by Kaizuka, see [16]. Most of the proof consists in proving a
+uniform estimate for the Poisson transform. More precisely, the following was proved by Kaizuka [[16], Theorem 3.3]:
+Let F be a joint eigenfunction with eigenvalue corresponding to a real and regular spectral parameter λ . Then F is
+the Poisson transform by Pλ of some f ∈ L2(K/M) if and only if
+sup
+R>1
+1
+Rr
+�
+B(0,R)
+| F(x) |2 dx < ∞.
+Moreover there exists a positive constant C independent of such λ,
+C−1 | c(λ) |2∥ f ∥2
+L2(K/M)≤ sup
+R>1
+1
+Rr
+�
+B(0,R)
+| Pλf(x) |2 dx ≤ C | c(λ) |2∥ f ∥2
+L2(K/M) .
+The generalization of these results to vector bundles setting has only just begin. In [8] we extend Kaizuka result to
+homogeneous line bundles over non-compact complex Grassmann manifolds (See also [4]).
+Our aim in this paper is to generalize theses results to a class of homogeneous vector bundles over the quaternionic
+hyperbolic space G/K, where G is the symplectic group Sp(n, 1) with maximal compact subgroup K = Sp(n)×Sp(1).
+To state our results in rough form, let us first introduce the class of the homogenous vector bundles that we consider
+in this paper. Let τν be a unitary irreducible representation of Sp(1) realized on a (ν + 1)-dimensional Hilbert space
+(V, (., .)ν). We extend τν to a representation of K by setting τν ≡ 1 on Sp(n). As usual the space of sections of the
+homogeneous vector bundle G ×K V associated with τν will be identified with the space Γ(G, τν) of vector valued
+functions F : G → Vν which are right K-covariant of type τν, i.e.,
+F(gk) = τν(k)−1F(g),
+∀g ∈ G,
+∀k ∈ K.
+(1.4)
+We denote by C∞(G, τν) and C∞
+c (G, τν) the elements of Γ(G, τν) that are respectively smooth, smooth with compact
+support in G, and by L2(G, τν) the elements of Γ(G, τν) such that
+∥ F ∥L2(G,τν)=
+��
+G/K
+∥ F(g) ∥2
+ν dgK
+� 1
+2
+< ∞.
+In above ∥ . ∥ν is the norm in Vν and ∥ F(gK) ∥ν=∥ F(g) ∥ν is well defined for F satisfying (1.4).
+Let σν denote the restriction of τν to the group M ≃ Sp(n−1)×Sp(1). Over K/M we have the associated homogeneous
+vector bundle K ×M Vν with L2-sections identified with L2(K, σν) the space of all functions f : K → Vν which are
+M-covariant of type σν and satisfy
+∥ f ∥2
+L2(K,σν)=
+�
+K
+∥ f(k) ∥2
+ν dk < ∞,
+2
+
+where dk is the normalized Haar measure of K.
+For λ ∈ C and f ∈ L2(K, σν), the Poisson transform Pν
+λf is defined by
+Pν
+λf(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k))f(k) dk
+Let Ω denote the Casimir element of the Lie algebra g of G, viewed as a differential operator acting on C∞(G, τ).
+Then the image Pν
+λ(L2(K, σν)) is a proper closed subspace of Eλ(G, τν) the space of all F ∈ C∞(G, τν) satisfying
+Ω F = −(λ2 + ρ2 − ν(ν + 2))F.
+For more details see section 2.
+For λ ∈ R \ {0}, we define a weighted L2-space E2
+λ(G, τν) consisting of all F in Eλ(G, τν) that satisfy
+∥ F ∥∗= sup
+R>1
+�
+1
+R
+�
+B(R)
+∥F(g)∥2
+ν dgK
+� 1
+2
+< ∞.
+Our first main result is an image characterization of the Poisson transform Pν
+λ of L2(K, σν) for λ ∈ R \ {0}.
+Theorem 1.1. Let λ ∈ R\{0} and ν a nonnegative integer.
+(i) There exists a positive constant Cν independent of λ such that for f ∈ L2(K, σν) we have
+C−1
+ν |cν(λ)| ∥f∥L2(K,σν) ≤ ∥Pν
+λf∥∗ ≤ Cν| | cν(λ) | ∥f∥L2(K,σν),
+(1.5)
+with
+cν(λ) = 2ρ−iλ
+Γ(ρ − 1)Γ(iλ)
+Γ( iλ+ν+ρ
+2
+)Γ( iλ+ρ−ν−2
+2
+)
+.
+Furthermore we have the following Plancherel type formula for the Poisson transform
+lim
+R→+∞
+1
+R
+�
+B(R)
+∥Pν
+λf(g)∥2
+ν dgK = 2 | cν(λ) |2 ∥f∥2
+L2(K,σν) .
+(1.6)
+ii) Pν
+λ is a topological isomorphism from L2(K, σν) onto E2
+λ(G, τν).
+This generalizes the result of Kaizuka [[16], (i) and (ii) in Theorem 3.3] which corresponds to τν trivial.
+Consequence
+For λ ∈ R we define the space
+E∗
+λ(G, τν) = {F ∈ Eλ(G, τν) : M(F) < ∞},
+where
+M(F) = lim sup
+R→∞
+�
+1
+R
+�
+B(R)
+| F(g) |2 dgK
+� 1
+2
+.
+Then as an immediate consequence of Theorem 1.1 we obtain the following result which generalizes a conjecture of
+W. Bray [10] which corresponds to τν trivial.
+Corollary 1.1. If λ ∈ R \ {0} then E∗
+λ(G, τν), M) is a Banach space.
+Remark 1.1. In the case of the trivial bundle (the scalar case) the conjecture of Bray was proved by Ionescu [15] for
+all rank one symmetric spaces . It was generalized to Riemannian symmetric spaces of higher rank by Kaizuka, see
+[16].
+3
+
+Next, let us introduce our second main result on the L2-range of the generalized spectral projections.
+For F ∈ C∞
+c (G, τν) the vector valued Helgason-Fourier transform FνF is given by (see [11])
+Fν F(λ, k) =
+�
+G
+e(iλ−ρ)H(g−1k)τν(κ(g−1k)−1)F(g) dg
+λ ∈ C,
+Then the following inversion formula holds (see section 4)
+F(g) = 1
+2π
+� ∞
+0
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k))FνF(λ, k) | cν(λ) |−2 dλ dk
++
+�
+λj∈Dν
+dν(λj)
+�
+K
+e−(iλj+ρ)H(g−1k)τν(κ(g−1k))FνF(λj, k) dk.
+(1.7)
+In above dν(λ) = −iResµ=λ(cν(µ)cν(−µ))−1, λ ∈ Dν and Dν is a finite set in {λ ∈ C; ℑ(λ) > 0} which parametrizes
+the τν-spherical functions arising from the discrete series of G. It is empty if ν ≤ ρ − 2.
+The formula (1.7) gives rise to the decomposition of L2(G, τν) into a continuous part and a discrete part:
+L2(G, τν) = L2
+cont(G, τν) ⊕ L2
+disc(G, τν)
+Our aim here is to study the operator Qν
+λ, λ ∈ R, defined for F ∈ L2
+cont(G, τν) ∩ C∞
+c (C, τν) by
+Qν
+λF(g) =| cν(λ) |−2 Pν
+λ[Fν F(λ, .)](g),
+(1.8)
+More precisely, following Strichartz idea, we are interested in the following question:
+Characterize those Fλ ∈ Eλ(G, τν) (λ ∈ (0, ∞)) for which there exists F ∈ L2
+cont(G, τν) such that Fλ = Qν
+λF.
+To do so, we introduce the space E2
++(G, τν) consisting of all Vτν-valued measurable functions ψ on (0, ∞) × G such
+that
+(i) Ω ψ(λ, .) = −(λ2 + ρ2 − ν(ν + 2)) ψ(λ, .) a.e. λ ∈ (0, ∞)
+(ii) ∥ ψ ∥+< ∞.
+where
+∥ ψ ∥2
++= sup
+R>1
+� ∞
+0
+1
+R
+�
+B(R)
+∥ ψ(λ, g) ∥2
+ν dgK dλ.
+The second main result we prove in this paper can be stated as follows
+Theorem 1.2.
+(i) There exists a positive constant C such that for F ∈ L2(G, τν) we have
+C−1 ∥ F ∥L2(G,τν)≤∥ Qν
+λF ∥+≤ C ∥ F ∥L2(G,τν)
+(1.9)
+Furthermore we have
+lim
+R→∞
+� ∞
+0
+1
+R
+�
+B(R)
+∥ Qν
+λF ∥2
+ν dgK dλ = 2 ∥ F ∥2
+L2(G,τν)
+(1.10)
+(ii) The linear map Qν
+λ is a topological isomorphism from L2
+cont(G, τν) onto E2
++(G, τν).
+This extends Kaizuka result [ [16], (i) and (ii) in Theorem 3.6] on the Strichartz conjecture (see [25] Conjecture
+4.5] to the class of vector bundles considered here.
+Before giving the outline of the paper, let us mention that a number of authors have obtained an image characterization
+for the Poisson transform Pλ (λ ∈ a∗ \ {0}) of L2-functions on K/M in the rank one case, see [[3], [5], [7], [15]].
+Nevertheless, the obtained characterization is weaker than the one conjectured by Strichartz. The approach taken in
+4
+
+the quoted papers is based on the theory of Calderon-Zygmund singular integrals (see also [21]). Using a different
+approach based on the techniques used in the scattering theory, Kaizuka [16] settled the Strichartz conjecture on
+Riemannian symmetric spaces of noncompact type, of arbitrary rank.
+We now describe the contents of this paper. The proofs of our results are a generalisation of Kaizuka’s method [16]. In
+section 2 we recall some basic facts on the quaternionc hyperbolic spaces and introduce the vector Poisson transforms.
+In section 3, we define the Helgason-Fourier transform on the vector bundles G ×K Vν and give the inversion and
+Plancherel Theorem. The proof of Theorem 1.2 follows from the Plancherel formula and Theorem 1.1. The main
+ingredients in proving Theorem 1.1 are a Fourier restriction estimate for the vector valued Helgason-Fourier transform
+(Proposition 4.1 in section 4) and an asymptotic formula for the vector Poisson transform in the framework of Agmon-
+Hörmander spaces [2] (Theorem 5.1). The proof of Theorem 5.1 will be derived from the Key lemma of this paper
+giving the asymptotic behaviour of the translate of the τν-spherical functions. Section 6 is devoted to the proof of our
+main results. In section 7 we prove the Key Lemma.
+2
+Preliminaries
+2.1
+The quaternionic hyperbolic space
+Let G = Sp(n, 1) be the group of all linear transformations of the right H-vector space Hn+1 which preserve the
+quadratic form
+n
+�
+j=1
+| uj |2 − | un+1 |2. Let K = Sp(n) × Sp(1) be the subgroup of G consisting of pairs (a, d)
+of unitaries.
+Then K is a maximal compact subgroup of G.
+The quaternionic hyperbolic space is the rank one
+symmetric space G/K of the noncompact type. It can be realized as the unit ball B(Hn) = {x ∈ Hn; | x |< 1}.
+The group G acts on B(Hn) by the fractional linear mappings x �→ g.x = (ax + b)(cx + d)−1, if g =
+�
+a
+b
+c
+d
+�
+, with
+a ∈ Hn×n, b ∈ Hn×1, c ∈ H1×n and d ∈ H.
+Denote by g the Lie algebra of G; g = k ⊕ p the Cartan decomposition of g, where p is a vector space of matrices of
+the form
+��
+0
+x
+x∗
+0
+�
+, x ∈ Hn
+�
+, and k =
+��
+X
+0
+0
+q
+�
+, X∗ + X = 0, q + q = 0
+�
+, where X∗ is the conjugate transpose
+of the matrix X and q ∈ H.
+Let H =
+�
+0n
+e1
+te1
+0
+�
+∈ p with te1 = (1, 0, · · · , 0). Then a = R H is a Cartan subspace in p, and the corresponding
+analytic subgroup A = {at = exp t H; t ∈ R}, where at =
+
+
+
+cht
+0
+sht
+0
+0n−1
+0
+sht
+0
+cht
+
+
+ . With A determined we then have that
+M =
+
+
+
+
+
+g =
+
+
+
+q
+0
+0
+0
+m
+0
+0
+0
+q
+
+
+ , m ∈ Sp(n − 1), | q |= 1
+
+
+
+
+
+≃ Sp(n − 1) × Sp(1).
+Let α ∈ a∗ be defined by α(H) = 1. Then a system Σ of restricted roots of the pair (g, a) is Σ = {±α, ±2α} if n ≥ 2
+and Σ = {±2α} if n = 1, with Weyl group W ≃ {±Id}. A positive subsystem of roots corresponding to the positive
+Weyl chamber a+ ≃ (0, ∞) in a is Σ+ = {α, 2α} if n ≥ 2 and Σ+ = {2α} if n = 1.
+Let n = gα + g2α be the direct sum of the positive root subspaces, with dim gα = 4(n − 1) and dim g2α = 3 and N the
+corresponding analytic subgroup of G. Then the half sum of the positive restricted roots with multiplicities counted
+ρ equals to (2n + 1)α, and shall be viewed as a real number ρ = 2n + 1 by the identification a∗
+c ≃ C via λα ↔ λ.
+Let A+ = {at ∈ A;
+t ≥ 0}. Then we have the Cartan decomposition G = KA+K, that is any g ∈ G can be written
+g = k1(g) eA+(g) k2(g),
+k1(g), k2(g) ∈ K and A+(g) ∈ a+.
+5
+
+If we write g ∈ G in (n + 1) × (n + 1) block notation as g =
+�
+a
+b
+c
+d
+�
+. Then a straightforward computation gives
+cosh A+(g) =| d |
+and
+H(g) = log | ce1 + d | .
+(2.1)
+We normalize the invariant measure dgK on G/K so that the following integral formula holds: for all h ∈ L1(G/K),
+�
+G/K
+h(gK)dgK =
+�
+G
+h(g.0)dg =
+�
+K
+� ∞
+0
+h(k at)∆(t) dk dt,
+(2.2)
+where dt is the Lebesgue measure, ∆(t) = (2 sinh t)4n−1(2 cosh t)3, and dk is the Haar measure of K with
+�
+K
+dk = 1.
+2.2
+The vector Poisson transform
+In this subsection we define the Poisson transform associated to the vector bundles G×KVν over Sp(n, 1)/Sp(n)×Sp(1)
+and derive some results referring to [23], [27], and [28] for more informations on the subject.
+Let σν denote the restriction of τν to M. For λ ∈ C we consider the representation σν,λ of P = MAN on Vν defined
+by σν,λ(man) = aρ−iλσν(m). Then σν,λ defines a principal series representations of G on the Hilbert space
+Hν,λ := {f : G → Vν | f(gman) = σ−1
+ν,λ(man)f(g) ∀man ∈ MAN, f|K ∈ L2},
+where G acts by the left regular representation. We shall denote by C−ω(G, σν,λ) the space of its hyperfunctions
+vectors. By the Iwasawa decomposition, the restriction map from G to K gives an isomorphism from Hν,λ onto the
+space L2(K, σν). This yields, the so-called compact picture of Hν,λ, with the group action given by
+πσν,λ(g)f(k) = e(iλ−ρ)H(g−1k)f(κ(g−1k)).
+By C−ω(K, σν) we denote the space of its hyperfunctions vectors.
+A Poisson transform is the continuous, linear, G-equivariant map Pν
+λ from C−ω(G, σν,λ) to C∞(G, τν) defined by
+Pν
+λ f(g) =
+�
+K
+τν(k)f(gk) dk.
+In the compact picture the Poisson transform is given by
+Pν
+λ f(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k)) f(k) dk.
+Let D(G, τν) denote the algebra of left invariant differential operators on C∞(G, τν). Let Eν,λ(G) be the space of all
+F ∈ C∞(G, τν) such that Ω F = −(λ2 + ρ2 − ν(ν + 2)) F.
+Proposition 2.1. (i) D(G, τν) is the algebra generated by the Casimir operator Ω of g.
+(ii) For λ ∈ C, ν ∈ N, the Poisson transform Pν
+λ maps C−ω(G, σν,λ) to Eν,λ(G).
+Proof. (i) Let U(a) be the universal enveloping algebra of the complexification of a. Since the restriction of τν to M
+is irreducible, then D(G, τν) ≃ U(a)W . As a is one dimensional, then D(G, τν) ≃ C[s2], symmetric functions of one
+variable . Thus D(G, τν) is generated by the Casimir element Ω of the Lie algebra g of G, viewed as a differential
+operator acting on C∞(G, τν).
+(ii) Since σν is irreducible, the image of Pν
+λ consists of joint eigenfunctions with respect to the action of Ω. Moreover
+Ω acts by the infinitesimal character of the the principal series representations πσν,λ. It follows from Proposition 8.22
+and Lemma 12.28 in [18], that
+πσν,λ(Ω) = −(λ2 + ρ2 − c(σν))Id
+on
+C−ω(G, σν,λ),
+(2.3)
+where c(σν) is the Casimir value of σν given by c(σν) = ν(ν + 2).
+6
+
+Let Φν,λ be the τν-spherical function associated to σν. Then Φν,λ admits the following Eisenstein integral repre-
+sentation (see [[11], Lemma 3.2]):
+Φν,λ(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k)k−1) dk.
+Note that Φν,λ lies in C∞(G, τν, τν) the space of smooth functions F : G → End(Vτν) satisfying
+F(k1gk2) = τν(k−1
+2 )F(g)τν(k−1
+1 ),
+the so called τν-radial functions. Being τν-radial, Φν,λ is completely determined by its restriction to A, by the Cartan
+decomposition G = KAK. Moreover, since σν is irreducible, it follows that Φν,λ(at) ∈ EndM(Vν) ≃ CIdVν, ∀at ∈ A.
+Therefore there exists ϕν : R → C such that Φν,λ(at) = ϕν(t).IdVν. We have
+ϕν,λ(t) =
+1
+ν + 1
+�
+K
+e−(iλ+ρ)H(g−1k)χν(κ(g−1k)k−1) dk,
+(2.4)
+where χν is the character of τν.
+This so-called trace τν-spherical function has been computed explicitly in [12] using the radial part of the Casimir
+operator Ω (see also [26] ). We have ϕν,λ(t) = (cosh t)νφ(ρ−2,ν+1)
+λ
+(t), where φ(ρ−2,ν+1)
+λ
+(t) is the Jacobi function (cf.
+[19])
+φ(ρ−2,ν+1)
+λ
+(t) = 2F1(iλ + ρ + ν
+2
+, −iλ + ρ + ν
+2
+; ρ − 1; − sinh2 t).
+We deduce from (A4) the asymptotic behaviour of ϕν,λ
+ϕλ,ν(at) = e(iλ−ρ)t[cν(λ) + ◦(1)], as t → ∞
+if
+ℑ(λ) < 0.
+(2.5)
+where
+cν(λ) =
+2ρ−iλΓ(ρ − 1)Γ(iλ)
+Γ( iλ+ρ+ν
+2
+)Γ( iλ+ρ−ν−2
+2
+)
+.
+(2.6)
+For λ ∈ C the c-function of Harish-Chandra associated to τν is defined by
+c(τν, λ) =
+�
+N
+e−(iλ+ρ)H(n)τν(κ(n)) dn.
+The integral converges for λ such that ℜ(iλ) > 0 and it has a meromorphic continuation to C.
+In above dn is the Haar measure of N = θ(N), θ being the Cartan involution.
+We may use formula (2.6) to give explicitly c(τν, λ). Indeed, one easily check that c(τν, λ) ∈ EndM(Vν) = CIdVν.
+Then using the following result on the behaviour of Φν,λ(at) ([28], Proposition 2.4)
+Φν,λ(at) = e(iλ−ρ)t(c(τν, λ) + ◦(1))as
+t → ∞,
+together with Φν,λ(at) = ϕν,λ(t).Id, we find then from (2.5) that c(τν, λ) = cν(λ)IdVν.
+We end this section by recalling a result of Olbrich [23] on the range of the Poisson transform on vector bundles which
+reads in our case as follows
+Theorem 2.1. [23] Let ν ∈ N and λ ∈ C such that
+(i) −2iλ /∈ N
+(ii) iλ + ρ /∈ −2N − ν ∪ −2N + ν + 2.
+Then the Poisson transform Pν
+λ is a K-isomorphism from C−ω(K, σν) onto Eν,λ(G).
+7
+
+3
+The vector-valued Helgason-Fourier transfrorm
+In this section we give the inversion and the Plancherel formulas for the Helgason-Fourier transform on the vector
+bundle G ×K Vν.
+According to [11] the vector-valued Helgason-Fourier transform of f ∈ C∞
+c (G, τν) is the Vν-valued function on C × K
+defined by:
+Fνf(λ, k) =
+�
+G
+eλ,ν(k−1g) f(g)dg,
+where eλ,ν is the vector valued function eλ,ν : G → End(Vν) given by
+eλ,ν(g) = e(iλ−ρ)H(g−1)τ −1
+ν (κ(g−1)).
+Notice that our sign on "λ" is the opposite of the one in [11].
+In order to state the next theorem, we introduce the finite set in {λ, ℑ(λ) ≥ 0}
+Dν = {λj = i(ν − ρ + 2 − 2j), j = 0, 1, · · · , ν − ρ + 2 − 2j > 0}.
+Note that Dν is empty if ν ≤ ρ − 2. It parametrizes the discrete series representation of G containing τν, see [12].
+Let
+dν(λj) = 2−2(ρ−ν−1)(ν − ρ − 2j + 2)(ρ − 2 + j)!(ν − j)!
+Γ2(ρ − 1)j!(ν − ρ − j + 2)!
+,
+λj ∈ Dν
+For λj ∈ Dν, we define the operators Qν
+j
+L2(G, τν) → Eν,λj(G, τν)
+F �→ dν(λj) Φν,λj ∗ F
+We denote the image by A2
+j. We set
+L2
+disc(G, τν) =
+�
+j; ν−ρ+2−2j>0
+A2
+j,
+and denote by L2
+cont(G, τν) its orthocomplement. Let L2
+σν(R+ × K, | cν(λ) |−2 dλ dk) be the space of vector functions
+φ : R+ × K → Vν satisfying
+(i) For each fixed λ, φ(λ, km) = σν(m)−1φ(λ, k), ∀m ∈ M
+(ii)
+�
+R+×K ∥ Fνφ(λ, k) ∥2 | cν(λ) |−2 dλ dk < ∞.
+Theorem 3.1. (i) For F ∈ C∞
+c (G, τν) we have the following inversion and Plancherel formulas
+F(g) = 1
+2π
+� ∞
+0
+�
+K
+e∗
+λ,ν(k−1g)FνF(λ, k) | cν(λ) |−2 dλ dk +
+�
+λj∈Dν
+dν(λj)
+�
+K
+e∗
+λj,ν(k−1g)FνF(λj, k) dk,
+(3.1)
+�
+G
+∥ F(g) ∥2
+ν dgK = 1
+2π
+� ∞
+0
+�
+K
+∥ FνF((λ, k) ∥2
+ν| cν(λ) |−2 dλ dk+
+�
+λj∈Dν
+dν(λj)
+�
+K
+< FνF(λj, k), FνF(−λj, k) >ν dk
+(3.2)
+(ii) The Fourier transform Fν extends to an isometry from L2
+cont(G, τν) onto the space L2
+σν(R+ ×K, | cν(λ) |−2 dλ dk).
+The first part of Theorem 3.1 can be easily deduced from the inversion and Plancherel formulas for the spherical
+transform.
+8
+
+Let C∞
+c (G, τν, τν) denote the space of smooth compactly supported τν-radial functions. The spherical transform of
+F ∈ C∞
+c (G, τν, τν) is the C-valued function HνF defined by:
+HνF(λ) =
+1
+ν + 1
+�
+G
+T r[Φν,λ(g−1)F(g))]dg,
+λ ∈ C.
+The inversion and the Plancherel formulas for the τ-spherical transform have been given explicitly in [12]. For the
+convenience of the reader we give an elementary proof by using the Jacobi transform.
+Theorem 3.2. For F ∈ C∞
+c (G, τν, τν) we have the following inversion and Plancherel formulas
+F(g) = 1
+2π
+� +∞
+0
+Φν,λ(g)HνF(λ) | cν(λ) |−2 dλ +
+�
+λj∈Dν
+Φν,λj(g)Hνf(λj) dν(λj),
+(3.3)
+�
+G
+∥ F(g) ∥2
+HS dg = ν + 1
+2π
+� +∞
+0
+| HνF((λ) |2| cν(λ) |−2 dλ + (ν + 1)
+�
+λj∈Dν
+dν(λj) | HνF((λj) |2,
+(3.4)
+In above ∥ ∥HS stands for the Hilbert-Schmidt norm.
+Proof. Let F ∈ C∞
+c (G, τν, τν) and let fν be its scalar component.
+Using the integral formula (2.2), the identity
+Φν,λ(at) = Φν,λ(a−t) = (cosh t)νφ(ρ−2,ν+1)
+λ
+(t) and the fact that ∆(t) = (2 cosh t)−2ν∆ρ−2,ν+1, we have
+HνF(λ) =
+� ∞
+0
+fν(t)(cosh t)νφ(ρ−2,ν+1)
+λ
+(t) ∆(t) dt
+=
+� ∞
+0
+fν(t)(22 cosh t)−νφ(ρ−2,ν+1)
+λ
+(t) ∆ρ−2,ν+1(t) dt.
+(3.5)
+Thus the τν-spherical transform HνF may be written in terms of the Jacobi transform J α,β, with α = ρ − 2 and
+β = ν + 1. Namely, we have
+HνF(λ) = J ρ−2,ν+1[(22 cosh t)−νfν](λ).
+We refer to (A5) in the Appendix for the definition of the Jacobi transform.
+Now the theorem follows from the inversion and the Plancherel formulas for the Jacobi transform (A6), (A6’) and
+(A7) in the Appendix.
+For the proof of the surjectivity statement in Theorem 3.1 we shall need the following result
+Proposition 3.1. Let F ∈ C∞
+c (G, τν) and Φ ∈ C∞(G, τν, τν). Then we have
+Fν(F ∗ Φ)(λ, k) = HνΦ(λ)FνF(λ, k),
+λ ∈ C, k ∈ K,
+where the convolution is defined by
+(Φ ∗ F)(g) =
+�
+G
+Φν,λ(x−1g)F(x) dx.
+Proof. Let Φ ∈ C∞(G, τν, τν), v ∈ Vν, and set Fv = Φ(. )v. Then we have the following relation between the Fourier
+transform and the spherical transform
+FνFv(λ, k) = HνΦ(λ)τ(k−1)v.
+(3.6)
+By definition
+Fν(F ∗ Φ)(λ, k) =
+�
+G
+�
+G
+eν
+λ(k−1g)Φ(x−1g)F(x)dxdg
+=
+�
+G
+dx
+�
+G
+eν
+λ(k−1xy)Φ(y)F(x)dy
+9
+
+Using the following cocycle relations for the Iwasawa function H(x)
+H(xy) = H(xκ(y)) + H(y),
+and
+κ(xy) = κ(xκ(y)),
+for all x, y ∈ G, we get the following identity
+eν
+λ(k−1xy) = e(iλ−ρ)H(x−1k)eν
+λ(κ−1(x−1k)y),
+from which we obtain
+Fν(Φ ∗ F)(λ, k) =
+�
+G
+e(iλ−ρ)H(x−1k)
+��
+G
+eλ,ν(κ−1(x−1k)y)Φ(y)F(x) dy
+�
+dx.
+Next, put hv(y) = Φ(y)v, v ∈ Vτν. Then (3.6) implies
+�
+G
+eλ,ν(κ−1(x−1k)y)Φ(y)F(x) dy = Fν(hF (x))(λ, κ−1(x−1k))
+= H(Φ)(λ)τν(κ−1(x−1k))F(x),
+from which we deduce
+Fν(Φ ∗ F)(λ, k) = H(Φ)(λ)
+�
+G
+e(iλ−ρ)H(x−1k)τν(κ−1(x−1k))F(x)dx,
+and the proposition follows.
+We now come to the proof of Theorem 3.1.
+Proof. (i) We may follow the same method as in [11] to prove the inversion formula (3.1) and the Plancherel formula
+(3.2) from Theorem 3.2. We give an outline of the proof.
+Let F ∈ C∞
+c (G, τν) and consider the τν-radial function defined for any g ∈ G by
+Fg,v(x).w =
+�
+K
+< τν(k)w, v >ν F(gkx) dk,
+v being a fixed vector in Vν. Then a straightforward calculation shows that
+HνFg,v(λ) =
+1
+ν + 1 < (Φν,λ ∗ F)(g), v >ν .
+The inversion formula for the spherical transform together with T rFg,v(e) =< F(g), v >ν imply
+F(g) = 1
+2π
+� ∞
+0
+(Φν,λ ∗ F)(g) | cν(λ) |−2 dλ +
+�
+λj∈Dν
+(Φν,λj ∗ F)(g)dν(λj).
+To conclude use the following result for the translated spherical function ( see [11] Proposition 3.3)
+Φν,λ(x−1y) =
+�
+K
+e−(iλ+ρ)H(y−1k)e(iλ−rho)H(x−1k)τν(κ(y−1k))τν(κ−1(x−1k)) dk,
+(3.7)
+to get
+(Φν,λ ∗ F)(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k))FνF(λ, k) dk,
+10
+
+and the inversion formula (3.1) follows.
+The proof of the Plancherel formula (3.2) is essentially the same as in the scalar case, so we omit it.
+Note that as a consequence of the Plancherel formula not involving the discrete series, we have
+�
+G
+∥ F(g) ∥2 dgK = 1
+π
+� ∞
+0
+�
+K
+∥ FνF(λ, k) ∥2 | cν(λ) |−2 dλ dk,
+for every F ∈ L2
+cont(G, τν).
+(ii) We prove the surjectivity statement. Suppose that there exists a function f in L2
+σν(R+ × K, | cν(λ) |−2 dλ dk)
+such that
+� ∞
+0
+�
+K
+< f(λ, k), FνF(λ, k) >| cν(λ) |−2 dλ dk = 0
+for all F ∈ C∞
+c (G, τν). Changing F into F ∗ Φ where Φ ∈ C∞(G, τν, τν) and using Proposition 3.1, we have
+� ∞
+0
+�
+K
+< f(λ, k), FνF(λ, k) > Hνφ(λ) | cν(λ) |−2 dλ dk = 0
+By the Stone-Weierstrass theorem, the algebra {HνΦ, Φ ∈ C∞(G, τν, τν)} is dense in C∞
+e (R) the space of even
+continuous functions on R vanishing at infinity. Therefore for every F ∈ C∞
+c (G, τν) there is a set EF of measure zero
+in R such that
+�
+K
+< f(λ, k), FνF(λ, k) > dk = 0
+for all λ not in EF . The rest of the proof is based on an adaptation of the arguments given in [14] Theorem 1.5, for
+the scalar case, and the proof of Theorem 3.1 is completed.
+4
+Fourier restriction estimate
+The main result of this section is the following uniform continuity estimate for the Fourier-Helgason restriction operator.
+Proposition 4.1. Let ν ∈ N. There exists a positive constant Cν such that for λ ∈ R\{0} and R > 1, we have
+� �
+K
+∥FνF(λ, k)∥2
+νdk
+�1/2
+≤ Cν|cν(λ)|R1/2
+� �
+G/K
+∥F(g)∥2
+ν dgK
+�1/2
+,
+(4.1)
+for every F ∈ L2(G, τν) with suppF ⊂ B(R).
+To prove this result we shall need estimates of the Harish-Chandra c-function.
+To this end we introduce the
+function bν(λ) defined on R by
+bν(λ) =
+
+
+
+cν(λ)
+if
+ν−ρ+2
+2
+∈ Z+
+λ cν(λ)
+if
+ν−ρ+2
+2
+/∈ Z+
+Lemma 4.1. Assume ν > ρ − 2.
+(i) The function bν(λ) has no zero in R.
+(ii) There exists a positive constant C such that for λ ∈ R, we have
+C−1(1 + λ2)
+2ρ−4−ε(ν)
+4
+≤| bν(λ) |−1≤ C(1 + λ2)
+2ρ−4−ε(ν)
+4
+,
+(4.2)
+11
+
+with ε(ν) = ±1 according to ν−ρ+2
+2
+/∈ Z+ or ν−ρ+2
+2
+∈ Z+
+Proof.
+(i) If ν−ρ+2
+2
+/∈ Z+, then bν(λ) = 2ρ+ν−iλΓ(ρ−1)Γ(iλ+1)
+Γ( iλ+ρ+ν
+2
+)Γ( iλ+ρ−ν−2
+2
+), and clearly bν(λ) has no zero on R.
+If ν−ρ+2
+2
+∈ Z+ then bν(λ) a priori can have zero and pole at λ = 0. This is not the case, since
+lim
+λ→0 bν(λ) = (−1)
+ν−ρ+2
+2
+2ρ+νΓ(ρ − 1)( ν−ρ+2
+2
+)!
+Γ( ρ+ν
+2 )
+.
+(ii) To prove the estimate (4.2) we shall use the following property of the Γ-function
+lim
+|z|→∞
+Γ(z + a)
+Γ(z)
+z−a = 1, | arg(z) |< π − δ,
+(4.3)
+where a is any complex number, and log is the principal value of the logarithm and δ > 0.
+Assume first that ν−ρ+2
+2
+/∈ Z+. Using the duplicata formula for the function gamma
+Γ(2z) = 22z−2
+√π Γ(z)Γ(z + 1
+2),
+we rewrite bν(λ) as
+bν(λ) = 2ρ+ν−1
+√π
+Γ( iλ+1
+2
+)Γ( iλ+2
+2
+)
+Γ( iλ+ρ+ν
+2
+)Γ( iλ+ρ−ν−2
+2
+)
+.
+It follows from (4.3) that for every λ ∈ R, we have
+| bν(λ) |≤ C(1 + λ2)− 2ρ−5
+4
+and
+| bν(λ) |−1≤ C(1 + λ2)
+2ρ−5
+4 .
+The proof for the case ν−ρ+2
+2
+∈ Z+ follows the same line as in the case ν−ρ+2
+2
+/∈ Z+, so we omit it.
+This finishes the proof of the Lemma.
+Let us recall from [1] an auxiliary lemma which will be useful for the proof of Proposition 4.1.
+Let η be a positive Schwartz function on R whose Fourier transform has a compact support. For m ∈ R, set
+ηm(x) =
+�
+R
+η(t)(1 + |t − x|)m/2 dt.
+Lemma 4.2.
+i) ηm is a positive C∞-function with
+C−1(1 + t2)
+m
+2 ≤ ηm(t) ≤ C(1 + t2)
+m
+2 ,
+(4.4)
+for some positive constant C.
+ii) The Fourier transform of ηm has a compact support.
+In order to prove the Fourier restriction Theorem, we need to introduce the bundle valued Radon transform, see
+[9] for more informations.
+The Radon transform for F ∈ C∞
+c (G, τν) is defined by
+RF(g) = eρH(g)
+�
+N
+F(gn)dn.
+12
+
+We set RF(t, k) = RF(kat). Then, using the Iwaswa decomposition G = NAK, we may rewrite the Helgason-Fourier
+transform as
+FνF(λ, k) = FR(RF(·, k))(λ),
+where
+FRφ(λ) =
+�
+R
+e−iλtφ(t) dt,
+is the Euclidean Fourier transform of φ a Vν-valued smooth function with compact support in R.
+We define on p the scalar product < X, Y >= 1
+2T r(XY ) and denote by | | the corresponding norm. It induces a distance
+function d on G/K. By the Cartan decomposition G = K exp p, any g ∈ G may be written uniquely as g = k exp X,
+so that d(0, gK) =| X |. Define the open ball centred at 0 and of radius R by B(R) = {gK ∈ G/K;
+d(0, gK) < R}.
+Lemma 4.3. Let F ∈ C∞
+0 (G, τν). If supp F ⊂ B(R), then supp RF ⊂ [−R, R] × K.
+Proof. As (see [[13], page 476]
+d(0, ketHnK) ≥| t |,
+k ∈ K, n ∈ N, t ∈ R
+it follows that supp RF ⊂ [−R, R] × K if supp F ⊂ B(R)
+Proof of Proposition 4.1. It suffices to prove the estimate (4.1) for functions F ∈ C∞
+c (G, τν) supported in B(R).
+It follows from the Plancherel formula (3.2) that
+�
+B(R)
+∥ F(g) ∥2
+ν dgK ≥
+�
+K
+�
+R
+∥ FνF(λ, k) ∥2
+ν | cν(λ) |−2 dλ dk
+Therefore it is sufficient to show
+�
+K
+�
+R
+∥ FνF(λ, k) ∥2
+ν | cν(λ) |−2 dλ dk ≥ C | cν(λ) |−2
+R
+�
+R
+∥ FνF(λ, k) ∥2
+ν dk,
+(4.5)
+fir some positive constant C.
+By (4.2) we have | cν(λ) |−1≍ η 2ρ−3
+2 (λ). Therefore (4.5) is equivalent to
+η 2ρ−3
+2 (λ)
+R
+�
+K
+∥ FνF(λ, k) ∥2
+ν dk ≤
+�
+K
+�
+R
+∥ FνF(λ, k) ∥2
+ν η 2ρ−3
+2 (λ)dλ dk
+(4.6)
+Let T be the tempered distribution on R defined by T := F−1
+R η 2ρ−3
+2 . By Lemma 4.2, T is compactly supported . Let
+R0 > 1 such that supp T ⊂ [−R0, R0]. Then (4.6) is equivalent to
+�
+K
+∥ FR(T ∗ RF(. , k))(λ) ∥2
+ν dk ≤ CR
+�
+K
+�
+R
+FR(T ∗ RF(. , k))(λ) ∥2
+ν dλ dk,
+(4.7)
+where ∗ denotes the convolution on R.
+From suppT ⊂ [−R0, R0] and Lemma 4.3, it follows that for any k ∈ K, supp (T ∗ RF(. , k)) ⊂ [−(R + R0), R + R0].
+Thus
+�
+K
+∥ FR(T ∗ RF(. , k)(λ) ∥2
+ν dk ≤ 2(R + R0)
+�
+K
+�
+R
+∥ (T ∗ RF(. k))(t) ∥2
+ν dt dk
+Next use the Euclidean Plancherel formula to get (4.7), and the proof is finished.
+As a consequence of Proposition 4.1, we obtain the uniform continuity estimate for the Poisson transform Pν
+λ.
+Corollary 4.1. Let ν ∈ N. There exists a positive constant Cν such that for λ ∈ R\{0}, we have
+sup
+R>1
+�
+1
+R
+�
+B(R)
+∥ Pν
+λf(g) ∥2
+ν dgK
+�1/2
+≤ Cν |cν(λ)| ∥ f ∥L2(K,σν)
+(4.8)
+for every f ∈ L2(K, σν).
+13
+
+Proof. Let F ∈ L2(G, τν) with supp F ⊂ B(R), and let f ∈ L2(K, σν). Since λ is real and τν is unitary, the Poisson
+transform and the restriction Fourier transform are related by the following formula
+�
+B(R)
+< Pν
+λf(g), F(g) >ν dg =
+�
+K
+< f(k), FνF(λ, k) >ν dk.
+Thus
+|
+�
+B(R)
+< Pν
+λf(g), F(g) >ν dg | ≤ ∥f∥L2(K,σν)(
+�
+K
+∥ FνF(λ, k) ∥2
+ν dk)
+1
+2
+≤ Cν|cν(λ)|R1/2 ∥ f ∥L2(K,τν)∥ F ∥L2(G,τν),
+by the restriction Fourier theorem. Taking the supermum over all F with ∥ F ∥L2(G,τν)= 1, the corollary follows.
+5
+Asymptotic expansion for the Poisson transform
+In this section we give an asymptotic expansion for the Poisson transform.
+We first start by establishing some
+intermediate results.
+Let L2
+λ(K, σν) denote the finite linear span of the functions
+f g
+λ,v : k �−→ f g
+λ,v(k) = e(iλ−ρ)H(g−1k)τ −1
+ν (κ(g−1k))v,
+g ∈ G, v ∈ Vν.
+Lemma 5.1. For λ ∈ R \ {0}, ν ∈ N the space L2
+λ(K, σν) is a dense subspace of L2(K, σν).
+Proof. As λ ∈ R \ {0}, the density is just a reformulation of the injectivity of the Poisson transform Pν,λ.
+Lemma 5.2. Let λ ∈ R \ {0}, ν ∈ N. Then there exists a unique unitary isomorphism U ν
+λ on L2(K, σν) such that :
+U ν
+λ f g
+λ,v = f g
+−λ,v,
+g ∈ G.
+Moreover, for f1, f2 ∈ L2(K, σν), we have Pν
+λF1 = Pν
+−λF2 if and only if U ν
+λF1 = F2 ( i.e. U ν
+λ = (Pν
+−λ)−1 ◦ Pν
+λ).
+Proof. The proof is the same as in the scalar case so we omit it.
+We now introduce the function space B∗(G, τν) on G, consisting of functions F in L2
+loc(G, τν) satisfying
+∥ F ∥B∗(G,τν)= sup
+j∈N
+[2− j
+2
+�
+Aj
+∥ F(g) ∥2
+ν dgK] < ∞,
+where A0 = {g ∈ G; d(0, g.0) < 1} and Aj = {g ∈ G; 2j−1 ≤ d(0, g.0) < 2j}, for j ≥ 1.
+One could easily show that ∥ F ∥B∗(G,τν)≤∥ F ∥∗≤ 2 ∥ F ∥B∗(G,τν).
+We define an equivalent relation on B∗(G, τν). For F1, F2 ∈ B∗(G, τν) we write F1 ≃ F2 if
+lim
+R→+∞
+1
+R
+�
+B(R)
+∥ F1(g) − F2(g) ∥2
+ν dg = 0.
+Note that by using the polar decomposition we see that F1 ≃ F2 if
+lim
+R→+∞
+1
+R
+�
+K×[0,R]
+∥ F1(ketH)) − F2(ketH)) ∥2
+ν ∆(t) dt dk = 0.
+We now state the main result of this section
+14
+
+Theorem 5.1. Let ν ∈ N, λ ∈ R\{0}. For f ∈ L2(K, σν) we have the following asymptotic expansions for the Poisson
+transform in B∗(G, τν)
+Pλ,νf(x) ≃ τ −1
+ν (k2(x))[cν(λ)e(iλ−ρ)(A+(x)f(k1(x)) + cν(−λ)e(−iλ−ρ)(A+(x))U ν
+λf(k1(x))],
+(5.1)
+where x = k1(x)eA+(x)k2(x).
+Most of the proof of the above theorem consists in proving the following Key Lemma, giving the asymptotic ex-
+pansion for the translates of the τν-spherical function.
+KEY LEMMA. For λ ∈ R \ {0}, g ∈ G and v ∈ Vν, we have the following asymptotic expansion in B∗(G, τν)
+Φν,λ(g−1x). v ≃ τ −1
+ν (k2(x))
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)A+(x)f g
+sλ,v(k1(x)),
+x = k1(x)eA+(x)k2(x).
+Proof of Theorem 5.1. We first note that both side of (5.1) depend continuously on f ∈ L2(K, σν). This can
+be proved in the same manner as in [8]. Therefore we only have to prove that the asymptotic expansion (5.1) holds
+for f ∈ L2
+λ(K, σν). Let f = f g
+λ,v. Then according to [[11], Proposition 3.3], we have
+Pν
+λf(x) = Φν,λ(g−1x)v.
+The theorem follows from the Key lemma.
+As a consequence of Theorem 5.1 we obtain the following result giving the behaviour of the Poisson integrals.
+Proposition 5.1.
+1. For any f ∈ L2(K, σν) we have the Plancherel-Poisson formula
+lim
+R→+∞
+1
+R
+�
+B(R)
+∥ Pν
+λf(g) ∥2
+ν dgK = 2 | cν(λ) |2 ∥ f ∥2
+L2(K,σν)
+(5.2)
+2. Let ν ∈ N. There exists a positive constant Cν such that for any λ ∈ R \ {0}, we have
+C−1
+ν
+| cν(λ) | ∥ f ∥L2(K,σν)≤∥ Pλ
+ν f ∥∗≤ Cν | cν(λ) | ∥ f ∥L2(K,σν),
+(5.3)
+for every f ∈ L2(K, σν).
+Proof.
+1. We define for f ∈ L2(K, σν)
+Sν
+λf(x) := τ −1
+ν (k2(x))[cν(λ)e(iλ−ρ)(A+(x)f(k1(x)) + cν(−λ)e(−iλ−ρ)(A+(x))U ν
+λf(k1(x))],
+x = k1(x)eA+(x)k2(x).
+By the unitarity of Uλ, we have
+1
+R
+�
+B(R)
+∥Sν
+λf(g)∥2dgK = 2|cν(λ)|2∥f∥2
+L2(K,τν)
+�
+1
+R
+� R
+0
+e−2ρt∆(t)dt
+�
++ 2|cν(λ)|2ℜ
+�
+< f, Uλf >L2(K,σν)
+1
+R
+� R
+0
+e2(iλ−ρ)t∆(t)dt
+�
+.
+From
+lim
+R→+∞
+1
+R
+� R
+0
+e−2ρt∆(t)dt = 1, and
+lim
+R→+∞
+1
+R
+� R
+0
+e2(iλ−ρ)t∆(t)dt = 0, we deduce that
+lim
+R→+∞
+1
+R
+�
+B(R)
+∥ Sν
+λf(g) ∥2
+ν dgK = 2 | cν(λ) |2∥ f ∥2
+L2(K,σν) .
+(5.4)
+15
+
+Next write
+1
+R
+�
+B(R)
+∥ Pν
+λf(g) ∥2
+ν dgK = 1
+R
+�
+B(R)
+(∥ Sν
+λf(g) ∥2
+ν + ∥ Pν
+λf(g) − Sν
+λf(g) ∥2
+ν
++ 2Re[< Pν
+λf(g) − Sν
+λf(g), Sν
+λf(g) >])dgK.
+The estimate (5.2) then follows from (5.4), Theorem 5.1 and the Schwarz inequality.
+2. The right hand side of the estimate (5.3) has already been proved, see corollary 4.1.
+The left hand side of the estimate (5.3) obviously follows from the estimate (5.2). This finishes the proof of the
+proposition.
+Remark 5.1. Let f1, f2 ∈ L2(K, σν). Then using the polarization identity as well as the estimate (5.2), we get
+lim
+R→+∞
+1
+R
+�
+B(R)
+< Pν
+λf1(g), Pν
+λf2(g) >ν dgK = 2 | cν(λ) |2< f1, f2 >L2(K,σν)
+(5.5)
+6
+Proof of the main results
+In this section we shall prove Theorem 1.1 on the L2-range of the vector Poisson transform and Theorem 1.2 charac-
+terizing the image Qν
+λ(L2(G, τν).
+6.1
+The L2-range of the Poisson transform
+We first recall some results of harmonic analysis on the homogeneous vector bundle K ×M Vν associated to the
+representation σν of M.
+Let �K be the unitary dual of K. For δ ∈ �K let Vδ denote a representation space of δ with dδ = dim Vδ. We denote by
+�K(σν) the set of δ ∈ �K such that σν occurs in δ |M with multiplicity mδ > 0.
+The decomposition of L2(K, σν) under K (the group K acts by left translations on this space) is given by the Frobenius
+reciprocity law
+L2(K, σν) =
+�
+δ∈�
+K(σν)
+Vδ ⊗ HomM(Vν, Vδ),
+where v ⊗L, for v ∈ Vδ, L ∈ HomM(Vν, Vδ) is identified with the function (v ⊗L)(k) = L∗(δ(k−1)v), where L∗ denotes
+the adjoint of L.
+For each δ ∈ �K(σν) let (Lj)mδ
+j=1 be an orthonormal basis of HomM(Vν, Vδ) with respect to the inner product
+< L1, L2 >=
+1
+ν + 1T r(L1L∗
+2).
+Let {v1, · · · , vdδ} be an orhonormal basis of Vδ. Then
+f δ
+ij : k →
+�
+dδ
+ν + 1L∗
+i δ(k−1)vj,
+1 ≤ i ≤ mδ,
+1 ≤ j ≤ dδ,
+δ ∈ �K(σ)
+form an orthonormal basis of L2(K, σν).
+For f ∈ L2(K, σν) we have the Fourier series expansion f(k) =
+�
+δ∈�
+K(σ)
+mδ
+�
+i=1
+dδ
+�
+j=1
+aδ
+ijf δ
+ij(k) with
+∥ f ∥2
+L2(K,σ)=
+�
+δ∈�
+K(σ)
+mδ
+�
+i=1
+dδ
+�
+j=1
+| aδ
+ij |2 .
+16
+
+We define for δ ∈ �K(σ) and λ ∈ C, the generalized Eisenstein integral
+ΦL
+λ,δ(g) =
+�
+K
+e−(iλ+ρ)H(g−1k)τν(κ(g−1k))L∗δ(k−1)dk,
+L ∈ HomM(Vν, Vδ).
+It is easy to see that ΦL
+λ,δ satisfies the following identity
+ΦL
+λ,δ(k1gk2) = τν(k−1
+2 )ΦL
+λ,δ(g)δ(k−1
+1 ),
+k1, k2 ∈ K, g ∈ G.
+We now prove an asymptotic estimate for the generalized Eisenstein integrals.
+Proposition 6.1. Let ν ∈ N, λ ∈ R \ {0}. Then for δ ∈ �K(σν), T, S ∈ HomM(Vν, Vδ) we have
+lim
+R→+∞
+1
+R
+�
+B(R)
+Tr
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+dgK = 2 | cν(λ) |2 Tr(T S∗).
+(6.1)
+Proof. By definition we have
+lim
+R→+∞
+1
+R
+�
+B(R)
+Tr
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+dgK =
+dδ
+�
+j=1
+lim
+R→+∞
+1
+R
+�
+B(R)
+< ΦS
+λ,δ(g)vj, ΦT
+λ,δ(g)vj >ν dgK
+Noting that ΦT
+λ,δ(g)vj is the Poisson transform of the function k �→ L∗δ(k−1)vj and using (5.5), we get
+lim
+R→+∞
+1
+R
+�
+B(R)
+Tr
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+dgK = 2 | cν(λ) |2
+dδ
+�
+j=1
+�
+K
+< S∗δ(k−1)vj, T ∗δ(k−1)vj >ν dk.
+Hence Schur Lemma lead us to conclude that
+lim
+R→+∞
+1
+R
+�
+B(R)
+Tr
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+dgK = 2 | cν(λ) |2 Tr(T S∗), and
+the proof is finished.
+Remark 6.1. Noting that
+T r(
+�
+ΦT
+λ,δ(g)∗ΦS
+λ,δ(g)
+�
+= T r(
+�
+ΦT
+λ,δ(a)∗ΦS
+λ,δ(a)
+�
+,
+g = k1 a k2,
+it follows from (6.1) that
+lim
+R→+∞
+1
+R
+� R
+0
+T r
+�
+ΦT
+λ,δ(at)∗ΦS
+λ,δ(at)
+�
+∆(t)dt =| cν(λ) |2 Tr(T S∗).
+(6.2)
+Proof of Theorem 1.1.
+(i) The estimate (5.3) implies that the Poisson transform Pλ,ν maps L2(K, σν) into Eλ(G, τν) and that the estimate
+(1.5) holds.
+(ii) We now prove that the Poisson transform maps L2(K, σν) onto E2
+λ(G, τν). Let F ∈ E2
+λ(G, τν). Since λ ∈ R \ {0},
+we know by Theorem 2.1 that there exists a hyperfunction f ∈ C−ω(K, σν) such that F = Pλ,νf.
+Let f =
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+mδ
+�
+i=1
+aδ
+ijf δ
+ij, be the Fourier series expansion of f. Then we have
+F(g) =
+�
+δ∈�
+K(σ)
+�
+dδ
+ν + 1
+dδ
+�
+j=1
+mδ
+�
+i=1
+aδ
+ijΦLi
+λ,δ(g)vj
+in
+C∞(G, V ).
+By the Schur relations, we have
+�
+K
+< ΦLi
+λ,δ(kat)vj, ΦLm
+λ,δ′(kat)vn >ν dk =
+�
+0
+if δ ≁ δ′
+1
+dδ T r(ΦLm
+λ,δ′ (at))∗ΦLi
+λ,δ(at) < vj, vn > if
+δ′ = δ
+17
+
+Therefore
+�
+K
+∥ F(kat) ∥2 dk =
+1
+ν + 1
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+�
+1≤i,j≤mδ
+aδ
+ijaδ
+mjT r[(ΦLm
+λ,δ (at))∗ΦLi
+λ,δ(at)]
+=
+1
+ν + 1
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+T r
+
+
+�
+1≤i,m≤mδ
+(aδ
+mjΦLm
+λ,δ (at))∗(aδ
+ijΦLi
+λ,δ(at)
+
+
+=
+1
+ν + 1
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+∥
+mδ
+�
+i=1
+aδ
+ijΦLi
+λ,δ(at) ∥2
+HS,
+Let Λ be a finite subset in �K(σ). Since ∥ F ∥∗< ∞, it follows that, for any R > 1 we have
+∞ >∥ F ∥2
+∗≥
+1
+ν + 1
+�
+δ∈Λ
+dδ
+�
+j=1
+1
+R
+� R
+0
+∥
+mδ
+�
+i=1
+aδ
+ijΦLi
+λ,δ(at) ∥2
+HS ∆(t) dt
+By (6.2) we have
+lim
+R→∞
+1
+R
+� R
+0
+∥
+mδ
+�
+i=1
+aδ
+ijΦLi
+λ,δ(at) ∥2
+HS ∆(t) dt = lim
+R→∞
+�
+1≤i,m≤mδ
+aδ
+ijaδ
+mj
+1
+R
+� R
+0
+T r[(ΦLm
+λ,δ (at))∗ΦLi
+λ,δ(at)] ∆(t)dt
+= 2 | cν(λ) |2
+�
+1≤i,m≤mδ
+aδ
+ijaδ
+mjT r(LiL∗
+m)
+= 2(ν + 1) | cν(λ) |2
+mδ
+�
+i=1
+| aδ
+ij |2 .
+Thus ∞ >∥ F ∥2
+∗≥| cν(λ) |2 �
+δ∈Λ
+dδ
+�
+j=1
+mδ
+�
+i=1
+| aδ
+ij |2. Since Λ is arbitrary, it follows that
+| cν(λ) |2
+�
+δ∈�
+K(σ)
+dδ
+�
+j=1
+mδ
+�
+i=1
+| aδ
+ij |2≤∥ F ∥2
+∗ .
+This shows that f ∈ L2(K, σν) with | cν(λ) |∥ f ∥L2(K,σν)≤∥ Pν
+λf ∥∗ and the proof of the theorem is completed.
+6.2
+The L2-range of the generalized spectral projections
+We now proceed to the poof of the second main result of this paper.
+Proof of Theorem 1.2.
+Let F ∈ L2
+c(G, τν) ∩ C∞(G, τν). It follows from the definition ( see (1.8)) that the operator Qν
+λ may be written as
+Qν
+λF(g) =| cν(λ) |−2 Pν
+λ(FνF(λ, .))(g).
+(6.3)
+Using Theorem 1.1 we deduce that
+sup
+R>1
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2
+ν dgK ≤ Cν | cν(λ) |−2
+�
+K
+∥ FνF(λ, k) ∥2
+ν dk.
+The above inequality and the Plancherel formula (3.4) imply
+� ∞
+0
+(sup
+R>1
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2
+ν dgK) dλ ≤ Cν
+� ∞
+0
+�
+K
+∥ FνF(λ, k) ∥2
+ν| cν(λ) |−2 dk dλ
+≤ Cν ∥ F ∥2
+L2(G,τ) .
+18
+
+This prove the right hand side of the inequality (1.9).
+From (6.3) and (1.6) we have
+lim
+R→∞
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2
+ν dgK = 2 | cν(λ) |−2
+�
+K
+∥ FνF(λ, k) ∥2
+ν dk,
+and since for all R > 1
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2 dgK ≤ Cν | cν(λ) |−2
+�
+K
+∥ FνF(λ, k) ∥2 dk,
+a.e. λ ∈ (0, ∞),
+we may apply the Lebesgue’s dominated convergence theorem to get
+lim
+R→∞
+� ∞
+0
+�
+1
+R
+�
+B(R)
+∥ Qν
+λF(g) ∥2
+ν dgK
+�
+dλ = 2 ∥ F ∥2
+L2(G,τν) .
+It follows from the above equality that
+C ∥ F ∥2
+L2(G,τν)≤
+� ∞
+0
+(sup
+R>1
+�
+B(R)
+∥ Qν
+λF(x) ∥2 dx) dλ.
+This complete the proof of the inequality (1.9).
+We now prove that Qν
+λ maps L2
+c(G, τν) onto E2
+λ(G, τν). Let Fλ ∈ E2
+λ(G, τν). Then we have
+sup
+R>1
+1
+R
+�
+B(R)
+∥ Fλ(g) ∥2
+ν dgK < ∞,
+for a.e.
+λ ∈ (0, ∞).
+By Theorem 1.1, there exists fλ ∈ L2(K, σν) such that Fλ(g) =| cν(λ) |−2 Pν
+λfλ(g) with
+sup
+R>1
+1
+R
+�
+B(R)
+∥ Fλ(g) ∥2
+ν dgK ≥ C−1
+ν
+| cν(λ) |−2
+�
+K
+∥ fλ(k) ∥2 dk
+Integrating the both side of the above inequality over (0, ∞), we get
+∞ >∥ Fλ ∥2
+∗≥ C−1
+ν
+� ∞
+O
+�
+K
+∥ fλ(k) ∥2
+ν | cν(λ) |−2 dk dλ.
+It now follows from Theorem 3.1, that there exists F ∈ L2
+c(G, τν) such that FνF(λ, k) = fλ(k).
+Henceforth Fλ(g) =| cν(λ) |−2 Pλ,ν(FνF(λ, .)(g). This finishes the proof of Theorem 1.2.
+7
+Proof of the Key Lemma
+In this section we prove the Key Lemma of this paper. To this end we need to establish some auxiliary results. We
+first prove an asymptotic formula for the τν-spherical function.
+Proposition 7.1. Let λ ∈ R \ {0}. For any v ∈ Vν we have
+Φν,λ(g). v ≃
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)A+(g)τ −1
+ν (κ1(g)κ2(g)). v,
+(7.1)
+g = κ1(g)eA+(g)κ2(g)
+Proof. Since ∆(t) ≤ e2ρ t, we get
+1
+R
+�
+B(R)
+∥ e(iλ−ρ)A+(g)τ −1
+ν (κ1(g)κ2(g)). v ∥2 dg = 1
+R ∥ v ∥2
+� R
+0
+e−2ρ t∆(t)dt
+≤∥ v ∥2 .
+19
+
+This shows that the right hand side of (7.1) belongs to B∗(G, τν).
+Since λ ∈ R \ {0}, we may use the identity (A3) to write
+ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t =
+�
+s∈{±1}
+cν(sλ)
+�
+(2 cosh t)νΨρ−2,ν+1
+sλ
+(t) − e(isλ−ρ)t�
+=
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t �
+(1 + e−2t)νe(ρ+ν−isλ)tΨρ−2,ν+1
+sλ
+(t) − 1
+�
+.
+It follows from (A2’) that
+ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t =
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t �
+(1 + e−2t)ν − 1) + e−2tEsλ(t)
+�
+,
+where | Esλ(t) |≤ 2νC if t ≥ 1. Therefore
+| ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t |≤ Cν,λe−ρe−2t,
+if t ≥ 1. This together with
+| ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t |≤ Cν,λe−ρt,
+for t ∈ [0, 1], imply that
+lim
+R→∞
+1
+R
+�
+B(R)
+∥ Φν,λ(g). v −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)A+(g)τ −1(κ1(g)κ2(g)). v ∥2
+ν dgK =
+=∥ v ∥2 lim
+R→∞
+1
+R
+� R
+0
+| ϕν,λ(t) −
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)t |2 ∆(t) dt = 0,
+and the proof is finished.
+Lemma 7.1. Let g ∈ G, k ∈ K and t a non negative real number . Then we have
+0 ≤ A+(g−1k exp(tH)) − H(g−1k exp(tH)) ≤ 1+ | g.0 |
+1− | g.0 |e−2t,
+(7.2)
+Proof. Let g−1 =
+�
+a
+b
+c
+d,
+�
+and k ==
+�
+u
+0
+O
+v,
+�
+, where a, b, c and d are n×n, n×1, 1×n and 1×1 matrices respectively.
+A direct computation yields
+g−1k exp(tH) =
+�
+∗
+∗ ∗
+c1
+d1
+�
+,
+where c1 = c u
+�
+cosh t
+0
+0
+In−1
+�
+and d1 = sinh t cue1 + cosh t dv.
+By (2.1) we have
+eH(g−1k exp(tH)) = et | cue1 + dv |,
+and
+eA+(g−1k exp(tH)) =| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1)
+1
+2 .
+20
+
+From
+eA+(g−1k exp(tH))−H(g−1k exp(tH)) =
+e−t
+| cue1 + dv |[| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1)
+1
+2 ],
+together with
+| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1)
+1
+2 ≤ 2 | sinh t cue1v−1 + cosh t d |
+≤| cue1v−1 + d | et+ | d − cue1v−1 | e−t
+we deduce that
+e(A+(g−1k exp(tH))−H(g−1k exp(tH)) ≤ 1 + | d − cue1v−1 |
+| cue1v−1 + d |e−2t.
+Noting that (g.0)∗ = −(d−1c), and k.e1 = ue1v−1, we get
+e(A+(g−1k exp(tH))−H(g−1k exp(tH)) ≤ 1 + | 1+ < g.0, k.e1 >|
+| 1− < g.0, k.e1 >|e−2t
+≤ 1 + 1+ | g.0 |
+1− | g.0 |e−2t,
+from which we deduce (7.2), and the proof of the lemma is finished.
+Proof of the Key Lemma. Since B∗(G, τν) is G-invariant, we may apply Proposition 7.1 to get
+Φν,λ(g−1x)v ≃ τ −1
+ν (κ1(g−1x)κ2(g−1x)
+�
+s∈{±}
+cν(sλ)e(isλ−ρ)A+(g−1x)v.
+Thus it suffices to show that
+τ −1
+ν (κ1(g−1x)κ2(g−1x)
+�
+s∈{±}
+cν(sλ)e(isλ−ρ)A+(g−1x)v ≃ τ −1
+ν (k2(x))
+�
+s∈{±1}
+cν(sλ)e(isλ−ρ)A+(x)f g
+sλ,v(k1(x)),
+(7.3)
+Note that
+τ −1
+ν [k1(g−1k1(x)eA+(x)k2(x))k2(g−1k1(x)eA+(x)k2(x))] = τ −1
+ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))k2(x))],
+x = k1(x)eA+(x)k2(x).
+Henceforth (7.3) is equivalent to
+τ −1
+ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))]
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)A+(g−1k1(x)eA+(x)) v
+≃
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)A+(x)f g
+sλ,v(k1(x))
+(7.4)
+We write the left hand side of (7.4) as
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)A+(x)f g
+sλ,v(k1(x)) + rg(x)v,
+where
+rg(x) =τ −1
+ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))]
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)A+(g−1k1(x)eA+(x))
+−
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)[A+(x)+H(g−1k1(x))]τ −1
+ν (κ(g−1k1(x)),
+x ∈ G
+(7.5)
+21
+
+To finish the proof we show that for each g ∈ G, rg ≃ 0.
+Noting that
+H(g−1k1(x)eA+(x)) = H(g−1k1(x)) + A+(x),
+we rewrite rg as
+rg(x) = [τ −1
+ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))) − τ−1
+ν (κ(g−1k1(x))]
+�
+s∈{±1}
+cν(λ)e(isλ−ρ)H(g−1k1(x)eA+(x))
++ τ −1
+ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x)))
+
+ �
+s∈{±1}
+cν(λ)[e(isλ−ρ)A+(g−1k1(x)eA+(x))) − e(isλ−ρ)H((g−1k1(x)eA+(x))]
+
+
+=: Ig(x) + Jg(x).
+Using the following
+Lemma 7.2. Let g =
+�
+a
+b
+c
+d
+�
+∈ Sp(n, 1). Then we have
+τν(κ1(g)κ2(g)) = τν( d
+| d |)
+(7.6)
+τν(κ(g)) = τν( ce1 + d
+| ce1 + d |)
+(7.7)
+lim
+R→∞ τν(κ1(g exp(RH))κ2(g exp(RH))) = τν(κ(g)).
+(7.8)
+we easily see that Igv ≃ 0.
+We have
+Jg(x) =τ −1
+ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x)))e(isλ−ρ)H(g−1k1(x)eA+(x))
+�
+s∈{±1}
+cν(λ)
+�
+e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1
+�
+As τν is unitary we have
+1
+R
+�
+K×[0,R]
+∥ Jg(ketH)v ∥2
+ν ∆(t)dt dk
+≤∥ v ∥2 2 | cν(λ) |2
+R
+�
+K×[0,R]
+e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2
+From
+| e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |≤ C(| λ | +ρ) | A+(g−1k1(x)eA+(x)) − H(g−1k1(x)eA+(x) |
+together with Lemma 7.2 we get
+�
+K×[0,R]
+e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2
+≤
+�
+C(| λ | +ρ)1+ | g.0 |
+1− | g.0 |
+�2 1
+R
+�
+K×[0,R]
+e−2ρH(g−1k)e−2(ρ+2t)∆(t) dk dt.
+22
+
+As
+�
+K
+e−2ρH(g−1k) dk = 1 and ∆(t) ≤ 2ρe2ρt we obtain
+lim
+R→∞
+1
+R
+�
+K×[0,R]
+e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2= 0.
+This shows that Jg ≃ 0. Therefore we have proved that for each g ∈ G, rg ≃ 0 as to be shown.
+It remain to prove Lemma 7.2.
+Proof of Lemma 7.2. If g =
+�
+a
+b
+c
+d
+�
+=
+�
+u1
+0
+0
+v1
+�
+at
+�
+u2
+0
+0
+v2
+�
+with respect to the Cartan decomposition G =
+KAK. Then we easily see that d = cosh t v1v2 and (7.6) follows. Analogously if g =
+�
+a
+b
+c
+d
+�
+=
+�
+u
+0
+0
+v
+�
+at n with
+respect to the Iwasawa decomposition. Then from g.e1 =
+�
+ae1 + b
+ce1 + d
+�
+= et
+�
+u
+v
+�
+we get et v = ce1 + d and (7.7) follows.
+We have
+g exp(RH) =
+�
+∗
+∗∗
+∗ ∗ ∗
+sinh Re1 + cosh Rd
+�
+Then (7.6) imply that τν(κ1(g)κ2(g)) = τν( tanh Rce1+d
+|tanh Rce1+d|). Thus limR→∞ τν(κ1(g)κ2(g)) = τν( ce1+d
+|ce1+d|). This finishes
+the proof of Lemma 7.2, and the proof of the Key Lemma is completed.
+8
+Appendix
+In this section we collect some results on the Jacobi functions, referring to [19] for more details.
+For α, β, λ ∈ C; α ̸= −1, −2, · · · and t ∈ R, the Jacobi function is defined by
+φ(α,β)
+λ
+(t) = 2F1(iλ + ρα,β
+2
+, −iλ + ρα,β
+2
+; α + 1; − sinh2 t),
+where 2F1 is the Gauss hypergeometric function and ρα,β = α + β + 1.
+The Jacobi function φ(α,β)
+λ
+is the unique even smooth function on R which satisfy φ(α,β)
+λ
+(0) = 1 and the differential
+equation
+{ d2
+dt2 + [(2α + 1) coth t + (2β + 1) tanh t] d
+dt + λ2 + ρ2
+α,β}φ(α,β)
+λ
+(t) = 0.
+(A1)
+For λ /∈ −iN another solution Ψα,β
+λ
+of (A1) such that
+Ψα,β
+λ
+(t) = e(iλ−ρα,β)t(1 + ◦(1)),
+as
+t → ∞
+(A2)
+is given by
+Ψα,β
+λ
+(t) = (2 sinh t)iλ−ρα,β 2F1(ρα,β − iλ
+2
+, β − α + 1 − iλ
+2
+; 1 − iλ; −
+1
+sinh2 t).
+Moreover there exists a constant C > 0 such that for all λ ∈ R and all t ≥ 1 we have
+Ψα,β
+λ
+(t) = e(iλ−ρα,β)t(1 + e−2tΘλ(t)),
+with
+| Θλ(t) |≤ C.
+(A2’)
+For λ /∈ iZ, we have
+φ(α,β)
+λ
+(t) =
+�
+s=±1
+cα,β(sλ)Ψα,β
+sλ (t)
+(A3)
+23
+
+where
+cα,β(λ) = 2ρα,β−iλ Γ(α + 1)Γ(iλ)
+Γ( iλ+ρα,β
+2
+)Γ( iλ+α−β+1
+2
+)
+.
+For ℜ(iλ) > 0, the asymptotic behaviour of φ(α,β)
+λ
+as t → ∞ is then given by
+lim
+t→∞ e(ρα,β−iλ)tφ(α,β)
+λ
+(t) = cα,β(λ).
+(A4)
+Let De(R) denote the space of even smooth function with compact support on R. For f ∈ De(R), the Fourier-Jacobi
+transform J α,βf (λ ∈ C) is defined by
+J α,βf(λ) =
+� ∞
+0
+f(t)φ(α,β)
+λ
+(t)∆α,β(t) dt,
+(A5)
+where ∆α,β(t) = (2 sinh t)2α+1(2 cosh t)2β+1.
+In the sequel, we assume that α > −1, β ∈ R. Then the meromorphic function cα,β(−λ)−1 has only simple poles for
+ℑλ ≥ 0 which occur in the set
+Dα,β = {λk = i(| β | −α − 1 − 2k); k = 0, 1, · · · , | β | −α − 1 − 2k > 0}.
+(If | β |≤ α + 1, then Dα,β is empty).
+The following inversion and Plancherel formulas for the Jacobi transform hold for every f ∈ De(R):
+f(t) = 1
+2π
+� ∞
+0
+(J α,βf)(λ) φ(α,β)
+λ
+(t) | cα,β(λ) |−2 dλ +
+�
+λk∈Dα,β
+dk(J α,βf)(λk) φ(α,β)
+λk
+(t),
+(A6)
+� ∞
+0
+| f(t) |2 ∆(t) dt = 1
+2π
+� ∞
+0
+| (J α,βf)(λ) |2 | cα,β(λ) |−2 dλ +
+�
+λk∈Dα,β
+dk | (J α,βf)(λk) |2
+(A6’)
+where dk = −i Resλ=λk(cα,β(λ)cα,β(−λ))−1, is given explicitly by
+dk = (β − α − 2k − 1)2−2(α+β)Γ(α + k + 1)Γ(β − k)
+Γ2(α + 1)Γ(β − α − k)k!
+.
+(A7)
+References
+[1] Anker,J. P.: A basis inequality for Scattering Theory for Riemannian Symmetric Spaces of the Noncompact Type.
+Amer. J. Math. 113 (3), 391-398 (1991)
+[2] A. Agmon, L. Hormander, Asymptotic properties of solutions of differential equations with simple characteristics,
+J. Analyse Math, 30 (1976), 1-38.
+[3] A. Boussejra, A. Intissar, Caractérisation des integrales de Poisson-Szego de L2(∂Bn) dans la boule de Bergman
+Bn, n ≥ 2. C. R. Acad. Sci. 318 (1994).
+[4] A. Boussejra, A. Intissar, L2-concrete Spectral Analysis of the Invariant Laplacian in the Unit Complex Ball. J.
+Funct. Anal. 160 (1998), 115-140.
+[5] A. Boussejra, H. Sami Characterization of the Lp-range of the Poisson transform in Hyperbolic spaces. J. Lie
+Theory. 12 (2002), 1-14.
+24
+
+[6] A. Boussejra, Boundary behavior of Poisson integrals on Boundaries of Symmetric Spaces, J. Lie Theory. 21
+(2011), 243-261.
+[7] A. Boussejra, N. Ourchane, Characterization of the Lp-range of the Poisson Transform On the Octonionic Plane.
+J. Lie Theory. 28 (2018) 805-828.
+[8] A. Boussejra, N. Imesmad, and A. Ouald Chaib, L2-Poisson integral representations of eigensections of invariant
+differential operators on a homogeneous line bundle over the complex Grassmann manifold SU(r, r +b)/S(U(r)×
+U(r + b)). Ann Glob Anal Geom 61, 399–426 (2022).
+[9] T. P. Branson, G. Olafsson, and H. Schlichtkrull, A bundle-valued Radon transform with applications to invariant
+wave equations, Quart. J. Math. Oxford (2) 45 (1994), 429-461.
+[10] W. O. Bray, Aspects of harmonic analysis on real hyperbolic space. Fourier analysis (Orono, ME, 1992), Lecture
+Notes in Pure and Appl. Math., vol. 157, Dekker, New York, 1994, pp. 77-102.
+[11] R. Camporesi, The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric
+spaces, Pacific J. Math. 179 (1997) 263–300.
+[12] G. van Djick, A. Pasquale, Harmonic Analysis on Vector bundles over Sp(1, n)/Sp(1) × Sp(n), L’Enseignement
+Mathématique, t. 45 (1999), 219-252.
+[13] S. Helgason, Groups and Geometric Analysis; Integral geometry, Invariant Differential operators and Spherical
+Functions. Academic Press, New York 1984.
+[14] S. Helgason, Groups and Geometric Analysis, volume 83 of Mathematical Surveys and Monographs. Amer. Math.
+Soc., Providence, RI, 2000.
+[15] A. D. Ionescu, On the Poisson transform on symmetric spaces of rank one, J. Funct. Anal. 174 (2000), no 2,513-523.
+[16] K. Kaizuka, A characterization of the L2-range of the Poisson transform related to Strichartz conjecture on
+symmetric spaces of noncompact type, Adv. Math. 303 (2016) 464-501.
+[17] M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima, M. Tanaka, Eigenfunctions of invariant
+differential operators on a symmetric space, Ann. of Math. (2) 111 (1980), no. 3, 589-608.
+[18] A. W. Knapp, Representation Theory of Semisimple Groups. An overview based on Examples, Princeton Math.
+Ser.36, Princeton Univ. Press, Princeton, NJ, 1986.
+[19] T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups. In: Askey, R.A., Koorn-
+winder, T.H., Schempp, N. (eds.), Special functions: Group theoretical aspects and applications. Dordrecht:
+Reidel Publishing Company, 1984, pp. 1–85
+[20] P. Kumar, S. K. Ray, and R. P. Sarkar, Characterization of almost Lp-eigenfunctions of the Laplace-Beltrami
+operator, Trans. Amer. Math. Soc. 366 (2014), 3191-3225.
+[21] P. Kumar, Fourier restriction theorem and characterization on weak L2-eigenfunctions of the Laplace-Beltrami
+operator, J. Funct. Anal. 266 (2014) 5584–5597.
+[22] N. Lohoué and Th. Rychner, Some function spaces on symmetric spaces related to convolution operators, J.
+Funct. Anal. 55 (1984), no. 2, 200-219.
+25
+
+[23] M. Olbrich, Die Poisson-transformation für homogene Vektorbündel. PhD thesis, Humboldt-Unversität zu Berlin,
+1995.
+[24] P. Sjögren, Characterization of Poisson integrals on symmaetric spaces, Math. Scand. 49(1981), no 2, 229-249.
+[25] R S. Strichartz, Harmonic Analysis as Spectral Theory of Laplacians, J. Funct. Anal. 87 (1991) 51-148.
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+harmonique, Lecture Notes in Mathematics 404. Springer Verlag, Berlin 218-238.
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+26
+
diff --git a/9NE4T4oBgHgl3EQf3Q1N/content/tmp_files/load_file.txt b/9NE4T4oBgHgl3EQf3Q1N/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4adec651f6a1014bd3ef401a7a901ac0f4561a3a
--- /dev/null
+++ b/9NE4T4oBgHgl3EQf3Q1N/content/tmp_files/load_file.txt
@@ -0,0 +1,842 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf,len=841
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='05304v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='RT]  12 Jan 2023  A characterization of the L2-range of the Poisson transforms on a class of  vector bundles over the quaternionic hyperbolic spaces  Abdelhamid Boussejra ∗Achraf Ouald Chaib†  Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Faculty of Sciences  University Ibn Tofail,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Kénitra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Morocco  Abstract  We study the L2-boundedness of the Poisson transforms associated to the homogeneous vector bundles  Sp(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' 1)×Sp(n)×Sp(1) Vτ over the quaternionic hyperbolic spaces Sp(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' 1)/Sp(n)× Sp(1) associated with irreducible  representations τ of Sp(n)×Sp(1) which are trivial on Sp(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' As a consequence, we describe the image of the section  space L2(Sp(n, 1)×Sp(n)×Sp(1) Vτ) under the generalized spectral projections associated to a family of eigensections  of the Casimir operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Keywords: Vector Poisson transform, Fourier restriction estimate, Strichartz conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  1  Introduction  Let G be a connected real semisimple noncompact Lie group with finite center, and K a maximal compact subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Then X = G/K is a Riemannian symmetric space of noncompact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let G = KAN be an Iwasawa decomposition  of G, and let M be the centralizer of A in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We write g = κ(g)eH(g)n(g), for each g ∈ G according to G = KAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  A central result in harmonic analysis (see [17]) asserts that all joint eigenfunctions F of the algebra D(X) of invariant  differential operators, are Poisson integrals  F(g) = Pλf(g) :=  �  K  e(iλ+ρ)H(g−1k)f(k) dk,  of a hyperfunction f on K/M, for a generic λ ∈ a∗  c (the complexification of a∗ the real dual of a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Since then a characterization of the Lp-range of the Poisson transform was developed in several articles such as [3],  [5], [6], [7], [15], [20], [21], [22], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The problem of characterizing the image of the Poisson transform Pλ of L2(K/M) with real and regular spectral  parameter λ is intimately related to Strichartz conjecture [[25], Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5] on the uniform L2-boundedness of  the generalized spectral projections associated with D(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  To be more specific, consider the generalized spectral  projections Qλ defined initially for F ∈ C∞  c (X) by  QλF(x) =| c(λ) |−2 Pλ(FF(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' )(x),  λ ∈ a∗,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1)  where FF is the Helgason Fourier transform of F and c(λ) is the Harish-Chandra c-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Conjecture (Strichartz [[25], Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' There exists a positive constant C such that for any Fλ = QλF with  ∗e-mail: boussejra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='abdelhamid@uit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='ma  †e-mail:achraf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='oualdchaib@uit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='ma  1  F ∈ L2(X) we have  C−1 ∥ F ∥2  L2(X)≤  sup  R>0,y∈X  �  a∗  +  1  Rr  �  B(y,R)  | Fλ(x) |2 dx dλ ≤ C ∥ F ∥2  L2(X),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2)  and  ∥ F ∥2  L2(X)= γr lim  R→∞  �  a∗  +  1  Rr  �  B(y,R)  | Fλ(x) |2 dx dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3)  Conversely, if Fλ is any family of joint eigenfunctions for which the right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3) is finite, then there  exists F ∈ L2(X) such that Fλ = QλF for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' λ ∈ a∗  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Here r = rank X, and B(y, R) denotes the open ball in X of radius R about y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The constant γr depends on the  normalizations of the measures dx and dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The strichartz conjecture has been recently settled by Kaizuka, see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Most of the proof consists in proving a  uniform estimate for the Poisson transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' More precisely, the following was proved by Kaizuka [[16], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3]:  Let F be a joint eigenfunction with eigenvalue corresponding to a real and regular spectral parameter λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then F is  the Poisson transform by Pλ of some f ∈ L2(K/M) if and only if  sup  R>1  1  Rr  �  B(0,R)  | F(x) |2 dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Moreover there exists a positive constant C independent of such λ,  C−1 | c(λ) |2∥ f ∥2  L2(K/M)≤ sup  R>1  1  Rr  �  B(0,R)  | Pλf(x) |2 dx ≤ C | c(λ) |2∥ f ∥2  L2(K/M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The generalization of these results to vector bundles setting has only just begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' In [8] we extend Kaizuka result to  homogeneous line bundles over non-compact complex Grassmann manifolds (See also [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Our aim in this paper is to generalize theses results to a class of homogeneous vector bundles over the quaternionic  hyperbolic space G/K, where G is the symplectic group Sp(n, 1) with maximal compact subgroup K = Sp(n)×Sp(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  To state our results in rough form, let us first introduce the class of the homogenous vector bundles that we consider  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let τν be a unitary irreducible representation of Sp(1) realized on a (ν + 1)-dimensional Hilbert space  (V, (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' )ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We extend τν to a representation of K by setting τν ≡ 1 on Sp(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' As usual the space of sections of the  homogeneous vector bundle G ×K V associated with τν will be identified with the space Γ(G, τν) of vector valued  functions F : G → Vν which are right K-covariant of type τν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=',  F(gk) = τν(k)−1F(g),  ∀g ∈ G,  ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4)  We denote by C∞(G, τν) and C∞  c (G, τν) the elements of Γ(G, τν) that are respectively smooth, smooth with compact  support in G, and by L2(G, τν) the elements of Γ(G, τν) such that  ∥ F ∥L2(G,τν)=  ��  G/K  ∥ F(g) ∥2  ν dgK  � 1  2  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  In above ∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' ∥ν is the norm in Vν and ∥ F(gK) ∥ν=∥ F(g) ∥ν is well defined for F satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let σν denote the restriction of τν to the group M ≃ Sp(n−1)×Sp(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Over K/M we have the associated homogeneous  vector bundle K ×M Vν with L2-sections identified with L2(K, σν) the space of all functions f : K → Vν which are  M-covariant of type σν and satisfy  ∥ f ∥2  L2(K,σν)=  �  K  ∥ f(k) ∥2  ν dk < ∞,  2  where dk is the normalized Haar measure of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  For λ ∈ C and f ∈ L2(K, σν), the Poisson transform Pν  λf is defined by  Pν  λf(g) =  �  K  e−(iλ+ρ)H(g−1k)τν(κ(g−1k))f(k) dk  Let Ω denote the Casimir element of the Lie algebra g of G, viewed as a differential operator acting on C∞(G, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Then the image Pν  λ(L2(K, σν)) is a proper closed subspace of Eλ(G, τν) the space of all F ∈ C∞(G, τν) satisfying  Ω F = −(λ2 + ρ2 − ν(ν + 2))F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  For more details see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  For λ ∈ R \\ {0}, we define a weighted L2-space E2  λ(G, τν) consisting of all F in Eλ(G, τν) that satisfy  ∥ F ∥∗= sup  R>1  �  1  R  �  B(R)  ∥F(g)∥2  ν dgK  � 1  2  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Our first main result is an image characterization of the Poisson transform Pν  λ of L2(K, σν) for λ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let λ ∈ R\\{0} and ν a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (i) There exists a positive constant Cν independent of λ such that for f ∈ L2(K, σν) we have  C−1  ν |cν(λ)| ∥f∥L2(K,σν) ≤ ∥Pν  λf∥∗ ≤ Cν| | cν(λ) | ∥f∥L2(K,σν),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5)  with  cν(λ) = 2ρ−iλ  Γ(ρ − 1)Γ(iλ)  Γ( iλ+ν+ρ  2  )Γ( iλ+ρ−ν−2  2  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Furthermore we have the following Plancherel type formula for the Poisson transform  lim  R→+∞  1  R  �  B(R)  ∥Pν  λf(g)∥2  ν dgK = 2 | cν(λ) |2 ∥f∥2  L2(K,σν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6)  ii) Pν  λ is a topological isomorphism from L2(K, σν) onto E2  λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  This generalizes the result of Kaizuka [[16], (i) and (ii) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3] which corresponds to τν trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Consequence  For λ ∈ R we define the space  E∗  λ(G, τν) = {F ∈ Eλ(G, τν) : M(F) < ∞},  where  M(F) = lim sup  R→∞  �  1  R  �  B(R)  | F(g) |2 dgK  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Then as an immediate consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 we obtain the following result which generalizes a conjecture of  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Bray [10] which corresponds to τν trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' If λ ∈ R \\ {0} then E∗  λ(G, τν), M) is a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' In the case of the trivial bundle (the scalar case) the conjecture of Bray was proved by Ionescu [15] for  all rank one symmetric spaces .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' It was generalized to Riemannian symmetric spaces of higher rank by Kaizuka, see  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  3  Next, let us introduce our second main result on the L2-range of the generalized spectral projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  For F ∈ C∞  c (G, τν) the vector valued Helgason-Fourier transform FνF is given by (see [11])  Fν F(λ, k) =  �  G  e(iλ−ρ)H(g−1k)τν(κ(g−1k)−1)F(g) dg  λ ∈ C,  Then the following inversion formula holds (see section 4)  F(g) = 1  2π  � ∞  0  �  K  e−(iλ+ρ)H(g−1k)τν(κ(g−1k))FνF(λ, k) | cν(λ) |−2 dλ dk  +  �  λj∈Dν  dν(λj)  �  K  e−(iλj+ρ)H(g−1k)τν(κ(g−1k))FνF(λj, k) dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='7)  In above dν(λ) = −iResµ=λ(cν(µ)cν(−µ))−1, λ ∈ Dν and Dν is a finite set in {λ ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' ℑ(λ) > 0} which parametrizes  the τν-spherical functions arising from the discrete series of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' It is empty if ν ≤ ρ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='7) gives rise to the decomposition of L2(G, τν) into a continuous part and a discrete part:  L2(G, τν) = L2  cont(G, τν) ⊕ L2  disc(G, τν)  Our aim here is to study the operator Qν  λ, λ ∈ R, defined for F ∈ L2  cont(G, τν) ∩ C∞  c (C, τν) by  Qν  λF(g) =| cν(λ) |−2 Pν  λ[Fν F(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' )](g),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='8)  More precisely, following Strichartz idea, we are interested in the following question:  Characterize those Fλ ∈ Eλ(G, τν) (λ ∈ (0, ∞)) for which there exists F ∈ L2  cont(G, τν) such that Fλ = Qν  λF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  To do so, we introduce the space E2  +(G, τν) consisting of all Vτν-valued measurable functions ψ on (0, ∞) × G such  that  (i) Ω ψ(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=') = −(λ2 + ρ2 − ν(ν + 2)) ψ(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=') a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' λ ∈ (0, ∞)  (ii) ∥ ψ ∥+< ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  where  ∥ ψ ∥2  += sup  R>1  � ∞  0  1  R  �  B(R)  ∥ ψ(λ, g) ∥2  ν dgK dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The second main result we prove in this paper can be stated as follows  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (i) There exists a positive constant C such that for F ∈ L2(G, τν) we have  C−1 ∥ F ∥L2(G,τν)≤∥ Qν  λF ∥+≤ C ∥ F ∥L2(G,τν)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='9)  Furthermore we have  lim  R→∞  � ∞  0  1  R  �  B(R)  ∥ Qν  λF ∥2  ν dgK dλ = 2 ∥ F ∥2  L2(G,τν)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='10)  (ii) The linear map Qν  λ is a topological isomorphism from L2  cont(G, τν) onto E2  +(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  This extends Kaizuka result [ [16], (i) and (ii) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6] on the Strichartz conjecture (see [25] Conjecture  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5] to the class of vector bundles considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Before giving the outline of the paper, let us mention that a number of authors have obtained an image characterization  for the Poisson transform Pλ (λ ∈ a∗ \\ {0}) of L2-functions on K/M in the rank one case, see [[3], [5], [7], [15]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Nevertheless, the obtained characterization is weaker than the one conjectured by Strichartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The approach taken in  4  the quoted papers is based on the theory of Calderon-Zygmund singular integrals (see also [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Using a different  approach based on the techniques used in the scattering theory, Kaizuka [16] settled the Strichartz conjecture on  Riemannian symmetric spaces of noncompact type, of arbitrary rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We now describe the contents of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The proofs of our results are a generalisation of Kaizuka’s method [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' In  section 2 we recall some basic facts on the quaternionc hyperbolic spaces and introduce the vector Poisson transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  In section 3, we define the Helgason-Fourier transform on the vector bundles G ×K Vν and give the inversion and  Plancherel Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2 follows from the Plancherel formula and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The main  ingredients in proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 are a Fourier restriction estimate for the vector valued Helgason-Fourier transform  (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 in section 4) and an asymptotic formula for the vector Poisson transform in the framework of Agmon-  Hörmander spaces [2] (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 will be derived from the Key lemma of this paper  giving the asymptotic behaviour of the translate of the τν-spherical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Section 6 is devoted to the proof of our  main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' In section 7 we prove the Key Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  2  Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1  The quaternionic hyperbolic space  Let G = Sp(n, 1) be the group of all linear transformations of the right H-vector space Hn+1 which preserve the  quadratic form  n  �  j=1  | uj |2 − | un+1 |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let K = Sp(n) × Sp(1) be the subgroup of G consisting of pairs (a, d)  of unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Then K is a maximal compact subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The quaternionic hyperbolic space is the rank one  symmetric space G/K of the noncompact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' It can be realized as the unit ball B(Hn) = {x ∈ Hn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' | x |< 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The group G acts on B(Hn) by the fractional linear mappings x �→ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='x = (ax + b)(cx + d)−1, if g =  �  a  b  c  d  �  , with  a ∈ Hn×n, b ∈ Hn×1, c ∈ H1×n and d ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Denote by g the Lie algebra of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' g = k ⊕ p the Cartan decomposition of g, where p is a vector space of matrices of  the form  ��  0  x  x∗  0  �  , x ∈ Hn  �  , and k =  ��  X  0  0  q  �  , X∗ + X = 0, q + q = 0  �  , where X∗ is the conjugate transpose  of the matrix X and q ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let H =  �  0n  e1  te1  0  �  ∈ p with te1 = (1, 0, · · · , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then a = R H is a Cartan subspace in p, and the corresponding  analytic subgroup A = {at = exp t H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' t ∈ R}, where at =  \uf8eb  \uf8ec  \uf8ed  cht  0  sht  0  0n−1  0  sht  0  cht  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' With A determined we then have that  M =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  g =  \uf8eb  \uf8ec  \uf8ed  q  0  0  0  m  0  0  0  q  \uf8f6  \uf8f7  \uf8f8 , m ∈ Sp(n − 1), | q |= 1  \uf8fc  \uf8f4  \uf8fd  \uf8f4  \uf8fe  ≃ Sp(n − 1) × Sp(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let α ∈ a∗ be defined by α(H) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then a system Σ of restricted roots of the pair (g, a) is Σ = {±α, ±2α} if n ≥ 2  and Σ = {±2α} if n = 1, with Weyl group W ≃ {±Id}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' A positive subsystem of roots corresponding to the positive  Weyl chamber a+ ≃ (0, ∞) in a is Σ+ = {α, 2α} if n ≥ 2 and Σ+ = {2α} if n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let n = gα + g2α be the direct sum of the positive root subspaces, with dim gα = 4(n − 1) and dim g2α = 3 and N the  corresponding analytic subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then the half sum of the positive restricted roots with multiplicities counted  ρ equals to (2n + 1)α, and shall be viewed as a real number ρ = 2n + 1 by the identification a∗  c ≃ C via λα ↔ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let A+ = {at ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  t ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then we have the Cartan decomposition G = KA+K, that is any g ∈ G can be written  g = k1(g) eA+(g) k2(g),  k1(g), k2(g) ∈ K and A+(g) ∈ a+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  5  If we write g ∈ G in (n + 1) × (n + 1) block notation as g =  �  a  b  c  d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then a straightforward computation gives  cosh A+(g) =| d |  and  H(g) = log | ce1 + d | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1)  We normalize the invariant measure dgK on G/K so that the following integral formula holds: for all h ∈ L1(G/K),  �  G/K  h(gK)dgK =  �  G  h(g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0)dg =  �  K  � ∞  0  h(k at)∆(t) dk dt,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2)  where dt is the Lebesgue measure, ∆(t) = (2 sinh t)4n−1(2 cosh t)3, and dk is the Haar measure of K with  �  K  dk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2  The vector Poisson transform  In this subsection we define the Poisson transform associated to the vector bundles G×KVν over Sp(n, 1)/Sp(n)×Sp(1)  and derive some results referring to [23], [27], and [28] for more informations on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let σν denote the restriction of τν to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For λ ∈ C we consider the representation σν,λ of P = MAN on Vν defined  by σν,λ(man) = aρ−iλσν(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then σν,λ defines a principal series representations of G on the Hilbert space  Hν,λ := {f : G → Vν | f(gman) = σ−1  ν,λ(man)f(g) ∀man ∈ MAN, f|K ∈ L2},  where G acts by the left regular representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We shall denote by C−ω(G, σν,λ) the space of its hyperfunctions  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' By the Iwasawa decomposition, the restriction map from G to K gives an isomorphism from Hν,λ onto the  space L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' This yields, the so-called compact picture of Hν,λ, with the group action given by  πσν,λ(g)f(k) = e(iλ−ρ)H(g−1k)f(κ(g−1k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  By C−ω(K, σν) we denote the space of its hyperfunctions vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  A Poisson transform is the continuous, linear, G-equivariant map Pν  λ from C−ω(G, σν,λ) to C∞(G, τν) defined by  Pν  λ f(g) =  �  K  τν(k)f(gk) dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  In the compact picture the Poisson transform is given by  Pν  λ f(g) =  �  K  e−(iλ+ρ)H(g−1k)τν(κ(g−1k)) f(k) dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let D(G, τν) denote the algebra of left invariant differential operators on C∞(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let Eν,λ(G) be the space of all  F ∈ C∞(G, τν) such that Ω F = −(λ2 + ρ2 − ν(ν + 2)) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' (i) D(G, τν) is the algebra generated by the Casimir operator Ω of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (ii) For λ ∈ C, ν ∈ N, the Poisson transform Pν  λ maps C−ω(G, σν,λ) to Eν,λ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' (i) Let U(a) be the universal enveloping algebra of the complexification of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Since the restriction of τν to M  is irreducible, then D(G, τν) ≃ U(a)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' As a is one dimensional, then D(G, τν) ≃ C[s2], symmetric functions of one  variable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Thus D(G, τν) is generated by the Casimir element Ω of the Lie algebra g of G, viewed as a differential  operator acting on C∞(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (ii) Since σν is irreducible, the image of Pν  λ consists of joint eigenfunctions with respect to the action of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Moreover  Ω acts by the infinitesimal character of the the principal series representations πσν,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' It follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='22  and Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='28 in [18], that  πσν,λ(Ω) = −(λ2 + ρ2 − c(σν))Id  on  C−ω(G, σν,λ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3)  where c(σν) is the Casimir value of σν given by c(σν) = ν(ν + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  6  Let Φν,λ be the τν-spherical function associated to σν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then Φν,λ admits the following Eisenstein integral repre-  sentation (see [[11], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2]):  Φν,λ(g) =  �  K  e−(iλ+ρ)H(g−1k)τν(κ(g−1k)k−1) dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Note that Φν,λ lies in C∞(G, τν, τν) the space of smooth functions F : G → End(Vτν) satisfying  F(k1gk2) = τν(k−1  2 )F(g)τν(k−1  1 ),  the so called τν-radial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Being τν-radial, Φν,λ is completely determined by its restriction to A, by the Cartan  decomposition G = KAK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Moreover, since σν is irreducible, it follows that Φν,λ(at) ∈ EndM(Vν) ≃ CIdVν, ∀at ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Therefore there exists ϕν : R → C such that Φν,λ(at) = ϕν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='IdVν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We have  ϕν,λ(t) =  1  ν + 1  �  K  e−(iλ+ρ)H(g−1k)χν(κ(g−1k)k−1) dk,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4)  where χν is the character of τν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  This so-called trace τν-spherical function has been computed explicitly in [12] using the radial part of the Casimir  operator Ω (see also [26] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We have ϕν,λ(t) = (cosh t)νφ(ρ−2,ν+1)  λ  (t), where φ(ρ−2,ν+1)  λ  (t) is the Jacobi function (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  [19])  φ(ρ−2,ν+1)  λ  (t) = 2F1(iλ + ρ + ν  2  , −iλ + ρ + ν  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' ρ − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' − sinh2 t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We deduce from (A4) the asymptotic behaviour of ϕν,λ  ϕλ,ν(at) = e(iλ−ρ)t[cν(λ) + ◦(1)], as t → ∞  if  ℑ(λ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5)  where  cν(λ) =  2ρ−iλΓ(ρ − 1)Γ(iλ)  Γ( iλ+ρ+ν  2  )Γ( iλ+ρ−ν−2  2  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6)  For λ ∈ C the c-function of Harish-Chandra associated to τν is defined by  c(τν, λ) =  �  N  e−(iλ+ρ)H(n)τν(κ(n)) dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The integral converges for λ such that ℜ(iλ) > 0 and it has a meromorphic continuation to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  In above dn is the Haar measure of N = θ(N), θ being the Cartan involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We may use formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6) to give explicitly c(τν, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Indeed, one easily check that c(τν, λ) ∈ EndM(Vν) = CIdVν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Then using the following result on the behaviour of Φν,λ(at) ([28], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4)  Φν,λ(at) = e(iλ−ρ)t(c(τν, λ) + ◦(1))as  t → ∞,  together with Φν,λ(at) = ϕν,λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='Id, we find then from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5) that c(τν, λ) = cν(λ)IdVν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We end this section by recalling a result of Olbrich [23] on the range of the Poisson transform on vector bundles which  reads in our case as follows  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' [23] Let ν ∈ N and λ ∈ C such that  (i) −2iλ /∈ N  (ii) iλ + ρ /∈ −2N − ν ∪ −2N + ν + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Then the Poisson transform Pν  λ is a K-isomorphism from C−ω(K, σν) onto Eν,λ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  7  3  The vector-valued Helgason-Fourier transfrorm  In this section we give the inversion and the Plancherel formulas for the Helgason-Fourier transform on the vector  bundle G ×K Vν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  According to [11] the vector-valued Helgason-Fourier transform of f ∈ C∞  c (G, τν) is the Vν-valued function on C × K  defined by:  Fνf(λ, k) =  �  G  eλ,ν(k−1g) f(g)dg,  where eλ,ν is the vector valued function eλ,ν : G → End(Vν) given by  eλ,ν(g) = e(iλ−ρ)H(g−1)τ −1  ν (κ(g−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Notice that our sign on "λ" is the opposite of the one in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  In order to state the next theorem, we introduce the finite set in {λ, ℑ(λ) ≥ 0}  Dν = {λj = i(ν − ρ + 2 − 2j), j = 0, 1, · · · , ν − ρ + 2 − 2j > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Note that Dν is empty if ν ≤ ρ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' It parametrizes the discrete series representation of G containing τν, see [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let  dν(λj) = 2−2(ρ−ν−1)(ν − ρ − 2j + 2)(ρ − 2 + j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' (ν − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Γ2(ρ − 1)j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' (ν − ρ − j + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  ,  λj ∈ Dν  For λj ∈ Dν, we define the operators Qν  j  L2(G, τν) → Eν,λj(G, τν)  F �→ dν(λj) Φν,λj ∗ F  We denote the image by A2  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We set  L2  disc(G, τν) =  �  j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' ν−ρ+2−2j>0  A2  j,  and denote by L2  cont(G, τν) its orthocomplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let L2  σν(R+ × K, | cν(λ) |−2 dλ dk) be the space of vector functions  φ : R+ × K → Vν satisfying  (i) For each fixed λ, φ(λ, km) = σν(m)−1φ(λ, k), ∀m ∈ M  (ii)  �  R+×K ∥ Fνφ(λ, k) ∥2 | cν(λ) |−2 dλ dk < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' (i) For F ∈ C∞  c (G, τν) we have the following inversion and Plancherel formulas  F(g) = 1  2π  � ∞  0  �  K  e∗  λ,ν(k−1g)FνF(λ, k) | cν(λ) |−2 dλ dk +  �  λj∈Dν  dν(λj)  �  K  e∗  λj,ν(k−1g)FνF(λj, k) dk,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1)  �  G  ∥ F(g) ∥2  ν dgK = 1  2π  � ∞  0  �  K  ∥ FνF((λ, k) ∥2  ν| cν(λ) |−2 dλ dk+  �  λj∈Dν  dν(λj)  �  K  < FνF(λj, k), FνF(−λj, k) >ν dk  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2)  (ii) The Fourier transform Fν extends to an isometry from L2  cont(G, τν) onto the space L2  σν(R+ ×K, | cν(λ) |−2 dλ dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The first part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 can be easily deduced from the inversion and Plancherel formulas for the spherical  transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  8  Let C∞  c (G, τν, τν) denote the space of smooth compactly supported τν-radial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The spherical transform of  F ∈ C∞  c (G, τν, τν) is the C-valued function HνF defined by:  HνF(λ) =  1  ν + 1  �  G  T r[Φν,λ(g−1)F(g))]dg,  λ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The inversion and the Plancherel formulas for the τ-spherical transform have been given explicitly in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For the  convenience of the reader we give an elementary proof by using the Jacobi transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For F ∈ C∞  c (G, τν, τν) we have the following inversion and Plancherel formulas  F(g) = 1  2π  � +∞  0  Φν,λ(g)HνF(λ) | cν(λ) |−2 dλ +  �  λj∈Dν  Φν,λj(g)Hνf(λj) dν(λj),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3)  �  G  ∥ F(g) ∥2  HS dg = ν + 1  2π  � +∞  0  | HνF((λ) |2| cν(λ) |−2 dλ + (ν + 1)  �  λj∈Dν  dν(λj) | HνF((λj) |2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4)  In above ∥ ∥HS stands for the Hilbert-Schmidt norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let F ∈ C∞  c (G, τν, τν) and let fν be its scalar component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Using the integral formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2), the identity  Φν,λ(at) = Φν,λ(a−t) = (cosh t)νφ(ρ−2,ν+1)  λ  (t) and the fact that ∆(t) = (2 cosh t)−2ν∆ρ−2,ν+1, we have  HνF(λ) =  � ∞  0  fν(t)(cosh t)νφ(ρ−2,ν+1)  λ  (t) ∆(t) dt  =  � ∞  0  fν(t)(22 cosh t)−νφ(ρ−2,ν+1)  λ  (t) ∆ρ−2,ν+1(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5)  Thus the τν-spherical transform HνF may be written in terms of the Jacobi transform J α,β, with α = ρ − 2 and  β = ν + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Namely, we have  HνF(λ) = J ρ−2,ν+1[(22 cosh t)−νfν](λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We refer to (A5) in the Appendix for the definition of the Jacobi transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Now the theorem follows from the inversion and the Plancherel formulas for the Jacobi transform (A6), (A6’) and  (A7) in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  For the proof of the surjectivity statement in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 we shall need the following result  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let F ∈ C∞  c (G, τν) and Φ ∈ C∞(G, τν, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then we have  Fν(F ∗ Φ)(λ, k) = HνΦ(λ)FνF(λ, k),  λ ∈ C, k ∈ K,  where the convolution is defined by  (Φ ∗ F)(g) =  �  G  Φν,λ(x−1g)F(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let Φ ∈ C∞(G, τν, τν), v ∈ Vν, and set Fv = Φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' )v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then we have the following relation between the Fourier  transform and the spherical transform  FνFv(λ, k) = HνΦ(λ)τ(k−1)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6)  By definition  Fν(F ∗ Φ)(λ, k) =  �  G  �  G  eν  λ(k−1g)Φ(x−1g)F(x)dxdg  =  �  G  dx  �  G  eν  λ(k−1xy)Φ(y)F(x)dy  9  Using the following cocycle relations for the Iwasawa function H(x)  H(xy) = H(xκ(y)) + H(y),  and  κ(xy) = κ(xκ(y)),  for all x, y ∈ G, we get the following identity  eν  λ(k−1xy) = e(iλ−ρ)H(x−1k)eν  λ(κ−1(x−1k)y),  from which we obtain  Fν(Φ ∗ F)(λ, k) =  �  G  e(iλ−ρ)H(x−1k)  ��  G  eλ,ν(κ−1(x−1k)y)Φ(y)F(x) dy  �  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Next, put hv(y) = Φ(y)v, v ∈ Vτν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6) implies  �  G  eλ,ν(κ−1(x−1k)y)Φ(y)F(x) dy = Fν(hF (x))(λ, κ−1(x−1k))  = H(Φ)(λ)τν(κ−1(x−1k))F(x),  from which we deduce  Fν(Φ ∗ F)(λ, k) = H(Φ)(λ)  �  G  e(iλ−ρ)H(x−1k)τν(κ−1(x−1k))F(x)dx,  and the proposition follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We now come to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' (i) We may follow the same method as in [11] to prove the inversion formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1) and the Plancherel formula  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We give an outline of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let F ∈ C∞  c (G, τν) and consider the τν-radial function defined for any g ∈ G by  Fg,v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='w =  �  K  < τν(k)w, v >ν F(gkx) dk,  v being a fixed vector in Vν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then a straightforward calculation shows that  HνFg,v(λ) =  1  ν + 1 < (Φν,λ ∗ F)(g), v >ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The inversion formula for the spherical transform together with T rFg,v(e) =< F(g), v >ν imply  F(g) = 1  2π  � ∞  0  (Φν,λ ∗ F)(g) | cν(λ) |−2 dλ +  �  λj∈Dν  (Φν,λj ∗ F)(g)dν(λj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  To conclude use the following result for the translated spherical function ( see [11] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3)  Φν,λ(x−1y) =  �  K  e−(iλ+ρ)H(y−1k)e(iλ−rho)H(x−1k)τν(κ(y−1k))τν(κ−1(x−1k)) dk,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='7)  to get  (Φν,λ ∗ F)(g) =  �  K  e−(iλ+ρ)H(g−1k)τν(κ(g−1k))FνF(λ, k) dk,  10  and the inversion formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The proof of the Plancherel formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2) is essentially the same as in the scalar case, so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Note that as a consequence of the Plancherel formula not involving the discrete series, we have  �  G  ∥ F(g) ∥2 dgK = 1  π  � ∞  0  �  K  ∥ FνF(λ, k) ∥2 | cν(λ) |−2 dλ dk,  for every F ∈ L2  cont(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (ii) We prove the surjectivity statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Suppose that there exists a function f in L2  σν(R+ × K, | cν(λ) |−2 dλ dk)  such that  � ∞  0  �  K  < f(λ, k), FνF(λ, k) >| cν(λ) |−2 dλ dk = 0  for all F ∈ C∞  c (G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Changing F into F ∗ Φ where Φ ∈ C∞(G, τν, τν) and using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1, we have  � ∞  0  �  K  < f(λ, k), FνF(λ, k) > Hνφ(λ) | cν(λ) |−2 dλ dk = 0  By the Stone-Weierstrass theorem, the algebra {HνΦ, Φ ∈ C∞(G, τν, τν)} is dense in C∞  e (R) the space of even  continuous functions on R vanishing at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Therefore for every F ∈ C∞  c (G, τν) there is a set EF of measure zero  in R such that  �  K  < f(λ, k), FνF(λ, k) > dk = 0  for all λ not in EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The rest of the proof is based on an adaptation of the arguments given in [14] Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5, for  the scalar case, and the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  4  Fourier restriction estimate  The main result of this section is the following uniform continuity estimate for the Fourier-Helgason restriction operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' There exists a positive constant Cν such that for λ ∈ R\\{0} and R > 1, we have  � �  K  ∥FνF(λ, k)∥2  νdk  �1/2  ≤ Cν|cν(λ)|R1/2  � �  G/K  ∥F(g)∥2  ν dgK  �1/2  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1)  for every F ∈ L2(G, τν) with suppF ⊂ B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  To prove this result we shall need estimates of the Harish-Chandra c-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  To this end we introduce the  function bν(λ) defined on R by  bν(λ) =  \uf8f1  \uf8f2  \uf8f3  cν(λ)  if  ν−ρ+2  2  ∈ Z+  λ cν(λ)  if  ν−ρ+2  2  /∈ Z+  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Assume ν > ρ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (i) The function bν(λ) has no zero in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (ii) There exists a positive constant C such that for λ ∈ R, we have  C−1(1 + λ2)  2ρ−4−ε(ν)  4  ≤| bν(λ) |−1≤ C(1 + λ2)  2ρ−4−ε(ν)  4  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2)  11  with ε(ν) = ±1 according to ν−ρ+2  2  /∈ Z+ or ν−ρ+2  2  ∈ Z+  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (i) If ν−ρ+2  2  /∈ Z+, then bν(λ) = 2ρ+ν−iλΓ(ρ−1)Γ(iλ+1)  Γ( iλ+ρ+ν  2  )Γ( iλ+ρ−ν−2  2  ), and clearly bν(λ) has no zero on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  If ν−ρ+2  2  ∈ Z+ then bν(λ) a priori can have zero and pole at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' This is not the case, since  lim  λ→0 bν(λ) = (−1)  ν−ρ+2  2  2ρ+νΓ(ρ − 1)( ν−ρ+2  2  )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Γ( ρ+ν  2 )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (ii) To prove the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2) we shall use the following property of the Γ-function  lim  |z|→∞  Γ(z + a)  Γ(z)  z−a = 1, | arg(z) |< π − δ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3)  where a is any complex number, and log is the principal value of the logarithm and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Assume first that ν−ρ+2  2  /∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Using the duplicata formula for the function gamma  Γ(2z) = 22z−2  √π Γ(z)Γ(z + 1  2),  we rewrite bν(λ) as  bν(λ) = 2ρ+ν−1  √π  Γ( iλ+1  2  )Γ( iλ+2  2  )  Γ( iλ+ρ+ν  2  )Γ( iλ+ρ−ν−2  2  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3) that for every λ ∈ R, we have  | bν(λ) |≤ C(1 + λ2)− 2ρ−5  4  and  | bν(λ) |−1≤ C(1 + λ2)  2ρ−5  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The proof for the case ν−ρ+2  2  ∈ Z+ follows the same line as in the case ν−ρ+2  2  /∈ Z+, so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  This finishes the proof of the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let us recall from [1] an auxiliary lemma which will be useful for the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let η be a positive Schwartz function on R whose Fourier transform has a compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For m ∈ R, set  ηm(x) =  �  R  η(t)(1 + |t − x|)m/2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  i) ηm is a positive C∞-function with  C−1(1 + t2)  m  2 ≤ ηm(t) ≤ C(1 + t2)  m  2 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4)  for some positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  ii) The Fourier transform of ηm has a compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  In order to prove the Fourier restriction Theorem, we need to introduce the bundle valued Radon transform, see  [9] for more informations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The Radon transform for F ∈ C∞  c (G, τν) is defined by  RF(g) = eρH(g)  �  N  F(gn)dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  12  We set RF(t, k) = RF(kat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then, using the Iwaswa decomposition G = NAK, we may rewrite the Helgason-Fourier  transform as  FνF(λ, k) = FR(RF(·, k))(λ),  where  FRφ(λ) =  �  R  e−iλtφ(t) dt,  is the Euclidean Fourier transform of φ a Vν-valued smooth function with compact support in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We define on p the scalar product < X, Y >= 1  2T r(XY ) and denote by | | the corresponding norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' It induces a distance  function d on G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' By the Cartan decomposition G = K exp p, any g ∈ G may be written uniquely as g = k exp X,  so that d(0, gK) =| X |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Define the open ball centred at 0 and of radius R by B(R) = {gK ∈ G/K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  d(0, gK) < R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let F ∈ C∞  0 (G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' If supp F ⊂ B(R), then supp RF ⊂ [−R, R] × K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' As (see [[13], page 476]  d(0, ketHnK) ≥| t |,  k ∈ K, n ∈ N, t ∈ R  it follows that supp RF ⊂ [−R, R] × K if supp F ⊂ B(R)  Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' It suffices to prove the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1) for functions F ∈ C∞  c (G, τν) supported in B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  It follows from the Plancherel formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2) that  �  B(R)  ∥ F(g) ∥2  ν dgK ≥  �  K  �  R  ∥ FνF(λ, k) ∥2  ν | cν(λ) |−2 dλ dk  Therefore it is sufficient to show  �  K  �  R  ∥ FνF(λ, k) ∥2  ν | cν(λ) |−2 dλ dk ≥ C | cν(λ) |−2  R  �  R  ∥ FνF(λ, k) ∥2  ν dk,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5)  fir some positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2) we have | cν(λ) |−1≍ η 2ρ−3  2 (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Therefore (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5) is equivalent to  η 2ρ−3  2 (λ)  R  �  K  ∥ FνF(λ, k) ∥2  ν dk ≤  �  K  �  R  ∥ FνF(λ, k) ∥2  ν η 2ρ−3  2 (λ)dλ dk  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6)  Let T be the tempered distribution on R defined by T := F−1  R η 2ρ−3  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2, T is compactly supported .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let  R0 > 1 such that supp T ⊂ [−R0, R0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6) is equivalent to  �  K  ∥ FR(T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' , k))(λ) ∥2  ν dk ≤ CR  �  K  �  R  FR(T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' , k))(λ) ∥2  ν dλ dk,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='7)  where ∗ denotes the convolution on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  From suppT ⊂ [−R0, R0] and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3, it follows that for any k ∈ K, supp (T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' , k)) ⊂ [−(R + R0), R + R0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Thus  �  K  ∥ FR(T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' , k)(λ) ∥2  ν dk ≤ 2(R + R0)  �  K  �  R  ∥ (T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' k))(t) ∥2  ν dt dk  Next use the Euclidean Plancherel formula to get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='7), and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  As a consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1, we obtain the uniform continuity estimate for the Poisson transform Pν  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' There exists a positive constant Cν such that for λ ∈ R\\{0}, we have  sup  R>1  �  1  R  �  B(R)  ∥ Pν  λf(g) ∥2  ν dgK  �1/2  ≤ Cν |cν(λ)| ∥ f ∥L2(K,σν)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='8)  for every f ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let F ∈ L2(G, τν) with supp F ⊂ B(R), and let f ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Since λ is real and τν is unitary, the Poisson  transform and the restriction Fourier transform are related by the following formula  �  B(R)  < Pν  λf(g), F(g) >ν dg =  �  K  < f(k), FνF(λ, k) >ν dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Thus  |  �  B(R)  < Pν  λf(g), F(g) >ν dg | ≤ ∥f∥L2(K,σν)(  �  K  ∥ FνF(λ, k) ∥2  ν dk)  1  2  ≤ Cν|cν(λ)|R1/2 ∥ f ∥L2(K,τν)∥ F ∥L2(G,τν),  by the restriction Fourier theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Taking the supermum over all F with ∥ F ∥L2(G,τν)= 1, the corollary follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  5  Asymptotic expansion for the Poisson transform  In this section we give an asymptotic expansion for the Poisson transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We first start by establishing some  intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let L2  λ(K, σν) denote the finite linear span of the functions  f g  λ,v : k �−→ f g  λ,v(k) = e(iλ−ρ)H(g−1k)τ −1  ν (κ(g−1k))v,  g ∈ G, v ∈ Vν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For λ ∈ R \\ {0}, ν ∈ N the space L2  λ(K, σν) is a dense subspace of L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' As λ ∈ R \\ {0}, the density is just a reformulation of the injectivity of the Poisson transform Pν,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let λ ∈ R \\ {0}, ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then there exists a unique unitary isomorphism U ν  λ on L2(K, σν) such that :  U ν  λ f g  λ,v = f g  −λ,v,  g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Moreover, for f1, f2 ∈ L2(K, σν), we have Pν  λF1 = Pν  −λF2 if and only if U ν  λF1 = F2 ( i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' U ν  λ = (Pν  −λ)−1 ◦ Pν  λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The proof is the same as in the scalar case so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We now introduce the function space B∗(G, τν) on G, consisting of functions F in L2  loc(G, τν) satisfying  ∥ F ∥B∗(G,τν)= sup  j∈N  [2− j  2  �  Aj  ∥ F(g) ∥2  ν dgK] < ∞,  where A0 = {g ∈ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' d(0, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0) < 1} and Aj = {g ∈ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' 2j−1 ≤ d(0, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0) < 2j}, for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  One could easily show that ∥ F ∥B∗(G,τν)≤∥ F ∥∗≤ 2 ∥ F ∥B∗(G,τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We define an equivalent relation on B∗(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For F1, F2 ∈ B∗(G, τν) we write F1 ≃ F2 if  lim  R→+∞  1  R  �  B(R)  ∥ F1(g) − F2(g) ∥2  ν dg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Note that by using the polar decomposition we see that F1 ≃ F2 if  lim  R→+∞  1  R  �  K×[0,R]  ∥ F1(ketH)) − F2(ketH)) ∥2  ν ∆(t) dt dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We now state the main result of this section  14  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let ν ∈ N, λ ∈ R\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For f ∈ L2(K, σν) we have the following asymptotic expansions for the Poisson  transform in B∗(G, τν)  Pλ,νf(x) ≃ τ −1  ν (k2(x))[cν(λ)e(iλ−ρ)(A+(x)f(k1(x)) + cν(−λ)e(−iλ−ρ)(A+(x))U ν  λf(k1(x))],  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1)  where x = k1(x)eA+(x)k2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Most of the proof of the above theorem consists in proving the following Key Lemma, giving the asymptotic ex-  pansion for the translates of the τν-spherical function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  KEY LEMMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For λ ∈ R \\ {0}, g ∈ G and v ∈ Vν, we have the following asymptotic expansion in B∗(G, τν)  Φν,λ(g−1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' v ≃ τ −1  ν (k2(x))  �  s∈{±1}  cν(sλ)e(isλ−ρ)A+(x)f g  sλ,v(k1(x)),  x = k1(x)eA+(x)k2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We first note that both side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1) depend continuously on f ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' This can  be proved in the same manner as in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Therefore we only have to prove that the asymptotic expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1) holds  for f ∈ L2  λ(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let f = f g  λ,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then according to [[11], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3], we have  Pν  λf(x) = Φν,λ(g−1x)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The theorem follows from the Key lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  As a consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 we obtain the following result giving the behaviour of the Poisson integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For any f ∈ L2(K, σν) we have the Plancherel-Poisson formula  lim  R→+∞  1  R  �  B(R)  ∥ Pν  λf(g) ∥2  ν dgK = 2 | cν(λ) |2 ∥ f ∥2  L2(K,σν)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' There exists a positive constant Cν such that for any λ ∈ R \\ {0}, we have  C−1  ν  | cν(λ) | ∥ f ∥L2(K,σν)≤∥ Pλ  ν f ∥∗≤ Cν | cν(λ) | ∥ f ∥L2(K,σν),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3)  for every f ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We define for f ∈ L2(K, σν)  Sν  λf(x) := τ −1  ν (k2(x))[cν(λ)e(iλ−ρ)(A+(x)f(k1(x)) + cν(−λ)e(−iλ−ρ)(A+(x))U ν  λf(k1(x))],  x = k1(x)eA+(x)k2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  By the unitarity of Uλ, we have  1  R  �  B(R)  ∥Sν  λf(g)∥2dgK = 2|cν(λ)|2∥f∥2  L2(K,τν)  �  1  R  � R  0  e−2ρt∆(t)dt  �  + 2|cν(λ)|2ℜ  �  < f, Uλf >L2(K,σν)  1  R  � R  0  e2(iλ−ρ)t∆(t)dt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  From  lim  R→+∞  1  R  � R  0  e−2ρt∆(t)dt = 1, and  lim  R→+∞  1  R  � R  0  e2(iλ−ρ)t∆(t)dt = 0, we deduce that  lim  R→+∞  1  R  �  B(R)  ∥ Sν  λf(g) ∥2  ν dgK = 2 | cν(λ) |2∥ f ∥2  L2(K,σν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4)  15  Next write  1  R  �  B(R)  ∥ Pν  λf(g) ∥2  ν dgK = 1  R  �  B(R)  (∥ Sν  λf(g) ∥2  ν + ∥ Pν  λf(g) − Sν  λf(g) ∥2  ν  + 2Re[< Pν  λf(g) − Sν  λf(g), Sν  λf(g) >])dgK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2) then follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4), Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 and the Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' The right hand side of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3) has already been proved, see corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The left hand side of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3) obviously follows from the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' This finishes the proof of the  proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let f1, f2 ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then using the polarization identity as well as the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2), we get  lim  R→+∞  1  R  �  B(R)  < Pν  λf1(g), Pν  λf2(g) >ν dgK = 2 | cν(λ) |2< f1, f2 >L2(K,σν)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5)  6  Proof of the main results  In this section we shall prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 on the L2-range of the vector Poisson transform and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2 charac-  terizing the image Qν  λ(L2(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1  The L2-range of the Poisson transform  We first recall some results of harmonic analysis on the homogeneous vector bundle K ×M Vν associated to the  representation σν of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let �K be the unitary dual of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For δ ∈ �K let Vδ denote a representation space of δ with dδ = dim Vδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We denote by  �K(σν) the set of δ ∈ �K such that σν occurs in δ |M with multiplicity mδ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The decomposition of L2(K, σν) under K (the group K acts by left translations on this space) is given by the Frobenius  reciprocity law  L2(K, σν) =  �  δ∈�  K(σν)  Vδ ⊗ HomM(Vν, Vδ),  where v ⊗L, for v ∈ Vδ, L ∈ HomM(Vν, Vδ) is identified with the function (v ⊗L)(k) = L∗(δ(k−1)v), where L∗ denotes  the adjoint of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  For each δ ∈ �K(σν) let (Lj)mδ  j=1 be an orthonormal basis of HomM(Vν, Vδ) with respect to the inner product  < L1, L2 >=  1  ν + 1T r(L1L∗  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let {v1, · · · , vdδ} be an orhonormal basis of Vδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then  f δ  ij : k →  �  dδ  ν + 1L∗  i δ(k−1)vj,  1 ≤ i ≤ mδ,  1 ≤ j ≤ dδ,  δ ∈ �K(σ)  form an orthonormal basis of L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  For f ∈ L2(K, σν) we have the Fourier series expansion f(k) =  �  δ∈�  K(σ)  mδ  �  i=1  dδ  �  j=1  aδ  ijf δ  ij(k) with  ∥ f ∥2  L2(K,σ)=  �  δ∈�  K(σ)  mδ  �  i=1  dδ  �  j=1  | aδ  ij |2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  16  We define for δ ∈ �K(σ) and λ ∈ C, the generalized Eisenstein integral  ΦL  λ,δ(g) =  �  K  e−(iλ+ρ)H(g−1k)τν(κ(g−1k))L∗δ(k−1)dk,  L ∈ HomM(Vν, Vδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  It is easy to see that ΦL  λ,δ satisfies the following identity  ΦL  λ,δ(k1gk2) = τν(k−1  2 )ΦL  λ,δ(g)δ(k−1  1 ),  k1, k2 ∈ K, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We now prove an asymptotic estimate for the generalized Eisenstein integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let ν ∈ N, λ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then for δ ∈ �K(σν), T, S ∈ HomM(Vν, Vδ) we have  lim  R→+∞  1  R  �  B(R)  Tr  �  ΦT  λ,δ(g)∗ΦS  λ,δ(g)  �  dgK = 2 | cν(λ) |2 Tr(T S∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' By definition we have  lim  R→+∞  1  R  �  B(R)  Tr  �  ΦT  λ,δ(g)∗ΦS  λ,δ(g)  �  dgK =  dδ  �  j=1  lim  R→+∞  1  R  �  B(R)  < ΦS  λ,δ(g)vj, ΦT  λ,δ(g)vj >ν dgK  Noting that ΦT  λ,δ(g)vj is the Poisson transform of the function k �→ L∗δ(k−1)vj and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5), we get  lim  R→+∞  1  R  �  B(R)  Tr  �  ΦT  λ,δ(g)∗ΦS  λ,δ(g)  �  dgK = 2 | cν(λ) |2  dδ  �  j=1  �  K  < S∗δ(k−1)vj, T ∗δ(k−1)vj >ν dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Hence Schur Lemma lead us to conclude that  lim  R→+∞  1  R  �  B(R)  Tr  �  ΦT  λ,δ(g)∗ΦS  λ,δ(g)  �  dgK = 2 | cν(λ) |2 Tr(T S∗), and  the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Noting that  T r(  �  ΦT  λ,δ(g)∗ΦS  λ,δ(g)  �  = T r(  �  ΦT  λ,δ(a)∗ΦS  λ,δ(a)  �  ,  g = k1 a k2,  it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1) that  lim  R→+∞  1  R  � R  0  T r  �  ΦT  λ,δ(at)∗ΦS  λ,δ(at)  �  ∆(t)dt =| cν(λ) |2 Tr(T S∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2)  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (i) The estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3) implies that the Poisson transform Pλ,ν maps L2(K, σν) into Eλ(G, τν) and that the estimate  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (ii) We now prove that the Poisson transform maps L2(K, σν) onto E2  λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let F ∈ E2  λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Since λ ∈ R \\ {0},  we know by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 that there exists a hyperfunction f ∈ C−ω(K, σν) such that F = Pλ,νf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let f =  �  δ∈�  K(σ)  dδ  �  j=1  mδ  �  i=1  aδ  ijf δ  ij, be the Fourier series expansion of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then we have  F(g) =  �  δ∈�  K(σ)  �  dδ  ν + 1  dδ  �  j=1  mδ  �  i=1  aδ  ijΦLi  λ,δ(g)vj  in  C∞(G, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  By the Schur relations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' we have  �  K  < ΦLi  λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='δ(kat)vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' ΦLm  λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='δ′(kat)vn >ν dk =  �  0  if δ ≁ δ′  1  dδ T r(ΦLm  λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='δ′ (at))∗ΦLi  λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='δ(at) < vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' vn > if  δ′ = δ  17  Therefore  �  K  ∥ F(kat) ∥2 dk =  1  ν + 1  �  δ∈�  K(σ)  dδ  �  j=1  �  1≤i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='j≤mδ  aδ  ijaδ  mjT r[(ΦLm  λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='δ (at))∗ΦLi  λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='δ(at)]  =  1  ν + 1  �  δ∈�  K(σ)  dδ  �  j=1  T r  \uf8ee  \uf8f0  �  1≤i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='m≤mδ  (aδ  mjΦLm  λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='δ (at))∗(aδ  ijΦLi  λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='δ(at)  \uf8f9  \uf8fb  =  1  ν + 1  �  δ∈�  K(σ)  dδ  �  j=1  ∥  mδ  �  i=1  aδ  ijΦLi  λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='δ(at) ∥2  HS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let Λ be a finite subset in �K(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Since ∥ F ∥∗< ∞, it follows that, for any R > 1 we have  ∞ >∥ F ∥2  ∗≥  1  ν + 1  �  δ∈Λ  dδ  �  j=1  1  R  � R  0  ∥  mδ  �  i=1  aδ  ijΦLi  λ,δ(at) ∥2  HS ∆(t) dt  By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2) we have  lim  R→∞  1  R  � R  0  ∥  mδ  �  i=1  aδ  ijΦLi  λ,δ(at) ∥2  HS ∆(t) dt = lim  R→∞  �  1≤i,m≤mδ  aδ  ijaδ  mj  1  R  � R  0  T r[(ΦLm  λ,δ (at))∗ΦLi  λ,δ(at)] ∆(t)dt  = 2 | cν(λ) |2  �  1≤i,m≤mδ  aδ  ijaδ  mjT r(LiL∗  m)  = 2(ν + 1) | cν(λ) |2  mδ  �  i=1  | aδ  ij |2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Thus ∞ >∥ F ∥2  ∗≥| cν(λ) |2 �  δ∈Λ  dδ  �  j=1  mδ  �  i=1  | aδ  ij |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Since Λ is arbitrary, it follows that  | cν(λ) |2  �  δ∈�  K(σ)  dδ  �  j=1  mδ  �  i=1  | aδ  ij |2≤∥ F ∥2  ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  This shows that f ∈ L2(K, σν) with | cν(λ) |∥ f ∥L2(K,σν)≤∥ Pν  λf ∥∗ and the proof of the theorem is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2  The L2-range of the generalized spectral projections  We now proceed to the poof of the second main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Let F ∈ L2  c(G, τν) ∩ C∞(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' It follows from the definition ( see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='8)) that the operator Qν  λ may be written as  Qν  λF(g) =| cν(λ) |−2 Pν  λ(FνF(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' ))(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3)  Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 we deduce that  sup  R>1  1  R  �  B(R)  ∥ Qν  λF(g) ∥2  ν dgK ≤ Cν | cν(λ) |−2  �  K  ∥ FνF(λ, k) ∥2  ν dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The above inequality and the Plancherel formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4) imply  � ∞  0  (sup  R>1  1  R  �  B(R)  ∥ Qν  λF(g) ∥2  ν dgK) dλ ≤ Cν  � ∞  0  �  K  ∥ FνF(λ, k) ∥2  ν| cν(λ) |−2 dk dλ  ≤ Cν ∥ F ∥2  L2(G,τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  18  This prove the right hand side of the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6) we have  lim  R→∞  1  R  �  B(R)  ∥ Qν  λF(g) ∥2  ν dgK = 2 | cν(λ) |−2  �  K  ∥ FνF(λ, k) ∥2  ν dk,  and since for all R > 1  1  R  �  B(R)  ∥ Qν  λF(g) ∥2 dgK ≤ Cν | cν(λ) |−2  �  K  ∥ FνF(λ, k) ∥2 dk,  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' λ ∈ (0, ∞),  we may apply the Lebesgue’s dominated convergence theorem to get  lim  R→∞  � ∞  0  �  1  R  �  B(R)  ∥ Qν  λF(g) ∥2  ν dgK  �  dλ = 2 ∥ F ∥2  L2(G,τν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  It follows from the above equality that  C ∥ F ∥2  L2(G,τν)≤  � ∞  0  (sup  R>1  �  B(R)  ∥ Qν  λF(x) ∥2 dx) dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  This complete the proof of the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We now prove that Qν  λ maps L2  c(G, τν) onto E2  λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let Fλ ∈ E2  λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then we have  sup  R>1  1  R  �  B(R)  ∥ Fλ(g) ∥2  ν dgK < ∞,  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  λ ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1, there exists fλ ∈ L2(K, σν) such that Fλ(g) =| cν(λ) |−2 Pν  λfλ(g) with  sup  R>1  1  R  �  B(R)  ∥ Fλ(g) ∥2  ν dgK ≥ C−1  ν  | cν(λ) |−2  �  K  ∥ fλ(k) ∥2 dk  Integrating the both side of the above inequality over (0, ∞), we get  ∞ >∥ Fλ ∥2  ∗≥ C−1  ν  � ∞  O  �  K  ∥ fλ(k) ∥2  ν | cν(λ) |−2 dk dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  It now follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1, that there exists F ∈ L2  c(G, τν) such that FνF(λ, k) = fλ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Henceforth Fλ(g) =| cν(λ) |−2 Pλ,ν(FνF(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' )(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' This finishes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  7  Proof of the Key Lemma  In this section we prove the Key Lemma of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' To this end we need to establish some auxiliary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' We  first prove an asymptotic formula for the τν-spherical function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let λ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For any v ∈ Vν we have  Φν,λ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' v ≃  �  s∈{±1}  cν(sλ)e(isλ−ρ)A+(g)τ −1  ν (κ1(g)κ2(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' v,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1)  g = κ1(g)eA+(g)κ2(g)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Since ∆(t) ≤ e2ρ t, we get  1  R  �  B(R)  ∥ e(iλ−ρ)A+(g)τ −1  ν (κ1(g)κ2(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' v ∥2 dg = 1  R ∥ v ∥2  � R  0  e−2ρ t∆(t)dt  ≤∥ v ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  19  This shows that the right hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1) belongs to B∗(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Since λ ∈ R \\ {0}, we may use the identity (A3) to write  ϕν,λ(t) −  �  s∈{±1}  cν(sλ)e(isλ−ρ)t =  �  s∈{±1}  cν(sλ)  �  (2 cosh t)νΨρ−2,ν+1  sλ  (t) − e(isλ−ρ)t�  =  �  s∈{±1}  cν(sλ)e(isλ−ρ)t �  (1 + e−2t)νe(ρ+ν−isλ)tΨρ−2,ν+1  sλ  (t) − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  It follows from (A2’) that  ϕν,λ(t) −  �  s∈{±1}  cν(sλ)e(isλ−ρ)t =  �  s∈{±1}  cν(sλ)e(isλ−ρ)t �  (1 + e−2t)ν − 1) + e−2tEsλ(t)  �  ,  where | Esλ(t) |≤ 2νC if t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Therefore  | ϕν,λ(t) −  �  s∈{±1}  cν(sλ)e(isλ−ρ)t |≤ Cν,λe−ρe−2t,  if t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' This together with  | ϕν,λ(t) −  �  s∈{±1}  cν(sλ)e(isλ−ρ)t |≤ Cν,λe−ρt,  for t ∈ [0, 1], imply that  lim  R→∞  1  R  �  B(R)  ∥ Φν,λ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' v −  �  s∈{±1}  cν(sλ)e(isλ−ρ)A+(g)τ −1(κ1(g)κ2(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' v ∥2  ν dgK =  =∥ v ∥2 lim  R→∞  1  R  � R  0  | ϕν,λ(t) −  �  s∈{±1}  cν(sλ)e(isλ−ρ)t |2 ∆(t) dt = 0,  and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let g ∈ G, k ∈ K and t a non negative real number .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then we have  0 ≤ A+(g−1k exp(tH)) − H(g−1k exp(tH)) ≤ 1+ | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0 |  1− | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0 |e−2t,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let g−1 =  �  a  b  c  d,  �  and k ==  �  u  0  O  v,  �  , where a, b, c and d are n×n, n×1, 1×n and 1×1 matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  A direct computation yields  g−1k exp(tH) =  �  ∗  ∗ ∗  c1  d1  �  ,  where c1 = c u  �  cosh t  0  0  In−1  �  and d1 = sinh t cue1 + cosh t dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1) we have  eH(g−1k exp(tH)) = et | cue1 + dv |,  and  eA+(g−1k exp(tH)) =| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1)  1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  20  From  eA+(g−1k exp(tH))−H(g−1k exp(tH)) =  e−t  | cue1 + dv |[| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1)  1  2 ],  together with  | sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1)  1  2 ≤ 2 | sinh t cue1v−1 + cosh t d |  ≤| cue1v−1 + d | et+ | d − cue1v−1 | e−t  we deduce that  e(A+(g−1k exp(tH))−H(g−1k exp(tH)) ≤ 1 + | d − cue1v−1 |  | cue1v−1 + d |e−2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Noting that (g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0)∗ = −(d−1c), and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e1 = ue1v−1, we get  e(A+(g−1k exp(tH))−H(g−1k exp(tH)) ≤ 1 + | 1+ < g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e1 >|  | 1− < g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e1 >|e−2t  ≤ 1 + 1+ | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0 |  1− | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0 |e−2t,  from which we deduce (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2), and the proof of the lemma is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof of the Key Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Since B∗(G, τν) is G-invariant, we may apply Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='1 to get  Φν,λ(g−1x)v ≃ τ −1  ν (κ1(g−1x)κ2(g−1x)  �  s∈{±}  cν(sλ)e(isλ−ρ)A+(g−1x)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Thus it suffices to show that  τ −1  ν (κ1(g−1x)κ2(g−1x)  �  s∈{±}  cν(sλ)e(isλ−ρ)A+(g−1x)v ≃ τ −1  ν (k2(x))  �  s∈{±1}  cν(sλ)e(isλ−ρ)A+(x)f g  sλ,v(k1(x)),  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3)  Note that  τ −1  ν [k1(g−1k1(x)eA+(x)k2(x))k2(g−1k1(x)eA+(x)k2(x))] = τ −1  ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))k2(x))],  x = k1(x)eA+(x)k2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Henceforth (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='3) is equivalent to  τ −1  ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))]  �  s∈{±1}  cν(λ)e(isλ−ρ)A+(g−1k1(x)eA+(x)) v  ≃  �  s∈{±1}  cν(λ)e(isλ−ρ)A+(x)f g  sλ,v(k1(x))  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4)  We write the left hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='4) as  �  s∈{±1}  cν(λ)e(isλ−ρ)A+(x)f g  sλ,v(k1(x)) + rg(x)v,  where  rg(x) =τ −1  ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))]  �  s∈{±1}  cν(λ)e(isλ−ρ)A+(g−1k1(x)eA+(x))  −  �  s∈{±1}  cν(λ)e(isλ−ρ)[A+(x)+H(g−1k1(x))]τ −1  ν (κ(g−1k1(x)),  x ∈ G  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='5)  21  To finish the proof we show that for each g ∈ G, rg ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Noting that  H(g−1k1(x)eA+(x)) = H(g−1k1(x)) + A+(x),  we rewrite rg as  rg(x) = [τ −1  ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))) − τ−1  ν (κ(g−1k1(x))]  �  s∈{±1}  cν(λ)e(isλ−ρ)H(g−1k1(x)eA+(x))  + τ −1  ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x)))  \uf8eb  \uf8ed �  s∈{±1}  cν(λ)[e(isλ−ρ)A+(g−1k1(x)eA+(x))) − e(isλ−ρ)H((g−1k1(x)eA+(x))]  \uf8f6  \uf8f8  =: Ig(x) + Jg(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Using the following  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Let g =  �  a  b  c  d  �  ∈ Sp(n, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then we have  τν(κ1(g)κ2(g)) = τν( d  | d |)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6)  τν(κ(g)) = τν( ce1 + d  | ce1 + d |)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='7)  lim  R→∞ τν(κ1(g exp(RH))κ2(g exp(RH))) = τν(κ(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='8)  we easily see that Igv ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We have  Jg(x) =τ −1  ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x)))e(isλ−ρ)H(g−1k1(x)eA+(x))  �  s∈{±1}  cν(λ)  �  e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1  �  As τν is unitary we have  1  R  �  K×[0,R]  ∥ Jg(ketH)v ∥2  ν ∆(t)dt dk  ≤∥ v ∥2 2 | cν(λ) |2  R  �  K×[0,R]  e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2  From  | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |≤ C(| λ | +ρ) | A+(g−1k1(x)eA+(x)) − H(g−1k1(x)eA+(x) |  together with Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2 we get  �  K×[0,R]  e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2  ≤  �  C(| λ | +ρ)1+ | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0 |  1− | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='0 |  �2 1  R  �  K×[0,R]  e−2ρH(g−1k)e−2(ρ+2t)∆(t) dk dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  22  As  �  K  e−2ρH(g−1k) dk = 1 and ∆(t) ≤ 2ρe2ρt we obtain  lim  R→∞  1  R  �  K×[0,R]  e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  This shows that Jg ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Therefore we have proved that for each g ∈ G, rg ≃ 0 as to be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  It remain to prove Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' If g =  �  a  b  c  d  �  =  �  u1  0  0  v1  �  at  �  u2  0  0  v2  �  with respect to the Cartan decomposition G =  KAK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then we easily see that d = cosh t v1v2 and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Analogously if g =  �  a  b  c  d  �  =  �  u  0  0  v  �  at n with  respect to the Iwasawa decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then from g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='e1 =  �  ae1 + b  ce1 + d  �  = et  �  u  v  �  we get et v = ce1 + d and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='7) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  We have  g exp(RH) =  �  ∗  ∗∗  ∗ ∗ ∗  sinh Re1 + cosh Rd  �  Then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='6) imply that τν(κ1(g)κ2(g)) = τν( tanh Rce1+d  |tanh Rce1+d|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Thus limR→∞ τν(κ1(g)κ2(g)) = τν( ce1+d  |ce1+d|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' This finishes  the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='2, and the proof of the Key Lemma is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  8  Appendix  In this section we collect some results on the Jacobi functions, referring to [19] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  For α, β, λ ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' α ̸= −1, −2, · · · and t ∈ R, the Jacobi function is defined by  φ(α,β)  λ  (t) = 2F1(iλ + ρα,β  2  , −iλ + ρα,β  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' α + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' − sinh2 t),  where 2F1 is the Gauss hypergeometric function and ρα,β = α + β + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The Jacobi function φ(α,β)  λ  is the unique even smooth function on R which satisfy φ(α,β)  λ  (0) = 1 and the differential  equation  { d2  dt2 + [(2α + 1) coth t + (2β + 1) tanh t] d  dt + λ2 + ρ2  α,β}φ(α,β)  λ  (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (A1)  For λ /∈ −iN another solution Ψα,β  λ  of (A1) such that  Ψα,β  λ  (t) = e(iλ−ρα,β)t(1 + ◦(1)),  as  t → ∞  (A2)  is given by  Ψα,β  λ  (t) = (2 sinh t)iλ−ρα,β 2F1(ρα,β − iλ  2  , β − α + 1 − iλ  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' 1 − iλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' −  1  sinh2 t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Moreover there exists a constant C > 0 such that for all λ ∈ R and all t ≥ 1 we have  Ψα,β  λ  (t) = e(iλ−ρα,β)t(1 + e−2tΘλ(t)),  with  | Θλ(t) |≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (A2’)  For λ /∈ iZ, we have  φ(α,β)  λ  (t) =  �  s=±1  cα,β(sλ)Ψα,β  sλ (t)  (A3)  23  where  cα,β(λ) = 2ρα,β−iλ Γ(α + 1)Γ(iλ)  Γ( iλ+ρα,β  2  )Γ( iλ+α−β+1  2  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  For ℜ(iλ) > 0, the asymptotic behaviour of φ(α,β)  λ  as t → ∞ is then given by  lim  t→∞ e(ρα,β−iλ)tφ(α,β)  λ  (t) = cα,β(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (A4)  Let De(R) denote the space of even smooth function with compact support on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' For f ∈ De(R), the Fourier-Jacobi  transform J α,βf (λ ∈ C) is defined by  J α,βf(λ) =  � ∞  0  f(t)φ(α,β)  λ  (t)∆α,β(t) dt,  (A5)  where ∆α,β(t) = (2 sinh t)2α+1(2 cosh t)2β+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  In the sequel, we assume that α > −1, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Then the meromorphic function cα,β(−λ)−1 has only simple poles for  ℑλ ≥ 0 which occur in the set  Dα,β = {λk = i(| β | −α − 1 − 2k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' k = 0, 1, · · · , | β | −α − 1 − 2k > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (If | β |≤ α + 1, then Dα,β is empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  The following inversion and Plancherel formulas for the Jacobi transform hold for every f ∈ De(R):  f(t) = 1  2π  � ∞  0  (J α,βf)(λ) φ(α,β)  λ  (t) | cα,β(λ) |−2 dλ +  �  λk∈Dα,β  dk(J α,βf)(λk) φ(α,β)  λk  (t),  (A6)  � ∞  0  | f(t) |2 ∆(t) dt = 1  2π  � ∞  0  | (J α,βf)(λ) |2 | cα,β(λ) |−2 dλ +  �  λk∈Dα,β  dk | (J α,βf)(λk) |2  (A6’)  where dk = −i Resλ=λk(cα,β(λ)cα,β(−λ))−1, is given explicitly by  dk = (β − α − 2k − 1)2−2(α+β)Γ(α + k + 1)Γ(β − k)  Γ2(α + 1)Γ(β − α − k)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  (A7)  References  [1] Anker,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=': A basis inequality for Scattering Theory for Riemannian Symmetric Spaces of the Noncompact Type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' 113 (3), 391-398 (1991)  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Agmon, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Hormander, Asymptotic properties of solutions of differential equations with simple characteristics,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Analyse Math, 30 (1976), 1-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Boussejra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Intissar, Caractérisation des integrales de Poisson-Szego de L2(∂Bn) dans la boule de Bergman  Bn, n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' 318 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
+page_content=' Boussejra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'}
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+Climate change heterogeneity:
+A new quantitative approach
+∗
+Mar´ıa Dolores Gadea Rivas †
+University of Zaragoza
+Jes´us Gonzalo ‡
+U. Carlos III de Madrid
+July 10, 2022
+Abstract
+Climate change is a non-uniform phenomenon.
+This paper proposes a new
+quantitative methodology to characterize, measure and test the existence of
+climate change heterogeneity. It consists of three steps. First, we introduce a
+new testable warming typology based on the evolution of the trend of the whole
+temperature distribution and not only on the average. Second, we define the
+concepts of warming acceleration and warming amplification in a testable for-
+mat. And third, we introduce the new testable concept of warming dominance
+to determine whether region A is suffering a worse warming process than region
+B. Applying this three-step methodology, we find that Spain and the Globe ex-
+perience a clear distributional warming process (beyond the standard average)
+but of different types. In both cases, this process is accelerating over time and
+asymmetrically amplified. Overall, warming in Spain dominates the Globe in
+all the quantiles except the lower tail of the global temperature distribution
+that corresponds to the Artic region. Our climate change heterogeneity results
+open the door to the need for a non-uniform causal-effect climate analysis that
+goes beyond the standard causality in mean as well as for a more efficient design
+of the mitigation-adaptation policies. In particular, the heterogeneity we find
+suggests that these policies should contain a common global component and a
+clear local-regional element. Future climate agreements should take the whole
+temperature distribution into account.
+JEL classification: C31, C32, Q54
+Keywords:
+Climate change; Climate heterogeneity; Global-Local Warming;
+Functional stochastic processes; Distributional characteristics; Trends; Quan-
+tiles; Temperature distributions.
+∗The authors gratefully acknowledge the financial support from the Gobierno de Aragon and FEDER
+funds (grant, LMP71-18), the Spanish Ministerio de Ciencia y Tecnolog´ıa, Agencia Espa˜nola de Investi-
+gaci´on (AEI) and European Regional Development Fund (ERDF, EU) under grants PID2019-104960GB-
+IOO, ECO2017-83255-C3-1-P (AEI/ERDF, EU) and ECO2016-81901-REDT, and Bank of Spain (ER grant
+program). We thank Rodrigo Gonzalez Laiz for excellent research assistance.
+† Department of Applied Economics, University of Zaragoza. Gran V´ıa, 4, 50005 Zaragoza (Spain). Tel:
++34 9767 61842, fax: +34 976 761840 and e-mail: lgadea@unizar.es
+‡ Department of Economics, University Carlos III, Madrid 126 28903 Getafe (Spain).
+Tel: +34 91
+6249853, fax: +34 91 6249329 and e-mail: jesus.gonzalo@uc3m.es (corresponding author)
+1
+arXiv:2301.02648v1  [econ.EM]  6 Jan 2023
+
+Climate change heterogeneity
+2
+1
+Introduction
+All the assessment reports (AR) published by the Intergovernmental Panel of Cli-
+mate Change (IPCC) show that there is overwhelming scientific evidence of the
+existence of global warming (GW). It is also well known that climate change (CC)
+is a non-uniform phenomenon. What is not so clear is the degree of heterogeneity
+across all the regions in our planet. In fact, an important part of the Sixth Assess-
+ment Report (AR6) published by the IPCC in 2021-2022 is dedicated to this issue:
+climate (warming) heterogeneity. This is reflected in the chapters studying regional
+climate change. Our paper introduces a new quantitative methodology that builds
+on that described in Gadea and Gonzalo 2020 (GG2020) to characterize, measure
+and test the existence of such climate change heterogeneity (CCH). This is done in
+three steps. First, we introduce a warming typology (W1, W2 and W3) based on
+the trending behavior of the quantiles of the temperature distribution of a given ge-
+ographical location. Second, we define in a testable format the concepts of warming
+acceleration and warming amplification. These concepts help to characterize (more
+ordinally than cardinally) the warming process of different regions. And third, we
+propose the new concept of warming dominance (WD) to establish when region A
+suffers a worse warming process than region B.
+We have chosen Spain as a benchmark geographical location because, as the AR6
+report states “. . . Spain is fully included in the Mediterranean (MED) Reference
+Region, but is one of the most climatically diverse countries in the world. . . ”.
+This fact opens up the possibility of studying warming heterogeneity (WH) from
+Spain to the Globe (outer heterogeneity, OWH) and also from Spain to some of its
+regions represented by Madrid and Barcelona (inner heterogeneity, IWH).
+The three steps rely on the results reported in GG2020, where the different
+distributional characteristics (moments, quantiles, inter quantile range, etc.) of the
+temperature distribution of a given geographical location are converted into time
+series objects. By doing this, we can easily implement and test all the concepts
+involved in the three steps.
+A summary of the results is as follows. Spain and the Globe present a clear
+warming process; but it evolves differently. Spain goes from a warming process where
+lower and upper temperatures share the same trend behavior (IQR is maintained
+constant over time, warming type W1) to one characterized by a larger increase in
+the upper temperatures (IQR increases over time, warming type W3). In contrast,
+
+Climate change heterogeneity
+3
+the Globe as a whole maintains a stable warming type process characterized by lower
+temperatures that increase more than the upper ones (IQR decreases in time).1 In
+our typology, this constitutes a case of warming type W2. Climate heterogeneity
+can go further.
+For instance, within Spain we find that Madrid is of type W3
+while the warming process of Barcelona is of type W1. This is in concordance with
+the Madrid climate being considered a Continental Mediterranean climate while
+Barcelona is more a pure Mediterranean one.
+The proposed warming typology (W1, W2 and W3), although dynamic, is more
+ordinal than cardinal. In this paper, the strength of a warming process is captured
+in the second step by analyzing its acceleration and its amplification with respect
+to a central tendency measure of the temperature distribution. Acceleration and
+amplification contribute to the analysis of warming heterogeneity. The acceleration
+in the Globe is present in all the quantiles above q30 while in Spain it already
+becomes significant above the 10th quantile. We find an asymmetric behavior of
+warming amplification; in Spain (in comparison with the Globe mean temperature)
+this is present in the upper temperatures (above the 80th and 90th quantiles) while
+in the Globe the opposite occurs (below the 20th and 30th quantiles). Within Spain,
+Madrid and Barcelona also behave differently in terms of acceleration and amplifi-
+cation. Overall, warming in Spain dominates that of the Globe in all the quantiles
+except for the lower quantile q05, and between Madrid and Barcelona there is a par-
+tial WD. Madrid WD Barcelona in the upper part of the distribution and Barcelona
+WD Madrid in the lower one.
+The existence of a clear heterogeneous warming process opens the door to the
+need of a new non-uniform causal (effect) research. One that goes beyond the stan-
+dard causality in mean analysis (see Tol, 2021). CCH also suggests that in order
+for the mitigation-adaptation policies to be as efficient as possible they should be
+designed following a type of common factor structure: a common global compo-
+nent plus an idiosyncratic local element.
+This goes in the line with the results
+found in Brock and Xepapadeas (2017), D’Autume et al. (2016) and Peng et al.
+(2021). Future climate agreements should clearly have this CCH into account. An
+important by-product of our warming heterogeneity results is the increase that this
+heterogeneity can generate in the public awareness of the GW process. A possible
+explanation for that can be found in the behavioral economics work by Malmendier
+1Similar results for Central England are found in GG2020 and for the US in Diebold and Rude-
+bush, 2022.
+
+Climate change heterogeneity
+4
+(2021), in the results of the European Social Survey analyzed in Nowakowski and
+Oswald (2020) or in the psychology survey by Maiella et al. (2020).
+The rest of the paper is organized as follows. Section 2 describes our basic climate
+econometrics methodology. Section 3 presents a brief description of the temperature
+data from Spain and the Globe. Section 4 addresses the application of our quantita-
+tive methodology in the cross-sectional version (temperatures measured monthly by
+stations in an annual interval) to Spain and (versus) the Globe. It also reports the
+results of applying the methodology using a purely temporal dimension (local daily
+temperature on an annual basis) for two representative stations in Spain (Madrid
+and Barcelona, empirical details in the Appendix). Section 5 offers a comparison
+and interpretation of the results. Finally, Section 6 concludes the paper.
+2
+Climate Econometrics Methodology
+In this section, we briefly summarize the novel econometric methodology introduced
+in GG2020 to analyze Global and Local Warming processes. Following GG2020,
+Warming is defined as an increasing trend in certain characteristics of the temper-
+ature distribution. More precisely:
+Definition 1. (Warming):
+Warming is defined as the existence of an increas-
+ing trend in some of the characteristics measuring the central tendency or position
+(quantiles) of the temperature distribution.
+An example is a deterministic trend with a polynomial function for certain val-
+ues of the β parameters Ct = β0 + β1t + β2t2 + ... + βktk.
+In GG2020 temperature is viewed as a functional stochastic process X = (Xt(ω), t ∈
+T), where T is an interval in R, defined in a probability space (Ω, ℑ, P). A conve-
+nient example of an infinite-dimensional discrete-time process consists of associating
+ξ = (ξn, n ∈ R+) with a sequence of random variables whose values are in an appro-
+priate function space. This may be obtained by setting
+Xt(n) = ξtN+n, 0 ≤ n ≤ N, t = 0, 1, 2, ..., T
+(1)
+so X = (Xt, t = 0, 1, 2, ..., T). If the sample paths of ξ are continuous, then we have
+a sequence X0, X1, .... of random variables in the space C[0, N]. The choice of the
+period or segment t will depend on the situation in hand. In our case, t will be the
+
+Climate change heterogeneity
+5
+period of a year, and N represents cross-sectional units or higher-frequency time
+series.
+We may be interested in modeling the whole sequence of G functions, for instance
+the sequence of state densities (f1(ω), f2(ω), ..., fT (ω) ) as in Chang et al. (2015,
+2016) or only certain characteristics (Ct(w)) of these G functions, for instance, the
+state mean, the state variance, the state quantile, etc. These characteristics can
+be considered time series objects and, therefore, all the econometric tools already
+developed in the time series literature can be applied to Ct(w). With this charac-
+teristic approach we go from Ω to RT , as in a standard stochastic process, passing
+through a G functional space:
+Ω
+(w)
+X
+−→
+G
+Xt(w)
+C−→
+R
+Ct(w)
+Going back to the convenient example and abusing notation, the stochastic struc-
+ture can be summarized in the following array:
+X10(w) = ξ0(w)
+X11(w) = ξ1(w)
+. . .
+X1N(w) = ξN(w)
+C1(w)
+X20(w) = ξN+1(w)
+X21(w) = ξN+2(w)
+. . .
+X2N(w) = ξ2N(w)
+C2(w)
+.
+.
+.
+.
+.
+.
+. . .
+. . .
+. . .
+.
+.
+.
+.
+.
+.
+XT0(w) = ξ(T−1)N+1(w)
+XT1(w) = ξ(T−1)N+2(w)
+. . .
+XTN(w) = ξTN(w)
+CT (w)
+(2)
+The objective of this section is to provide a simple test to detect the existence of
+a general unknown trend component in a given characteristic Ct of the temperature
+process Xt.
+To do this, we need to convert Definition 1 into a more practical
+definition.
+Definition 2. (Trend test): Let h(t) be an increasing function of t. A characteristic
+Ct of a functional stochastic process Xt contains a trend if β ̸= 0 in the regression
+Ct = α + βh(t) + ut, t = 1, ..., T.
+(3)
+The main problem of this definition is that the trend component in Ct as well
+as the function h(t) are unknown. Therefore this definition can not be easily imple-
+mented. If we assume that Ct does not have a trend component (it is I(0))2 and
+2Our definition of an I(0) process follows Johansen (1995). A stochastic process Yt that satisfies
+Yt − E(Yt) =
+∞
+�
+i=1
+Ψiεt−i is called I(0) if
+∞
+�
+i=1
+Ψ izi converges for |z| < 1 + δ, for some δ > 0 and
+∞
+�
+i=1
+Ψ
+i ̸= 0, where the condition εt ∼ iid(0,σ2) with σ2 > 0 is understood.
+
+Climate change heterogeneity
+6
+h(t) is linear, then we have the following well known result.
+Proposition 1. Let Ct = I(0). In the regression
+Ct = α + βt + ut
+(4)
+the OLS estimator
+�β =
+T�
+t=1
+(Ct − C)(t − t)
+T�
+t=1
+(t − t)2
+(5)
+satisfies
+T 3/2 �β = Op(1)
+(6)
+and asymptotically (T → ∞)
+tβ=0 is N(0, 1).
+In order to analyze the behavior of the t-statistic tβ = 0, for a general trend
+component in Ct, it is very convenient to use the concept of Summability (Berenguer-
+Rico and Gonzalo, 2014)
+Definition 3. (Order of Summability):
+A trend h(t) is said to be summable of
+order “δ” (S(δ)) if there exists a slowly varying function L(T),3 such that
+ST =
+1
+T 1+δ L(T)
+T
+�
+t=1
+h(t)
+(8)
+is O(1), but not o(1).
+Proposition 2. Let Ct = h(t) + I(0) such that h(t) is S(δ) with δ ≥ 0, and such
+that the function g(t) = h(t)t is S(δ + 1). In the regression
+Ct = α + βt + ut
+(9)
+the OLS �β estimator satisfies
+T (1−δ) �β = Op(1).
+(10)
+Assuming that the function h(t)2 is S(1 + 2δ − γ) with 0 ≤ γ ≤ 1 + δ, then
+3A positive Lebesgue measurable function, L, on (0, ∞) is slowly varying (in Karamata’s sense)
+at ∞ if
+L(λn)
+L(n) → 1 (n → ∞) ∀λ > 0.
+(7)
+(See Embrechts et al., 1999, p. 564).
+
+Climate change heterogeneity
+7
+tβ=0 =
+� Op(T γ/2) for 0 ≤ γ ≤ 1
+Op(T 1/2) for 1 ≤ γ ≤ 1 + δ
+(11)
+Examples of how this proposition applies for different particular Data Generat-
+ing Processes (DGP) can be found in GG.
+A question of great empirical importance is how our trend test (TT) of Proposi-
+tion 2 behaves when Ct = I(1) (accumulation of an I(0) process). Following Durlauf
+and Phillips (1988), T 1/2 �β = Op(1); however, tβ=0 diverges as T→∞. Therefore,
+our TT can detect the stochastic trend generated by an I(1) process. In fact, our
+test will detect trends generated by any of the three standard persistent processes
+considered in the literature (see Muller and Watson, 2008): (i) fractional or long-
+memory models; (ii) near-unit-root AR models; and (iii) local-level models. Let
+Ct = µ + zt, t = 1, ..., T.
+(12)
+In the first model, zt is a fractional process with 1/2 < d < 3/2. In the second
+model, zt follows an AR, with its largest root close to unity, ρT = 1 − c/T. In the
+third model, zt is decomposed into an I(1) and an I(0) component. Its simplest
+format is zt = υt + ϵt with υt = υt−1 +ηt, where ϵt is ID(0, q ∗ σ2), ηt is ID(0, σ2),
+σ2 > 0 and both disturbances are serially and mutually independent. Note that the
+pure unit-root process is nested in all three models: d = 1, c = 0, and q = 0.
+The long-run properties implied by each of these models can be characterized
+using the stochastic properties of the partial sum process for zt.
+The standard
+assumptions considered in the macroeconomics or finance literature assume the ex-
+istence of a “δ,” such that T −1/2+δ �T
+t=1 zt −→ σ H(.), where “δ” is a model-specific
+constant and H is a model-specific zero-mean Gaussian process with a given covari-
+ance kernel k(r, s). Then, it is clear that the process Ct = µ + zt is summable (see
+Berenguer-Rico and Gonzalo, 2014). This is the main reason why Proposition 3
+holds for these three persistent processes.
+Proposition 3. Let Ct = µ + zt, t = 1, ..., T, with zt any of the following three
+processes: (i) a fractional or long-memory model, with 1/2 < d < 3/2; (ii) a near-
+unit-root AR model; or (iii) a local-level model. Furthermore, T −1/2+δ �T
+t=1 zt −→ σ
+H(.), where “δ” is a model-specific constant and H is a model-specific zero-mean
+Gaussian process with a given covariance kernel k(r, s). Then, in the LS regression
+Ct = α + βt + ut,
+
+Climate change heterogeneity
+8
+the t-statistic diverges,
+tβ=0 = Op(T 1/2).
+After the development of the theoretical core, we are in a position to design
+tools to approach the empirical strategy. The following subsection describes each of
+them.
+2.1
+Empirical tools: definitions and tests
+From Propositions 2 and 3, Definition 2 can be simplified into the following testable
+and practical definition.
+Definition 4. (Practical definition 2):
+A characteristic Ct of a functional stochas-
+tic process Xt contains a trend if in the LS regression,
+Ct = α + βt + ut, t = 1, ..., T,
+(13)
+β = 0 is rejected.
+Several remarks are relevant with respect to this definition: (i) regression (13)
+has to be understood as the linear LS approximation of an unknown trend function
+h(t) (see White, 1980); (ii) the parameter β is the plim of �βols; (iii) if the regression
+(13) is the true data-generating process, with ut ∼ I(0), then the OLS �β estimator
+is asymptotically equivalent to the GLS estimator (see Grenander and Rosenblatt,
+1957); (iv) in practice, in order to test β = 0, it is recommended to use a robust
+HAC version of tβ=0 (see Busetti and Harvey, 2008); and (v) this test only detects
+the existence of a trend but not the type of trend.
+For all these reasons, in the empirical applications we implement Definition 4
+by estimating regression (13) using OLS and constructing a HAC version of tβ=0
+(Newey and West, 1987).
+These linear trends can be common across characteristics indicating similar pat-
+ters in the time evolution of these characteristics.
+Definition 5. (Co-trending): A set of m distributional characteristics (C1t,C2t,...,Cmt)
+do linearly co-trend if in the multivariate regression
+�
+�
+C1t
+...
+Cmt
+�
+� =
+�
+�
+α1
+...
+αm
+�
+� +
+�
+�
+β1
+...
+βm
+�
+� t +
+�
+�
+u1t
+...
+umt
+�
+�
+(14)
+
+Climate change heterogeneity
+9
+all the slopes are equal, β1 = β2 = ... = βm. 4
+This co-trending hypothesis can be tested by a standard Wald test.
+When m = 2 an alternative linear co-trending test can be obtained from the
+regression
+Cit − Cjt = α + βt + ut
+i ̸= j i, j = 1, ..., m by testing the null hypothesis of β = 0 vs β ̸= 0 using a simple
+tβ=0 test.
+Climate classification is a tool used to recognize, clarify and simplify the existent
+climate heterogeneity in the Globe. It also helps us to better understand the Globe’s
+climate and therefore to design more efficient global warming mitigation policies.
+The prevalent climate typology is that proposed by K¨oppen (1900) and later on
+modified in K¨oppen and Geiger (1930). It is an empirical classification that divides
+the climate into five major types, which are represented by the capital letters A
+(tropical zone), B (dry zone), C (temperate zone), D (continental zone), and E
+(polar zone). Each of these climate types except for B is defined by temperature
+criteria. More recent classifications can been found in the AR6 of the IPCC (2021,
+2022) but all of them share the spirit of the original one of K¨oppen (1900).
+The climate classification we propose in this section is also based on temperature
+data and it has three simple distinctive characteristics:
+• It considers the whole temperature distribution and not only the average
+• It has a dynamic nature: it is based on the evolution of the trend of the
+temperature quantiles (lower and upper).
+• It can be easily tested
+Definition 6. (Warming Typology): We define four types of warming processes:
+• W0: There is no trend in any of the quantiles (No warming).
+• W1: All the location distributional characteristics have the same positive trend
+(dispersion does not contain a trend)
+• W2: The Lower quantiles have a larger positive trend than the Upper quantiles
+(dispersion has a negative trend)
+• W3: The Upper quantiles have a larger positive trend than the Lower quantiles
+(dispersion has a positive trend).
+4This definition is slightly different from the one in Carrion-i-Silvestre and Kim (2019).
+
+Climate change heterogeneity
+10
+Climate is understood, unlike weather, as a medium and long-term phenomenon
+and, therefore, it is crucial to take trends into account. Notice that this typology
+can be used to describe macroclimate as well as microclimate locations.
+Most of the literature on Global or Local warming only considers the trend
+behavior of the central part of the distribution (mean or median). By doing this, we
+are losing very useful information that can be used to describe the whole warming
+process. This information is considered in the other elements of the typology W1,
+W2 and W3. This typology does not say anything about the intensity of the warming
+process and its dynamic. Part of this intensity is captured in the following definitions
+of warming acceleration and warming amplification.
+Definition 7. (Warming Acceleration):
+We say that there is warming acceler-
+ation in a distributional temperature characteristic Ct between the time periods
+t1 = (1, ..., s) and t2 = (s + 1, ..., T) if in the following two regressions:
+Ct = α1 + β1t + ut, t = 1, ..., s, ..., T,
+(15)
+Ct = α2 + β2t + ut, t = s + 1, ..., T,
+(16)
+the second trend slope is larger than the first one: β2 > β1.
+In practice, we implement this definition by testing in the previous system the
+null hypothesis β2 = β1 against the alternative β2 > β1 An alternative warming
+acceleration test can be formed by testing for a structural break at t = s. Neverthe-
+less, we prefer the approach of Definition 7 because it matches closely the existent
+narrative on warming acceleration in the climate literature.
+Definition 8. (Warming Amplification with respect to the mean):
+We say that
+there is a warming amplification in distributional characteristic Ct with respect the
+mean if in the following regression:
+Ct = β0 + β1meant + ϵt
+(17)
+the mean slope is greater than one: β1 > 1.
+When the mean, meant, and Ct come from the same distribution, we name this
+“inner” warming amplification. Otherwise, the mean may come from an external
+environment and, in that case, we call it “outer” warming amplification.
+Both concepts, acceleration and amplification, introduce a quantitative dimen-
+sion to the ordinarily defined classification. For example, the acceleration, which
+
+Climate change heterogeneity
+11
+has a dynamic character, allows us to observe the transition from one type of cli-
+mate to another. Amplification, on the other hand, makes it possible to compare
+the magnitude of the trends that define each type of climate. It should be noted
+that, although static in nature, it can be computed recursively at different points in
+time.
+In the previous definitions, we classify the warming process of different regions
+which is crucial in the design of local mitigation and adaptation policies. But we,
+also, need to compare the different climate change processes of two regions in order
+to characterize climate heterogeneity independently of the type of warming they are
+experimenting. For this purpose, we propose the following definition that shares the
+spirit of the stochastic dominance concept used in the economic-finance literature.
+Definition 9. (Warming Dominance (WD): We say that the temperature distri-
+butions of Region A warming dominates (WD) the temperature distributions of
+Region B if in the following regression
+qτt(A) − qτt(B) = ατ + βτt + uτt,
+(18)
+βτ ≥ 0 for all 0 < τ < 1 and there is at least one value τ ∗ for which a strict
+inequality holds.
+It is also possible to have only partial (WD). For instance, in the lower or upper
+quantiles.
+3
+The data
+3.1
+Spain
+The measurement of meteorological information in Spain started in the eighteenth
+century. However, it was not until the mid-nineteenth century that reliable and reg-
+ular data became available. In Spain, there are four main sources of meteorological
+information: the Resumen Anual, Bolet´ın Diario, Bolet´ın Mensual de Climatolog´ıa
+and Calendario Meteorol´ogico. These were first published in 1866, 1893, 1940 and
+1943, respectively. A detailed explanation of the different sources can be found in
+Carreras and Tafunell (2006).
+Currently, AEMET (Agencia Estatal de Meterolog´ıa) is the agency responsible
+for storing, managing and providing meteorological data to the public. Some of the
+historical publications, such as the Bolet´ın Diario and Calendario Meteorol´ogico can
+
+Climate change heterogeneity
+12
+be found in digital format in their respective archives for whose use it is necessary
+to use some kind of Optical Character Recognition (OCR) software.5
+In 2015, AEMET developed AEMET OpenData, an Application Programming
+Interface (API REST) that allows the dissemination and reuse of Spanish meteoro-
+logical and climatological information. To use it, the user needs to obtain an API
+key to allow access to the application. Then, either through the GUI or through
+a programming language such as Java or Python, the user can request data. More
+information about the use of the API can be found on their webpage.6
+In this paper, we are concerned with Spanish daily station data, specifically
+temperature data. Each station records the minimum, maximum and average tem-
+perature as well as the amount of precipitation, measured as liters per square meter.
+The data period ranges from 1920 to 2019. However, in 1920 there were only 13
+provinces (out of 52) who had stations available. It was not until 1965 that all the
+52 provinces had at least one working station. Moreover, it is important to keep in
+mind that the number of stations has increased substantially from only 14 stations in
+1920 to more than 250 in 2019. With this information in mind, we select the longest
+span of time that guarantees a wide sample of stations so that all the geographical
+areas of peninsular Spain are represented. For this reason, we decided to work with
+station data from 1950 to 2019. There are 30 stations whose geographical distri-
+bution is displayed in the map in Figure 1. The original daily data are converted
+into monthly data, so that we finally work with a total of 30x12 station-month units
+corresponding to peninsular Spain and, consequently, we have 360 observations each
+year with which to construct the annual distributional characteristics.
+3.2
+The Globe
+In the case of the Globe, we use the database of the Climate Research Unit (CRU)
+that offers monthly and yearly data of land and sea temperatures in both hemi-
+spheres from 1850 to the present, collected from different stations around the world.7
+Each station temperature is converted to an anomaly, taking 1961-1990 as the base
+5http : //www.aemet.es/es/conocermas/recursosenlinea/calendarios?n = todos and https :
+//repositorio.aemet.es/handle/20.500.11765/6290.
+6https : //opendata.aemet.es/centrodedescargas/inicio. The use of AEMET data is regulated
+in the following resolution https : //www.boe.es/boe/dias/2016/01/05/pdfs/BOE − A − 2016 −
+111.pdf.
+7We
+use
+CRUTEM
+version
+5.0.1.0,
+which
+can
+be
+downloaded
+from
+(https://crudata.uea.ac.uk/cru/data/temperature/).
+A recent revision of the methodology
+can be found in Jones et al. (2012).
+
+Climate change heterogeneity
+13
+period, and each grid-box value, on a five-degree grid, is the mean of all the station
+anomalies within that grid box. This database (in particular, the annual temper-
+ature of the Northern Hemisphere) has become one of the most widely used to
+illustrate GW from records of thermometer readings. These records form the blade
+of the well-known “hockey stick” graph, frequently used by academics and other
+institutions, such as, the IPCC. In this paper, we prefer to base our analysis on raw
+station data, as in GG2020.
+The database provides data from 1850 to nowadays, although due to the high
+variability at the beginning of the period it is customary in the literature to begin
+in 1880. In this work, we have selected the stations that are permanently present
+in the period 1950-2019 according to the concept of the station-month unit. In this
+way, the results are comparable with those obtained for Spain. Although there are
+10,633 stations on record, the effective number fluctuates each year and there are
+only 2,192 stations with data for all the years in the sample period, which yields
+19,284 station-month units each year (see this geographical distribution in the map
+in Figure 1).8 In summary, we analyze raw global data (stations instead of grids)
+for the period 1950 to 2019, compute station-month units that remain all the time
+and with these build the annual distributional characteristics.
+4
+Empirical strategy
+In this section we apply our three-step quantitative methodology to show the ex-
+istent climate heterogeneity between Spain and the Globe as well as within Spain,
+between Madrid and Barcelona. Because all our definitions are written in a test-
+ing format, it is straightforward to empirically apply them. First, we test for the
+existence of warming by testing the existence of a trend in a given distributional
+characteristic. How common are the trends of the different characteristics (revealed
+by a co-trending test) determine the warming typology. Second, the strength of
+the warming process is tested by testing the hypothesis of warming acceleration
+and warming amplification. And third, independently of the warming typology, we
+determine how the warming process of Spain compares with that of the Globe as
+a whole (we do the same for Madrid and Barcelona). This is done by testing for
+warming dominance.
+8In the CRU data there are 115 Spanish stations. However, after removing stations not present
+for the whole 1880 to 2019 period, only Madrid-Retiro, Valladolid and Soria remain. Since 1950,
+applying the same criteria, only 30 remain.
+
+Climate change heterogeneity
+14
+(a) Spain. Selected stations, AEMET data 1950-2019
+(b) The Globe. Selected stations, CRU data 1950-2019
+Figure 1
+Geographical distribution of stations
+The results are presented according to the following steps: first, we apply our
+trend test (see Definition 4) to determine the existence of local or global warming
+and test for any possible warming acceleration; second, we test different co-trending
+
+45
+18d:w 135°W 90
+45°
+S
+90Climate change heterogeneity
+15
+hypotheses to determine the type of warming of each area; thirdly, we test the
+warming amplification hypothesis for different quantiles with respect to the mean
+(of Spain as well as of the Globe): H0 : β1 = 1 versus Ha : β1 > 1 in (17); and
+finally, we compare the CC of different regions, for Spain and the Globe, and within
+Spain, between Madrid and Barcelona, with our warming dominance test (see 18).9
+4.1
+Local warming: Spain
+The cross-sectional analysis is approached under two assumptions. First, choosing a
+sufficiently long and representative period of the geographical diversity of the Span-
+ish Iberian Peninsula, 1950-2019. Second, we work with month-station units from
+daily observations to construct the annual observations of the time series object from
+the data supplied by the stations, following a methodology similar to that carried
+out for the whole planet in GG2020.10 The study comprises the steps described in
+the previous section. The density of the data and the evolution of characteristics
+are displayed, respectively in Figures 2 and 3.
+We find positive and significant trends in the mean, max, min and all the quan-
+tiles. Therefore from definition 1, we conclude there exists a clear local warming
+(see Table 1).
+The recursive evolution for the periods 1950-2019 and 1970-2019 shows a clear
+increase in the trends of the mean, some dispersion measures and higher quantiles
+(see the last column of Table 1). More precisely, there is a significant trend acceler-
+ation in most of the distributional characteristics except the lower quantiles (below
+q20). These quantiles, q05 and q10, remain stable.
+The co-trending tests for the full sample 1950-2019 show a similar evolution of
+the trend for all the quantiles with a constant iqr (see Table 2). This indicates
+that in this period the warming process of Spain can be considered a W1 type.
+More recently, 1970-2019, the co-trending tests (see Table 3) indicate the upper
+quantiles grow faster than the lower ones. This, together with a positive trend in
+the dispersion measured by the iqr shows that Spain has evolved from a W1 to a
+9Before testing for the presence of trends in the distributional characteristics of the data, we
+test for the existence of unit roots. To do so, we use the well-known Augmented Dickey-Fuller test
+(ADF; Dickey and Fuller, 1979), where the number of lags is selected in accordance with the SBIC
+criterion. The results, available from the authors on request, show that the null hypothesis of a
+unit root is rejected for all the characteristics considered.
+10The results with daily averages are very similar.
+The decision to work with monthly data
+instead of daily in the cross-sectional approach has been based on its compatibility with the data
+available for the Globe.
+
+Climate change heterogeneity
+16
+W3 warming type process
+Finally, no evidence of “inner” amplification during the period 1950-2019 is found
+in the lower quantiles. Regarding the upper quantiles, we found both “inner” and
+“outer” amplification in the second period, which supports the previous finding of
+a transition from type W1 to type W3 (see Table 4).
+Summing up, with our proposed tests for the evolution of the trend of the whole
+temperature distribution, we conclude that Spain has evolved from a W1 type to a
+much more dangerous W3 type. The results of acceleration and dynamic amplifi-
+cation reinforce the finding of this transition to type W3.
+Figure 2
+Spain annual temperature density calculated with monthly data across stations
+
+0.06
+0.05
+0.04
+0.03 -
+0.02 -
+0.01 ~
+0
+30.6096
+26.691
+22.7724
+18.8538
+14.9352
+11.0166
+7.09796
+3.17936
+1970
+-0.739249
+1960
+-4.65786
+1950
+temperature in degrees Celsius (month-station units)2010
+2000
+1990
+1980
+vears0.07Climate change heterogeneity
+17
+Figure 3
+Characteristics of temperature data in Spain with stations selected since 1950
+(monthly data across stations, AEMET, 1950-2019)
+
+30
+15
+mean
+max
+13
+25
+1950
+1970
+1990
+2010
+1950
+1970
+1990
+2010
+0
+-6
+min
+std
+1950
+1970
+1990
+2010
+1950
+1970
+1990
+2010
+35
+range
+30
+iqr
+8 
+25
+1950
+1970
+1990
+2010
+1950
+1970
+1990
+2010
+0.2
+2.5
+kurtosis
+skewness
+2
+0.2
+1950
+1970
+1990
+2010
+1950
+1970
+1990
+2010
+20 
+10
+0
+1950
+1970
+1990
+2010
+q5
+q10
+q20
+q30
+q40
+q50
+q60
+q70
+q80
+q90
+q95Climate change heterogeneity
+18
+Table 1
+Trend acceleration hypothesis (Spain monthly data across stations, AEMET,
+1950-2019)
+Trend test by periods
+Acceleration test
+names/periods
+1950-2019
+1970-2019
+1950-2019, 1970-2019
+mean
+0.0242
+0.0389
+3.0294
+(0.0000)
+(0.0000)
+(0.0015)
+max
+0.0312
+0.0526
+2.7871
+(0.0000)
+(0.0000)
+(0.0030)
+min
+0.0289
+0.0251
+-0.2557
+(0.0000)
+(0.0654)
+(0.6007)
+std
+0.0036
+0.0098
+1.7952
+(0.0518)
+(0.0021)
+(0.0374)
+iqr
+0.0051
+0.0158
+1.8197
+(0.1793)
+(0.0028)
+(0.0355)
+rank
+0.0023
+0.0276
+1.2705
+(0.8249)
+(0.1127)
+(0.1030)
+kur
+-0.0010
+-0.0018
+-0.9191
+(0.0203)
+(0.0198)
+(0.8202)
+skw
+0.0011
+-0.0002
+-1.5989
+(0.0271)
+(0.7423)
+(0.9439)
+q5
+0.0227
+0.0206
+-0.2559
+(0.0000)
+(0.0059)
+(0.6008)
+q10
+0.0200
+0.0203
+0.0406
+(0.0000)
+(0.0077)
+(0.4838)
+q20
+0.0209
+0.0300
+1.4158
+(0.0000)
+(0.0000)
+(0.0796)
+q30
+0.0221
+0.0333
+2.0100
+(0.0000)
+(0.0000)
+(0.0232)
+q40
+0.0213
+0.0366
+2.4867
+(0.0000)
+(0.0000)
+(0.0071)
+q50
+0.0211
+0.0404
+3.2496
+(0.0000)
+(0.0000)
+(0.0007)
+q60
+0.0246
+0.0446
+3.1147
+(0.0000)
+(0.0000)
+(0.0011)
+q70
+0.0273
+0.0478
+3.3143
+(0.0000)
+(0.0000)
+(0.0006)
+q80
+0.0275
+0.0471
+2.6949
+(0.0000)
+(0.0000)
+(0.0040)
+q90
+0.0321
+0.0548
+3.2441
+(0.0000)
+(0.0000)
+(0.0007)
+q95
+0.0335
+0.0526
+3.3568
+(0.0000)
+(0.0000)
+(0.0005)
+Note:
+OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:
+Ct = α + βt + ut, for two
+different time periods. For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, ..., s, ..., T, Ct =
+α2 + β2t + ut, t = s + 1, ..., T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1. We show the value of
+the t-statistic and its HAC p-value.
+Table 2
+Co-trending analysis (Spain monthly data across stations, AEMET, 1950-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+13.235
+0.211
+Lower quantiles (q05, q10, q20, q30)
+0.310
+0.958
+Medium quantiles (q40, q50, q60)
+0.438
+0.803
+Upper quantiles (q70, q80, q90, q95)
+1.515
+0.679
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+0.771
+0.993
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+8.331
+0.215
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+11.705
+0.111
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+-0.002
+0.786
+q95-q50
+0.012
+0.000
+q95-q05
+0.011
+0.096
+q75-q25 (iqr)
+0.005
+0.179
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+
+Climate change heterogeneity
+19
+Table 3
+Co-trending analysis (Spain monthly data across stations, AEMET, 1970-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+38.879
+0.000
+Lower quantiles (q05, q10, q20, q30)
+3.121
+0.373
+Medium quantiles (q40, q50, q60)
+1.314
+0.518
+Upper quantiles (q70, q80, q90, q95)
+1.719
+0.633
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+12.771
+0.047
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+10.675
+0.099
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+37.892
+0.000
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+0.020
+0.029
+q95-q50
+0.012
+0.050
+q55-q05
+0.032
+0.002
+q75-q25 (iqr)
+0.016
+0.003
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+Table 4
+Amplification hypothesis (Spain monthly data, AEMET 1950-2019
+periods/variables
+1950-2019
+1970-2019
+1950-2019
+1970-2019
+Inner
+Outer
+q05
+0.80
+0.56
+0.55
+0.39
+(0.866)
+(0.998)
+(0.990)
+(0.996)
+q10
+0.83
+0.65
+0.62
+0.52
+(0.899)
+(0.994)
+(0.992)
+(0.986)
+q20
+0.94
+0.90
+0.76
+0.81
+(0.816)
+(0.890)
+(0.993)
+(0.899)
+q30
+0.93
+0.91
+0.77
+0.87
+(0.935)
+(0.929)
+(0.997)
+(0.834)
+q40
+0.97
+1.03
+0.80
+0.97
+(0.744)
+(0.318)
+(0.978)
+(0.566)
+q50
+0.98
+1.10
+0.83
+1.12
+(0.612)
+(0.067)
+(0.944)
+(0.212)
+q60
+1.09
+1.15
+0.96
+1.23
+(0.103)
+(0.051)
+(0.619)
+(0.056)
+q70
+1.11
+1.16
+1.05
+1.30
+(0.040)
+(0.006)
+(0.350)
+(0.028)
+q80
+1.11
+1.14
+1.06
+1.29
+(0.083)
+(0.071)
+(0.325)
+(0.060)
+q90
+1.14
+1.16
+1.19
+1.45
+(0.101)
+(0.118)
+(0.078)
+(0.007)
+q95
+1.10
+1.09
+1.18
+1.36
+(0.089)
+(0.191)
+(0.051)
+(0.008)
+Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1
+in the regression: Cit = βi0 + βi1meant + ϵit. mean refers to the average of the Spanish Global
+temperature distribution for the “inner” and “outer”cases, respectively.
+
+Climate change heterogeneity
+20
+4.2
+Global warming: the Globe
+In this section, we carry out a similar analysis to that described in the previous
+subsection for Spain. Figures 4 and 5 show the time evolution of the Global temper-
+ature densities and their different distributional characteristics from 1950 to 2019.
+The data in both figures are obtained from stations that report data throughout the
+sample period.
+Table 5 shows a positive trend in the mean as well as in all the quantiles. This
+indicates the clear existence of Global warming, more pronounced (larger trend) in
+the lower part of the distribution (a negative trend in the dispersion measures). The
+warming process suffers an acceleration in all the quantiles above q30.
+From the co-trending analysis (see Tables 6 and 7) we can determine the type
+of warming process characterizing the whole Globe. Table 6 indicates that in the
+period 1950-2019 the Globe experimented a W2 warming type (the lower part of
+the temperature distribution grows faster than the middle and upper part, implying
+iqr and std have a negative trend). Similar results are maintained for the period
+1970-2019 (in this case only the dispersion measure std has a negative trend).
+The asymmetric amplification results shown in Table 8 reinforce the W2 typology
+for the whole Globe: an increase of one degree in the global mean temperature
+increases the lower quantiles by more than one degree. This does not occur with
+the upper part of the distribution. Notice that this amplification goes beyond the
+standard Artic amplification (q05) affecting also q10, q20 and q30.
+Summing up, the results from our different proposed tests for the evolution of
+the trend of the whole temperature distribution indicate that the Globe can be
+cataloged as a undergoing type W2 warming process.
+This warming type may
+have more serious consequences for ice melting, sea level increases, permafrost, CO2
+migration, etc. than the other types.
+
+Climate change heterogeneity
+21
+Figure 4
+Global annual temperature density calculated with monthly data across stations
+
+0.03
+0.025
+density
+0.02
+0.015
+0.01 -
+0.005 ~
+0
+33.369
+21.5248
+9.68063
+-2.16358
+-14.0078
+-25.852
+-37.6962
+-49.5404
+1970
+-61.3846
+1960
+-73.2288
+1950
+temperature in degrees Celsius (month-station units)2010
+2000
+1990
+1980
+vears0.04
+0.035Climate change heterogeneity
+22
+Figure 5
+Characteristics of temperature data in the Globe (monthly data across stations, CRU,
+1950-2019)
+
+12
+40
+11
+38
+10
+mean
+max
+1950196019701980199020002010
+1950196019701980
+199020002010
+13
+-44
+-46
+-48
+50
+-52
+std
+11
+1950196019701980199020002010
+1950196019701980199020002010
+8
+90
+85
+16
+iqr
+range
+80
+19501960 1970198019902000 2010
+1950196019701980199020002010
+G
+kur
+-0.8
+W
+-0.9
+skw
+195019601970:1980199020002010
+195019601970198019902000:2010
+20 
+10
+0
+10
+1950196019701980199020002010
+q5
+q10
+q20
+q30
+q40
+q50
+q60
+q70
+q80
+q90
+q95Climate change heterogeneity
+23
+Table 5
+Trend acceleration hypothesis (CRU monthly data across stations, 1950-2019)
+Trend test by periods
+Acceleration test
+names/periods
+1950-2019
+1970-2019
+1950-2019, 1970-2019
+mean
+0.0213
+0.0300
+2.2023
+(0.0000)
+(0.0000)
+(0.0147)
+max
+0.0361
+0.0523
+1.1217
+(0.0000)
+(0.0001)
+(0.1320)
+min
+0.0423
+-0.0109
+0.5016
+(0.0000)
+(0.5867)
+(0.3084)
+std
+-0.0070
+-0.0057
+0.1776
+(0.0000)
+(0.0570)
+(0.4296)
+iqr
+-0.0067
+-0.0043
+0.2454
+(0.0435)
+(0.4183)
+(0.4033)
+rank
+-0.0062
+0.0632
+0.2181
+(0.5876)
+(0.0005)
+(0.4138)
+kur
+-0.0010
+0.0001
+0.0445
+(0.5205)
+(0.9566)
+(0.4823)
+skw
+0.0006
+0.0003
+0.0301
+(0.0577)
+(0.5726)
+(0.4880)
+q5
+0.0404
+0.0468
+0.7035
+(0.0000)
+(0.0000)
+(0.2415)
+q10
+0.0305
+0.0406
+0.9273
+(0.0000)
+(0.0001)
+(0.1777)
+q20
+0.0253
+0.0342
+1.0156
+(0.0000)
+(0.0000)
+(0.1558)
+q30
+0.0215
+0.0280
+1.2056
+(0.0000)
+(0.0000)
+(0.1150)
+q40
+0.0192
+0.0293
+1.9873
+(0.0000)
+(0.0000)
+(0.0245)
+q50
+0.0179
+0.0268
+1.8614
+(0.0000)
+(0.0000)
+(0.0324)
+q60
+0.0185
+0.0291
+2.1971
+(0.0000)
+(0.0000)
+(0.0149)
+q70
+0.0185
+0.0288
+2.5770
+(0.0000)
+(0.0000)
+(0.0055)
+q80
+0.0160
+0.0257
+2.2460
+(0.0000)
+(0.0000)
+(0.0132)
+q90
+0.0146
+0.0243
+2.0848
+(0.0005)
+(0.0000)
+(0.0195)
+q95
+0.0143
+0.0239
+1.7520
+(0.0001)
+(0.0000)
+(0.0410)
+Note:
+OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:
+Ct = α + βt + ut, for two
+different time periods. For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, ..., s, ..., T, Ct =
+α2 + β2t + ut, t = s + 1, ..., T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1. We show the value of
+the t-statistic and its HAC p-value.
+Table 6
+Co-trending analysis (CRU montly data, 1950-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+25.143
+0.005
+Lower quantiles (q05, q10, q20, q30)
+9.545
+0.023
+Medium quantiles (q40, q50, q60)
+0.078
+0.962
+Upper quantiles (q70, q80, q90, q95)
+1.099
+0.777
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+17.691
+0.007
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+2.041
+0.916
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+24.683
+0.001
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+-0.022
+0.000
+q95-q50
+-0.004
+0.193
+q95-q05
+-0.026
+0.000
+q75-q25 (iqr)
+-0.007
+0.043
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+
+Climate change heterogeneity
+24
+Table 7
+Co-trending analysis (CRU montly data, 1970-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+18.478
+0.047
+Lower quantiles (q05, q10, q20, q30)
+5.523
+0.137
+Medium quantiles (q40, q50, q60)
+0.569
+0.752
+Upper quantiles (q70, q80, q90, q95)
+2.667
+0.446
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+7.606
+0.268
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+6.714
+0.348
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+14.520
+0.043
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+-0.020
+0.047
+q95-q50
+-0.003
+0.462
+q95-q05
+-0.023
+0.048
+q75-q25 (iqr)
+-0.004
+0.418
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+Table 8
+Amplification hypotheses (CRU monthly data across stations, 1950-2019)
+periods/variables
+1950-2019
+1970-2019
+q05
+2.00
+1.83
+(0.000)
+(0.000)
+q10
+1.79
+1.73
+(0.000)
+(0.001)
+q20
+1.41
+1.37
+(0.000)
+(0.000)
+q30
+1.07
+1.00
+(0.089)
+(0.502)
+q40
+0.88
+0.91
+(0.999)
+(0.973)
+q50
+0.74
+0.81
+(1.000)
+(0.997)
+q60
+0.74
+0.85
+(0.999)
+(0.973)
+q70
+0.77
+0.85
+(1.000)
+(0.988)
+q80
+0.72
+0.78
+(1.000)
+(1.000)
+q90
+0.69
+0.70
+(1.000)
+(1.000)
+q95
+0.60
+0.64
+(1.000)
+(1.000)
+Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1
+in the regression: Cit = βi0 +βi1meant +ϵit. mean refers to the average of the Global temperature
+distribution.
+
+Climate change heterogeneity
+25
+4.3
+Micro-local warming: Madrid and Barcelona
+The existence of warming heterogeneity implies that in order to design more ef-
+ficient mitigation policies, they have to be developed at different levels: global,
+country, region etc. How local we need to go will depend on the existing degree of
+micro-warming heterogeneity. In this subsection, we go to the smallest level, cli-
+mate station level . We analyze, within Spain, the warming process in two weather
+stations corresponding to two cities: Madrid (Retiro station) and Barcelona (Fabra
+station).
+11 Obviously, the data provided by these stations is not cross-sectional
+data but directly pure time series data. Our methodology can be easily applied to
+higher frequency time series, in this case daily data, to compute the distributional
+characteristics (see Figures A1 and A2)12.
+The results are shown in the Appendix. These two stations, Madrid-Retiro and
+Barcelona-Fabra clearly experience two different types of warming.
+First, there
+is evidence of micro-local warming, understood as the presence of significant and
+positive trends, in all the important temperature distributional characteristics of
+both stations. The acceleration phenomenon is also clearly detected, in other words,
+the warming increases as time passes (see Tables A1 and A5). Secondly, from the
+co-trending tests (Tables A2-A3 and A6-A7), it can be concluded that the warming
+process of Madrid-Retiro is type W3 while for Barcelona-Fabra it is type W1. In
+both cases the warming typology is stable through both sample periods (1950-2019
+and 1970-2019). Thirdly, as expected, Madrid-Retiro presents “inner” and “outer”
+amplification for the upper quantiles, while Barcelona-Fabra does so only for the
+center part of its temperature distribution (see Tables A4 and A8).
+Summing up, even within Spain we find evidence of warming heterogeneity.
+While Madrid (Continental Mediterranean climate) has a similar pattern as that
+of peninsular Spain (1970-2019) W3, Barcelona (Mediterranean coastline climate)
+maintains a W1 typology. Thus there are two different warming processes which
+require mitigation policies at the country as well as the very local level.
+11From Madrid and Barcelona there is data since 1920’s, nevertheless we began the study in 1950
+for consistency with the previous analysis of Spain and the Globe.
+12See the application to Central England in GG2020 and in Gadea and Gonzalo (2022) to Madrid,
+Zaragoza and Oxford.
+
+Climate change heterogeneity
+26
+5
+Comparing results
+The goal of this section is to show the existence of climate heterogeneity by com-
+paring the results obtained from applying our three-step methodology to different
+regions. These results are summarized in Table 10. It is clear that there is distribu-
+tional warming in all the analyzed areas; but this warming follows different patterns
+and sometimes the warming type is not even stable. In the case of Spain, it depends
+on the period under consideration. Figure 6 captures graphically the different trend
+behavior and intensity of the distributional characteristics by regions (Spain and the
+Globe and Madrid and Barcelona).13 The graphical results in this figure coincide
+with the results of the warming typology tests shown in Table 10.
+The middle of Table 10 shows that warming acceleration is detected in all the
+locations. This acceleration is more general in Spain than in the Globe (see also the
+heatmap in Figure 7) and in Barcelona than in Madrid. Apart from these differences,
+the acceleration shares certain similarities across regions. This is not the case for
+the warming amplification that is clearly asymmetric. Spain suffers an amplification
+in the upper quantiles while the Globe does so in the lower ones. Notice that the
+latter amplification goes beyond the standard results found in the literature for the
+Arctic region (q05). We detect amplification also for the regions corresponding to
+the quantiles q10-q30. In the case of Madrid and Barcelona, Madrid suffers a wider
+warming amplification than Barcelona.
+The results of the first two steps of our methodology are obtained region by region
+(Spain, the Globe, Madrid and Barcelona). It is the last step, via the warming
+dominance test (see the numerical results in Table 9) where we compare directly
+one region with another. Warming in Spain dominates that of the Globe in all the
+quantiles except the lower q05.14 This would support the idea held in European
+institutions and gathered in international reports on the greater intensity of climate
+13The analysis of other characteristics such as the third and fourth order moments can contribute
+to the temperature distributions. In the case of Spain, the kurtosis is always negative with a mean
+value of -0.8 and a significant negative trend, which means that we are dealing with a platykurtic
+distribution with tails less thick than Normal, a shape that is accelerating over time. However, it
+is ot possible to draw conclusions about symmetry given its high variability over time. Conversely,
+the temperature distribution in the Globe is clearly leptokurtic with an average kurtosis of 0.9
+and a negative but not significant trend. The global temperature observations are therefore more
+concentrated around the mean and their tails are thicker than in a Normal distribution.
+The
+skewness is clearly negative although a decreasing and significant trend points to a reduction of the
+negative skewness.
+14A more detailed analysis of the warming process suffered in the Artic region can be found in
+Gadea and Gonzalo (2021).
+
+Climate change heterogeneity
+27
+change in the Iberian Peninsula. Warming in Madrid dominates that of Barcelona
+in the upper quantiles, while the reverse is the case in the lower quantiles. This
+latter result coincides with the idea that regions close to the sea have milder upper
+temperatures.
+Further research (beyond the scope of this paper) will go in the direction of
+finding the possible causes behind the warming types W1, W2, and W3. Following
+the literature, on diurnal temperature asymmetry (Diurnal Temperature Range =
+DTR = Tmax − Tmin) we can suggest as possible causes for W2 the cloud coverage
+(Karl et al. 1993) and the planetary boundary layer (see Davy et al. 2017). For
+W3, the process of desertification (see Karl et al. 1993).
+Summarizing, in this section we describe, measure and test the existence of
+warming heterogeneity in different regions of the planet. It is important to note
+that these extensive results can not be obtained by the standard analysis of the
+average temperature.
+Table 9
+Warming dominance
+Spain-Globe
+Madrid-Barcelona
+Quantile
+β
+t-ratio
+β
+t-ratio
+q05
+-0.018
+(-2.770)
+-0.013
+(-3.730)
+q10
+-0.010
+(-1.504)
+-0.013
+(-4.215)
+q20
+-0.004
+(-0.950)
+-0.012
+(-2.988)
+q30
+0.001
+(0.180)
+-0.013
+(-4.164)
+q40
+0.002
+(0.788)
+-0.009
+(-2.909)
+q50
+0.003
+(1.025)
+-0.003
+(-0.701)
+q60
+0.006
+(1.933)
+-0.001
+(-0.219)
+q70
+0.009
+(3.266)
+0.006
+(1.252)
+q80
+0.012
+(3.203)
+0.016
+(3.331)
+q90
+0.017
+(3.862)
+0.010
+(1.869)
+q95
+0.019
+(4.930)
+0.014
+(1.993)
+Note: The slopes (t-statistic) of the following regression
+qτt(A) − qτt(B) = ατ + βτt + uτt
+In the first column A=Spain, B=Globe and in the second A=Madrid, B=Barcelona.
+
+Climate change heterogeneity
+28
+Table 10
+Summary of results
+Cross analysis
+Sample
+Period
+Type
+Acceleration
+Amplification
+Dominance
+Inner
+Outer
+Spain
+1950-2019
+W1
+[mean, std, iqr, rank,
+[q70, q80, q95]
+[q90, q95]
+[q60,..., q95]
+q20,..., q95]
+1970-2019
+W3
+[q50,..., q80]
+[q60,..., q95]
+The Globe
+1950-2019
+W2
+[mean
+[q05,..., q30]
+[q05]
+q40,..., q95]
+1970-2019
+W2
+[q05,..., q20]
+Time analysis
+Sample
+Period
+Type
+Acceleration
+Amplification
+Dominance
+Madrid, Retiro Station
+1950-2019
+W3
+[mean, std, rank,
+[q50,..., q95]
+[ q40,..., q95]
+[q80,..., q95]
+q40, ..., q95]
+1970-2019
+W3
+[q50,..., q95]
+[q40,..., q95]
+Barcelona, Fabra Station
+1950-2019
+W1
+[mean,
+-
+[q30,..., q90]
+[q05,..., q40]
+q20,..., q95]
+1970-2019
+W1
+[q60, q70]
+[q30,..., q70]
+Note: For Spain and the Globe we build characteristics from station-months units. For Madrid and Barcelona we use daily
+frequency time series. A significance level of 10% is considered for all tests and characteristics.
+
+Climate change heterogeneity
+29
+-0.01
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+mean  
+max   
+min   
+std   
+iqr   
+rank  
+kur   
+skw   
+q5    
+q10   
+q20   
+q30   
+q40   
+q50   
+q60   
+q70   
+q80   
+q90   
+q95   
+Globe-montly-1950
+Spain-montly-1950
+-0.01
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+mean  
+max   
+min   
+std   
+iqr   
+rank  
+kur   
+skw   
+q5    
+q10   
+q20   
+q30   
+q40   
+q50   
+q60   
+q70   
+q80   
+q90   
+q95   
+Spain-montly-1950
+Madrid-daily-1950
+Barcelona-daily-1950
+Note: The bars represent the intensity of the trends found in each characteristic measured through
+the value of the β-coefficient estimated in the regression Ct = α + βt + ut.
+Figure 6
+Trend evolution of different temperature distributional characteristics
+
+Climate change heterogeneity
+30
+1950-2019
+1955-2019
+1960-2019
+1965-2019
+1970-2019
+1975-2019
+1980-2019
+1985-2019
+1990-2019
+1995-2019
+2000-2019
+mean  
+max   
+min   
+std   
+iqr   
+rank  
+kur   
+skw   
+q5    
+q10   
+q20   
+q30   
+q40   
+q50   
+q60   
+q70   
+q80   
+q90   
+q95   
+-0.08
+-0.06
+-0.04
+-0.02
+    0
+ 0.02
+ 0.04
+ 0.06
+ 0.08
+  0.1
+(a) Globe
+Spain
+1950-2019
+1955-2019
+1960-2019
+1965-2019
+1970-2019
+1975-2019
+1980-2019
+1985-2019
+1990-2019
+1995-2019
+2000-2019
+mean  
+max   
+min   
+std   
+iqr   
+rank  
+kur   
+skw   
+q5    
+q10   
+q20   
+q30   
+q40   
+q50   
+q60   
+q70   
+q80   
+q90   
+q95   
+-0.08
+-0.06
+-0.04
+-0.02
+    0
+ 0.02
+ 0.04
+ 0.06
+ 0.08
+  0.1
+(b) Spain
+Note: The color scale on the right side of the figure shows the intensity of the trend, based on the
+value of the β-coefficient estimated in the regression Ct = α + βt + ut.
+Figure 7
+Comparing heatmaps
+
+Climate change heterogeneity
+31
+6
+Conclusions
+The existence of Global Warming is very well documented in all the scientific reports
+published by the IPCC. In the last one, the AR6 report (2022), special attention is
+dedicated to climate change heterogeneity (regional climate). Our paper presents a
+new quantitative methodology, based on the evolution of the trend of the whole tem-
+perature distribution and not only on the average, to characterize, to measure and
+to test the existence of such warming heterogeneity. It is found that the local warm-
+ing experienced by Spain (one of most climatically diverse areas) is very different
+from that of the Globe as a whole. In Spain, the upper-temperature quantiles tend
+to increase more than the lower ones, while in the Globe just the opposite occurs.
+In both cases the warming process is accelerating over time. Both regions suffer an
+amplification effect of an asymmetric nature: there is warming amplification in the
+lower quantiles of the Globe temperature (beyond the standard well-known results
+of the Arctic zone) and in the upper ones of Spain. Overall, warming in Spain domi-
+nates that of the Globe in all the quantiles except the lower q05. This places Spain in
+a very difficult warming situation compared to the Globe. Such a situation requires
+stronger mitigation-adaptation policies. For this reason, future climate agreements
+should take into consideration the whole temperature distribution and not only the
+average.
+Any time a novel methodology is proposed, new research issues emerge for future
+investigation. Among those which have been left out of this paper (some are part
+of our current research agenda), three points stand out as important:
+• There is a clear need for a new non-uniform causal-effect climate change anal-
+ysis beyond the standard causality in mean.
+• In order to improve efficiency, mitigation-adaptation policies should be de-
+signed containing a common global component and an idiosyncratic regional
+element.
+• The relation between warming heterogeneity and public awareness of climate
+change deserves to be analyzed.
+
+Climate change heterogeneity
+32
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+35
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+tion. International Economic Review 21, 149–170.
+
+Climate change heterogeneity
+36
+7
+Appendix: Climate change of Madrid and Barcelona
+7.1
+Madrid-Retiro
+Figure A1
+Characteristics of temperature data in Madrid-Retiro (AEMET daily data, 1950-2019)
+
+16
+32
+30
+mean
+28
+max
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+5
+8
+wmw
+min
+std
+5
+6
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+15
+35
+rank
+w
+30
+10
+igr
+25
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+kur
+0.4
+0.2
+1.5
+0
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+30
+20
+10
+0
+1950
+1970
+1990
+2010 2019
+q5
+q10
+q20
+q30
+q40
+q50
+q60
+q70
+q80
+q90
+q95Climate change heterogeneity
+37
+Table A1
+Trend acceleration hypothesis (Madrid, daily data, AEMET, 1950-2019)
+Trend test by periods
+Acceleration test
+names/periods
+1950-2019
+1970-2019
+1950-2019, 1970-2019
+mean
+0.0326
+0.0447
+2.0972
+(0.0000)
+(0.0000)
+(0.0189)
+max
+0.0477
+0.0636
+1.2043
+(0.0000)
+(0.0000)
+(0.1153)
+min
+0.0362
+0.0087
+-1.5077
+(0.0011)
+(0.5859)
+(0.9330)
+std
+0.0112
+0.0197
+2.1160
+(0.0000)
+(0.0000)
+(0.0181)
+iqr
+0.0270
+0.0399
+1.1110
+(0.0000)
+(0.0004)
+(0.1343)
+rank
+0.0115
+0.0549
+2.0160
+(0.3666)
+(0.0045)
+(0.0229)
+kur
+-0.0016
+-0.0022
+-0.4449
+(0.0278)
+(0.0660)
+(0.6714)
+skw
+0.0012
+-0.0013
+-1.7769
+(0.1538)
+(0.2695)
+(0.9611)
+q5
+0.0248
+0.0183
+-0.5712
+(0.0000)
+(0.0774)
+(0.7156)
+q10
+0.0220
+0.0174
+-0.5815
+(0.0000)
+(0.0162)
+(0.7191)
+q20
+0.0200
+0.0187
+-0.1777
+(0.0000)
+(0.0099)
+(0.5704)
+q30
+0.0181
+0.0235
+0.6959
+(0.0000)
+(0.0019)
+(0.2438)
+q40
+0.0236
+0.0362
+1.6625
+(0.0000)
+(0.0000)
+(0.0494)
+q50
+0.0299
+0.0545
+2.8801
+(0.0000)
+(0.0000)
+(0.0023)
+q60
+0.0334
+0.0604
+3.1655
+(0.0000)
+(0.0000)
+(0.0010)
+q70
+0.0388
+0.0550
+1.7385
+(0.0000)
+(0.0000)
+(0.0422)
+q80
+0.0519
+0.0712
+1.9750
+(0.0000)
+(0.0000)
+(0.0251)
+q90
+0.0494
+0.0687
+1.7956
+(0.0000)
+(0.0000)
+(0.0374)
+q95
+0.0527
+0.0710
+1.7839
+(0.0000)
+(0.0000)
+(0.0383)
+Note:
+OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:
+Ct = α + βt + ut, for two
+different time periods. For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, ..., s, ..., T, Ct =
+α2 + β2t + ut, t = s + 1, ..., T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1. We show the value of
+the t-statistic and its HAC p-value.
+Table A2
+Co-trending analysis (Madrid-Retiro daily data, AEMET 1950-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+77.046
+0.000
+Lower quantiles (q05, q10, q20, q30)
+1.360
+0.715
+Medium quantiles (q40, q50, q60)
+2.036
+0.361
+Upper quantiles (q70, q80, q90, q95)
+3.944
+0.268
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+6.707
+0.349
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+31.822
+0.000
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+74.967
+0.000
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+0.005
+0.505
+q95-q50
+0.023
+0.000
+q05-q95
+-0.028
+0.000
+q75-q25 (iqr)
+0.027
+0.000
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+
+Climate change heterogeneity
+38
+Table A3
+Co-trending analysis (Madrid-Retiro daily data, AEMET, 1970-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+81.371
+0.000
+Lower quantiles (q05, q10, q20, q30)
+0.424
+0.935
+Medium quantiles (q40, q50, q60)
+8.111
+0.017
+Upper quantiles (q70, q80, q90, q95)
+3.214
+0.360
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+45.687
+0.000
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+18.851
+0.004
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+71.094
+0.000
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+0.036
+0.004
+q95-q50
+0.017
+0.051
+q05-q95
+-0.053
+0.000
+q75-q25 (iqr)
+0.040
+0.000
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+Table A4
+Amplification hypothesis (Madrid daily data, AEMET 1950-2019)
+periods/variables
+1950-2019
+1970-2019
+1950-2019
+1970-2019
+Inner
+Outer
+q05
+0.66
+0.43
+0.83
+0.56
+(0.993)
+(1.000)
+(0.802)
+(0.990)
+q10
+0.58
+0.42
+0.73
+0.54
+(1.000)
+(1.000)
+(0.974)
+(1.000)
+q20
+0.66
+0.53
+0.81
+0.65
+(1.000)
+(1.000)
+(0.961)
+(0.999)
+q30
+0.72
+0.74
+0.94
+0.90
+(1.000)
+(0.996)
+(0.758)
+(0.836)
+q40
+0.90
+1.02
+1.15
+1.21
+(0.887)
+(0.436)
+(0.072)
+(0.041)
+q50
+1.08
+1.29
+1.38
+1.53
+(0.188)
+(0.001)
+(0.001)
+(0.000)
+q60
+1.14
+1.31
+1.44
+1.54
+(0.040)
+(0.000)
+(0.000)
+(0.000)
+q70
+1.22
+1.23
+1.46
+1.38
+(0.012)
+(0.019)
+(0.000)
+(0.002)
+q80
+1.45
+1.36
+1.70
+1.52
+(0.000)
+(0.003)
+(0.000)
+(0.002)
+q90
+1.31
+1.29
+1.48
+1.38
+(0.004)
+(0.041)
+(0.005)
+(0.064)
+q95
+1.31
+1.33
+1.46
+1.39
+(0.001)
+(0.021)
+(0.007)
+(0.073)
+Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1
+in the regression: Cit = βi0 + βi1meant + ϵit. mean refers to the average of the Madrid or Spanish
+temperature distribution for the “inner” and “outer”cases, respectively.
+
+Climate change heterogeneity
+39
+7.2
+Barcelona-Fabra
+Figure A2
+Characteristics of temperature data in Barcelona-Fabra (AEMET daily data,
+1950-2019)
+
+16
+30
+15
+14
+mean
+25
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+8
+5
+std
+www
+min
+-5
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+30
+25
+igr
+20
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+2.5
+kur
+0.4
+skw
+0.2
+M
+0
+-0.2
+1950
+1970
+1990
+2010 2019
+1950
+1970
+1990
+2010 2019
+20
+10
+0
+1950
+1970
+1990
+2010 2019
+q5
+q10
+q20
+q30
+q40
+q50
+q60
+q70
+q80
+q90
+q95Climate change heterogeneity
+40
+Table A5
+Trend acceleration hypothesis (Barcelona, daily data, AEMET, 1950-2019)
+Trend test by periods
+Acceleration test
+names/periods
+1950-2019
+1970-2019
+1950-2019, 1970-2019
+mean
+0.0340
+0.0512
+3.2979
+(0.0000)
+(0.0000)
+(0.0006)
+max
+0.0394
+0.0531
+0.7280
+(0.0000)
+(0.0038)
+(0.2339)
+min
+0.0397
+0.0231
+-0.7411
+(0.0011)
+(0.2654)
+(0.7700)
+std
+0.0013
+0.0057
+0.9146
+(0.6185)
+(0.1787)
+(0.1810)
+iqr
+0.0042
+0.0113
+0.7351
+(0.4418)
+(0.1892)
+(0.2318)
+rank
+-0.0004
+0.0300
+0.9299
+(0.9806)
+(0.3322)
+(0.1770)
+kur
+-0.0013
+-0.0018
+-0.2693
+(0.1555)
+(0.2075)
+(0.6060)
+skw
+0.0011
+-0.0022
+-1.7869
+(0.2678)
+(0.1942)
+(0.9619)
+q5
+0.0374
+0.0358
+-0.1381
+(0.0000)
+(0.0015)
+(0.5548)
+q10
+0.0350
+0.0385
+0.4361
+(0.0000)
+(0.0000)
+(0.3317)
+q20
+0.0317
+0.0439
+1.7009
+(0.0000)
+(0.0000)
+(0.0456)
+q30
+0.0308
+0.0488
+2.4813
+(0.0000)
+(0.0000)
+(0.0072)
+q40
+0.0324
+0.0537
+2.9244
+(0.0000)
+(0.0000)
+(0.0020)
+q50
+0.0325
+0.0548
+2.7535
+(0.0000)
+(0.0000)
+(0.0034)
+q60
+0.0344
+0.0636
+3.0915
+(0.0000)
+(0.0000)
+(0.0012)
+q70
+0.0330
+0.0583
+2.9241
+(0.0000)
+(0.0000)
+(0.0020)
+q80
+0.0357
+0.0551
+2.4081
+(0.0000)
+(0.0000)
+(0.0087)
+q90
+0.0394
+0.0567
+2.0957
+(0.0000)
+(0.0000)
+(0.0190)
+q95
+0.0390
+0.0525
+1.3435
+(0.0000)
+(0.0000)
+(0.0907)
+Note:
+OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:
+Ct = α + βt + ut, for two
+different time periods. For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, ..., s, ..., T, Ct =
+α2 + β2t + ut, t = s + 1, ..., T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1. We show the value of
+the t-statistic and its HAC p-value.
+Table A6
+Co-trending analysis (Barcelona-Fabra daily data, AEMET, 1950-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+3.368
+0.971
+Lower quantiles (q05, q10, q20, q30)
+1.036
+0.792
+Medium quantiles (q40, q50, q60)
+0.073
+0.964
+Upper quantiles (q70, q80, q90, q95)
+0.784
+0.853
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+1.171
+0.978
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+1.901
+0.929
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+2.969
+0.888
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+-0.005
+0.528
+q95-q50
+0.006
+0.233
+q05-q95
+-0.002
+0.856
+q75-q25 (iqr)
+0.004
+0.442
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+
+Climate change heterogeneity
+41
+Table A7
+Co-trending analysis (Barcelona-Fabra daily data, AEMET, 1970-2019)
+Joint hypothesis tests
+Wald test
+p-value
+All quantiles (q05, q10,...,q90, q95)
+13.165
+0.215
+Lower quantiles (q05, q10, q20, q30)
+1.904
+0.593
+Medium quantiles (q40, q50, q60)
+1.267
+0.531
+Upper quantiles (q70, q80, q90, q95)
+0.384
+0.943
+Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)
+10.103
+0.120
+Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)
+1.642
+0.949
+Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )
+9.693
+0.207
+Spacing hypothesis
+Trend-coeff.
+p-value
+q50-q05
+0.019
+0.192
+q95-q50
+-0.002
+0.821
+q05-q95
+-0.017
+0.241
+q75-q25 (iqr)
+0.011
+0.189
+Note: Annual distributional characteristics (quantiles) of temperature. The top panel shows the
+Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.
+In the bottom panel, the TT is applied to the difference between two representative quantiles.
+Table A8
+Amplification hypothesis (Barcelona daily data, AEMET 1950-2019)
+periods/variables
+1950-2019
+1970-2019
+1950-2019
+1970-2019
+Inner
+Outer
+q05
+0.99
+0.76
+1.19
+0.87
+(0.523)
+(0.918)
+(0.225)
+(0.720)
+q10
+0.90
+0.79
+1.10
+0.94
+(0.824)
+(0.980)
+(0.263)
+(0.668)
+q20
+0.89
+0.85
+1.09
+1.04
+(0.931)
+(0.964)
+(0.192)
+(0.318)
+q30
+0.96
+0.98
+1.22
+1.25
+(0.813)
+(0.585)
+(0.000)
+(0.000)
+q40
+0.99
+1.04
+1.27
+1.33
+(0.570)
+(0.300)
+(0.000)
+(0.000)
+q50
+1.01
+1.07
+1.27
+1.32
+(0.466)
+(0.224)
+(0.002)
+(0.003)
+q60
+1.09
+1.23
+1.29
+1.42
+(0.175)
+(0.005)
+(0.014)
+(0.001)
+q70
+1.09
+1.17
+1.26
+1.31
+(0.128)
+(0.012)
+(0.022)
+(0.008)
+q80
+1.06
+1.04
+1.22
+1.17
+(0.191)
+(0.338)
+(0.052)
+(0.117)
+q90
+1.09
+1.08
+1.22
+1.20
+(0.125)
+(0.241)
+(0.047)
+(0.121)
+q95
+1.06
+1.03
+1.16
+1.12
+(0.304)
+(0.432)
+(0.192)
+(0.298)
+Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1 in
+the regression: Cit = βi0 + βi1meant + ϵit. mean refers to the average of the Barcelona or Spanish
+temperature distribution for the “inner” and “outer”cases, respectively.
+
diff --git a/AdE0T4oBgHgl3EQfxgLI/content/tmp_files/load_file.txt b/AdE0T4oBgHgl3EQfxgLI/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1944 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf,len=1943
+page_content='Climate change heterogeneity:  A new quantitative approach  ∗  Mar´ıa Dolores Gadea Rivas †  University of Zaragoza  Jes´us Gonzalo ‡  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Carlos III de Madrid  July 10, 2022  Abstract  Climate change is a non-uniform phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  This paper proposes a new  quantitative methodology to characterize, measure and test the existence of  climate change heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It consists of three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' First, we introduce a  new testable warming typology based on the evolution of the trend of the whole  temperature distribution and not only on the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Second, we define the  concepts of warming acceleration and warming amplification in a testable for-  mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' And third, we introduce the new testable concept of warming dominance  to determine whether region A is suffering a worse warming process than region  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Applying this three-step methodology, we find that Spain and the Globe ex-  perience a clear distributional warming process (beyond the standard average)  but of different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In both cases, this process is accelerating over time and  asymmetrically amplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Overall, warming in Spain dominates the Globe in  all the quantiles except the lower tail of the global temperature distribution  that corresponds to the Artic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Our climate change heterogeneity results  open the door to the need for a non-uniform causal-effect climate analysis that  goes beyond the standard causality in mean as well as for a more efficient design  of the mitigation-adaptation policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In particular, the heterogeneity we find  suggests that these policies should contain a common global component and a  clear local-regional element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Future climate agreements should take the whole  temperature distribution into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  JEL classification: C31, C32, Q54  Keywords:  Climate change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Climate heterogeneity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Global-Local Warming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Functional stochastic processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Distributional characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Trends;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Quan-  tiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Temperature distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  ∗The authors gratefully acknowledge the financial support from the Gobierno de Aragon and FEDER  funds (grant, LMP71-18), the Spanish Ministerio de Ciencia y Tecnolog´ıa, Agencia Espa˜nola de Investi-  gaci´on (AEI) and European Regional Development Fund (ERDF, EU) under grants PID2019-104960GB-  IOO, ECO2017-83255-C3-1-P (AEI/ERDF, EU) and ECO2016-81901-REDT, and Bank of Spain (ER grant  program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' We thank Rodrigo Gonzalez Laiz for excellent research assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  † Department of Applied Economics, University of Zaragoza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Gran V´ıa, 4, 50005 Zaragoza (Spain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Tel:  +34 9767 61842, fax: +34 976 761840 and e-mail: lgadea@unizar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='es  ‡ Department of Economics, University Carlos III, Madrid 126 28903 Getafe (Spain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Tel: +34 91  6249853, fax: +34 91 6249329 and e-mail: jesus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='gonzalo@uc3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='es (corresponding author)  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='02648v1  [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='EM]  6 Jan 2023  Climate change heterogeneity  2  1  Introduction  All the assessment reports (AR) published by the Intergovernmental Panel of Cli-  mate Change (IPCC) show that there is overwhelming scientific evidence of the  existence of global warming (GW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It is also well known that climate change (CC)  is a non-uniform phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' What is not so clear is the degree of heterogeneity  across all the regions in our planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In fact, an important part of the Sixth Assess-  ment Report (AR6) published by the IPCC in 2021-2022 is dedicated to this issue:  climate (warming) heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This is reflected in the chapters studying regional  climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Our paper introduces a new quantitative methodology that builds  on that described in Gadea and Gonzalo 2020 (GG2020) to characterize, measure  and test the existence of such climate change heterogeneity (CCH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This is done in  three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' First, we introduce a warming typology (W1, W2 and W3) based on  the trending behavior of the quantiles of the temperature distribution of a given ge-  ographical location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Second, we define in a testable format the concepts of warming  acceleration and warming amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' These concepts help to characterize (more  ordinally than cardinally) the warming process of different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' And third, we  propose the new concept of warming dominance (WD) to establish when region A  suffers a worse warming process than region B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  We have chosen Spain as a benchmark geographical location because, as the AR6  report states “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Spain is fully included in the Mediterranean (MED) Reference  Region, but is one of the most climatically diverse countries in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  This fact opens up the possibility of studying warming heterogeneity (WH) from  Spain to the Globe (outer heterogeneity, OWH) and also from Spain to some of its  regions represented by Madrid and Barcelona (inner heterogeneity, IWH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The three steps rely on the results reported in GG2020, where the different  distributional characteristics (moments, quantiles, inter quantile range, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=') of the  temperature distribution of a given geographical location are converted into time  series objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' By doing this, we can easily implement and test all the concepts  involved in the three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  A summary of the results is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Spain and the Globe present a clear  warming process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' but it evolves differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Spain goes from a warming process where  lower and upper temperatures share the same trend behavior (IQR is maintained  constant over time, warming type W1) to one characterized by a larger increase in  the upper temperatures (IQR increases over time, warming type W3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In contrast,  Climate change heterogeneity  3  the Globe as a whole maintains a stable warming type process characterized by lower  temperatures that increase more than the upper ones (IQR decreases in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='1 In  our typology, this constitutes a case of warming type W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Climate heterogeneity  can go further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  For instance, within Spain we find that Madrid is of type W3  while the warming process of Barcelona is of type W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This is in concordance with  the Madrid climate being considered a Continental Mediterranean climate while  Barcelona is more a pure Mediterranean one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The proposed warming typology (W1, W2 and W3), although dynamic, is more  ordinal than cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In this paper, the strength of a warming process is captured  in the second step by analyzing its acceleration and its amplification with respect  to a central tendency measure of the temperature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Acceleration and  amplification contribute to the analysis of warming heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The acceleration  in the Globe is present in all the quantiles above q30 while in Spain it already  becomes significant above the 10th quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' We find an asymmetric behavior of  warming amplification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' in Spain (in comparison with the Globe mean temperature)  this is present in the upper temperatures (above the 80th and 90th quantiles) while  in the Globe the opposite occurs (below the 20th and 30th quantiles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Within Spain,  Madrid and Barcelona also behave differently in terms of acceleration and amplifi-  cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Overall, warming in Spain dominates that of the Globe in all the quantiles  except for the lower quantile q05, and between Madrid and Barcelona there is a par-  tial WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Madrid WD Barcelona in the upper part of the distribution and Barcelona  WD Madrid in the lower one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The existence of a clear heterogeneous warming process opens the door to the  need of a new non-uniform causal (effect) research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' One that goes beyond the stan-  dard causality in mean analysis (see Tol, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' CCH also suggests that in order  for the mitigation-adaptation policies to be as efficient as possible they should be  designed following a type of common factor structure: a common global compo-  nent plus an idiosyncratic local element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  This goes in the line with the results  found in Brock and Xepapadeas (2017), D’Autume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (2016) and Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Future climate agreements should clearly have this CCH into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' An  important by-product of our warming heterogeneity results is the increase that this  heterogeneity can generate in the public awareness of the GW process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' A possible  explanation for that can be found in the behavioral economics work by Malmendier  1Similar results for Central England are found in GG2020 and for the US in Diebold and Rude-  bush, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  4  (2021), in the results of the European Social Survey analyzed in Nowakowski and  Oswald (2020) or in the psychology survey by Maiella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Section 2 describes our basic climate  econometrics methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Section 3 presents a brief description of the temperature  data from Spain and the Globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Section 4 addresses the application of our quantita-  tive methodology in the cross-sectional version (temperatures measured monthly by  stations in an annual interval) to Spain and (versus) the Globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It also reports the  results of applying the methodology using a purely temporal dimension (local daily  temperature on an annual basis) for two representative stations in Spain (Madrid  and Barcelona, empirical details in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Section 5 offers a comparison  and interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Finally, Section 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  2  Climate Econometrics Methodology  In this section, we briefly summarize the novel econometric methodology introduced  in GG2020 to analyze Global and Local Warming processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Following GG2020,  Warming is defined as an increasing trend in certain characteristics of the temper-  ature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' More precisely:  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (Warming):  Warming is defined as the existence of an increas-  ing trend in some of the characteristics measuring the central tendency or position  (quantiles) of the temperature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  An example is a deterministic trend with a polynomial function for certain val-  ues of the β parameters Ct = β0 + β1t + β2t2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' + βktk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In GG2020 temperature is viewed as a functional stochastic process X = (Xt(ω), t ∈  T), where T is an interval in R, defined in a probability space (Ω, ℑ, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' A conve-  nient example of an infinite-dimensional discrete-time process consists of associating  ξ = (ξn, n ∈ R+) with a sequence of random variables whose values are in an appro-  priate function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This may be obtained by setting  Xt(n) = ξtN+n, 0 ≤ n ≤ N, t = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T  (1)  so X = (Xt, t = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' If the sample paths of ξ are continuous, then we have  a sequence X0, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='. of random variables in the space C[0, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The choice of the  period or segment t will depend on the situation in hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In our case, t will be the  Climate change heterogeneity  5  period of a year, and N represents cross-sectional units or higher-frequency time  series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  We may be interested in modeling the whole sequence of G functions, for instance  the sequence of state densities (f1(ω), f2(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', fT (ω) ) as in Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (2015,  2016) or only certain characteristics (Ct(w)) of these G functions, for instance, the  state mean, the state variance, the state quantile, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' These characteristics can  be considered time series objects and, therefore, all the econometric tools already  developed in the time series literature can be applied to Ct(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' With this charac-  teristic approach we go from Ω to RT , as in a standard stochastic process, passing  through a G functional space:  Ω  (w)  X  −→  G  Xt(w)  C−→  R  Ct(w)  Going back to the convenient example and abusing notation, the stochastic struc-  ture can be summarized in the following array:  X10(w) = ξ0(w)  X11(w) = ξ1(w)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  X1N(w) = ξN(w)  C1(w)  X20(w) = ξN+1(w)  X21(w) = ξN+2(w)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  X2N(w) = ξ2N(w)  C2(w)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  XT0(w) = ξ(T−1)N+1(w)  XT1(w) = ξ(T−1)N+2(w)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  XTN(w) = ξTN(w)  CT (w)  (2)  The objective of this section is to provide a simple test to detect the existence of  a general unknown trend component in a given characteristic Ct of the temperature  process Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  To do this, we need to convert Definition 1 into a more practical  definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (Trend test): Let h(t) be an increasing function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' A characteristic  Ct of a functional stochastic process Xt contains a trend if β ̸= 0 in the regression  Ct = α + βh(t) + ut, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  (3)  The main problem of this definition is that the trend component in Ct as well  as the function h(t) are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Therefore this definition can not be easily imple-  mented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' If we assume that Ct does not have a trend component (it is I(0))2 and  2Our definition of an I(0) process follows Johansen (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' A stochastic process Yt that satisfies  Yt − E(Yt) =  ∞  �  i=1  Ψiεt−i is called I(0) if  ∞  �  i=1  Ψ izi converges for |z| < 1 + δ, for some δ > 0 and  ∞  �  i=1  Ψ  i ̸= 0, where the condition εt ∼ iid(0,σ2) with σ2 > 0 is understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  6  h(t) is linear, then we have the following well known result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Let Ct = I(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In the regression  Ct = α + βt + ut  (4)  the OLS estimator  �β =  T�  t=1  (Ct − C)(t − t)  T�  t=1  (t − t)2  (5)  satisfies  T 3/2 �β = Op(1)  (6)  and asymptotically (T → ∞)  tβ=0 is N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In order to analyze the behavior of the t-statistic tβ = 0, for a general trend  component in Ct, it is very convenient to use the concept of Summability (Berenguer-  Rico and Gonzalo, 2014)  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (Order of Summability):  A trend h(t) is said to be summable of  order “δ” (S(δ)) if there exists a slowly varying function L(T),3 such that  ST =  1  T 1+δ L(T)  T  �  t=1  h(t)  (8)  is O(1), but not o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Let Ct = h(t) + I(0) such that h(t) is S(δ) with δ ≥ 0, and such  that the function g(t) = h(t)t is S(δ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In the regression  Ct = α + βt + ut  (9)  the OLS �β estimator satisfies  T (1−δ) �β = Op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  (10)  Assuming that the function h(t)2 is S(1 + 2δ − γ) with 0 ≤ γ ≤ 1 + δ, then  3A positive Lebesgue measurable function, L, on (0, ∞) is slowly varying (in Karamata’s sense)  at ∞ if  L(λn)  L(n) → 1 (n → ∞) ∀λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  (7)  (See Embrechts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', 1999, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' 564).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  7  tβ=0 =  � Op(T γ/2) for 0 ≤ γ ≤ 1  Op(T 1/2) for 1 ≤ γ ≤ 1 + δ  (11)  Examples of how this proposition applies for different particular Data Generat-  ing Processes (DGP) can be found in GG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  A question of great empirical importance is how our trend test (TT) of Proposi-  tion 2 behaves when Ct = I(1) (accumulation of an I(0) process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Following Durlauf  and Phillips (1988), T 1/2 �β = Op(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' however, tβ=0 diverges as T→∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Therefore,  our TT can detect the stochastic trend generated by an I(1) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In fact, our  test will detect trends generated by any of the three standard persistent processes  considered in the literature (see Muller and Watson, 2008): (i) fractional or long-  memory models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (ii) near-unit-root AR models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' and (iii) local-level models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Let  Ct = µ + zt, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  (12)  In the first model, zt is a fractional process with 1/2 < d < 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In the second  model, zt follows an AR, with its largest root close to unity, ρT = 1 − c/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In the  third model, zt is decomposed into an I(1) and an I(0) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Its simplest  format is zt = υt + ϵt with υt = υt−1 +ηt, where ϵt is ID(0, q ∗ σ2), ηt is ID(0, σ2),  σ2 > 0 and both disturbances are serially and mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Note that the  pure unit-root process is nested in all three models: d = 1, c = 0, and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The long-run properties implied by each of these models can be characterized  using the stochastic properties of the partial sum process for zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The standard  assumptions considered in the macroeconomics or finance literature assume the ex-  istence of a “δ,” such that T −1/2+δ �T  t=1 zt −→ σ H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' ), where “δ” is a model-specific  constant and H is a model-specific zero-mean Gaussian process with a given covari-  ance kernel k(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Then, it is clear that the process Ct = µ + zt is summable (see  Berenguer-Rico and Gonzalo, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This is the main reason why Proposition 3  holds for these three persistent processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Let Ct = µ + zt, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T, with zt any of the following three  processes: (i) a fractional or long-memory model, with 1/2 < d < 3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (ii) a near-  unit-root AR model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' or (iii) a local-level model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Furthermore, T −1/2+δ �T  t=1 zt −→ σ  H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' ), where “δ” is a model-specific constant and H is a model-specific zero-mean  Gaussian process with a given covariance kernel k(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Then, in the LS regression  Ct = α + βt + ut,  Climate change heterogeneity  8  the t-statistic diverges,  tβ=0 = Op(T 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  After the development of the theoretical core, we are in a position to design  tools to approach the empirical strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The following subsection describes each of  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='1  Empirical tools: definitions and tests  From Propositions 2 and 3, Definition 2 can be simplified into the following testable  and practical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (Practical definition 2):  A characteristic Ct of a functional stochas-  tic process Xt contains a trend if in the LS regression,  Ct = α + βt + ut, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T,  (13)  β = 0 is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Several remarks are relevant with respect to this definition: (i) regression (13)  has to be understood as the linear LS approximation of an unknown trend function  h(t) (see White, 1980);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (ii) the parameter β is the plim of �βols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (iii) if the regression  (13) is the true data-generating process, with ut ∼ I(0), then the OLS �β estimator  is asymptotically equivalent to the GLS estimator (see Grenander and Rosenblatt,  1957);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (iv) in practice, in order to test β = 0, it is recommended to use a robust  HAC version of tβ=0 (see Busetti and Harvey, 2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' and (v) this test only detects  the existence of a trend but not the type of trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  For all these reasons, in the empirical applications we implement Definition 4  by estimating regression (13) using OLS and constructing a HAC version of tβ=0  (Newey and West, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  These linear trends can be common across characteristics indicating similar pat-  ters in the time evolution of these characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (Co-trending): A set of m distributional characteristics (C1t,C2t,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=',Cmt)  do linearly co-trend if in the multivariate regression  �  �  C1t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Cmt  �  � =  �  �  α1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  αm  �  � +  �  �  β1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  βm  �  � t +  �  �  u1t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  umt  �  �  (14)  Climate change heterogeneity  9  all the slopes are equal, β1 = β2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' = βm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' 4  This co-trending hypothesis can be tested by a standard Wald test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  When m = 2 an alternative linear co-trending test can be obtained from the  regression  Cit − Cjt = α + βt + ut  i ̸= j i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', m by testing the null hypothesis of β = 0 vs β ̸= 0 using a simple  tβ=0 test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate classification is a tool used to recognize, clarify and simplify the existent  climate heterogeneity in the Globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It also helps us to better understand the Globe’s  climate and therefore to design more efficient global warming mitigation policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The prevalent climate typology is that proposed by K¨oppen (1900) and later on  modified in K¨oppen and Geiger (1930).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It is an empirical classification that divides  the climate into five major types, which are represented by the capital letters A  (tropical zone), B (dry zone), C (temperate zone), D (continental zone), and E  (polar zone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Each of these climate types except for B is defined by temperature  criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' More recent classifications can been found in the AR6 of the IPCC (2021,  2022) but all of them share the spirit of the original one of K¨oppen (1900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The climate classification we propose in this section is also based on temperature  data and it has three simple distinctive characteristics:  It considers the whole temperature distribution and not only the average  It has a dynamic nature: it is based on the evolution of the trend of the  temperature quantiles (lower and upper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  It can be easily tested  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (Warming Typology): We define four types of warming processes:  W0: There is no trend in any of the quantiles (No warming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  W1: All the location distributional characteristics have the same positive trend  (dispersion does not contain a trend)  W2: The Lower quantiles have a larger positive trend than the Upper quantiles  (dispersion has a negative trend)  W3: The Upper quantiles have a larger positive trend than the Lower quantiles  (dispersion has a positive trend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  4This definition is slightly different from the one in Carrion-i-Silvestre and Kim (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  10  Climate is understood, unlike weather, as a medium and long-term phenomenon  and, therefore, it is crucial to take trends into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Notice that this typology  can be used to describe macroclimate as well as microclimate locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Most of the literature on Global or Local warming only considers the trend  behavior of the central part of the distribution (mean or median).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' By doing this, we  are losing very useful information that can be used to describe the whole warming  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This information is considered in the other elements of the typology W1,  W2 and W3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This typology does not say anything about the intensity of the warming  process and its dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Part of this intensity is captured in the following definitions  of warming acceleration and warming amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (Warming Acceleration):  We say that there is warming acceler-  ation in a distributional temperature characteristic Ct between the time periods  t1 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', s) and t2 = (s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T) if in the following two regressions:  Ct = α1 + β1t + ut, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T,  (15)  Ct = α2 + β2t + ut, t = s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T,  (16)  the second trend slope is larger than the first one: β2 > β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In practice, we implement this definition by testing in the previous system the  null hypothesis β2 = β1 against the alternative β2 > β1 An alternative warming  acceleration test can be formed by testing for a structural break at t = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Neverthe-  less, we prefer the approach of Definition 7 because it matches closely the existent  narrative on warming acceleration in the climate literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (Warming Amplification with respect to the mean):  We say that  there is a warming amplification in distributional characteristic Ct with respect the  mean if in the following regression:  Ct = β0 + β1meant + ϵt  (17)  the mean slope is greater than one: β1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  When the mean, meant, and Ct come from the same distribution, we name this  “inner” warming amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Otherwise, the mean may come from an external  environment and, in that case, we call it “outer” warming amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Both concepts, acceleration and amplification, introduce a quantitative dimen-  sion to the ordinarily defined classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For example, the acceleration, which  Climate change heterogeneity  11  has a dynamic character, allows us to observe the transition from one type of cli-  mate to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Amplification, on the other hand, makes it possible to compare  the magnitude of the trends that define each type of climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It should be noted  that, although static in nature, it can be computed recursively at different points in  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In the previous definitions, we classify the warming process of different regions  which is crucial in the design of local mitigation and adaptation policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' But we,  also, need to compare the different climate change processes of two regions in order  to characterize climate heterogeneity independently of the type of warming they are  experimenting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For this purpose, we propose the following definition that shares the  spirit of the stochastic dominance concept used in the economic-finance literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (Warming Dominance (WD): We say that the temperature distri-  butions of Region A warming dominates (WD) the temperature distributions of  Region B if in the following regression  qτt(A) − qτt(B) = ατ + βτt + uτt,  (18)  βτ ≥ 0 for all 0 < τ < 1 and there is at least one value τ ∗ for which a strict  inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  It is also possible to have only partial (WD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For instance, in the lower or upper  quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  3  The data  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='1  Spain  The measurement of meteorological information in Spain started in the eighteenth  century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' However, it was not until the mid-nineteenth century that reliable and reg-  ular data became available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In Spain, there are four main sources of meteorological  information: the Resumen Anual, Bolet´ın Diario, Bolet´ın Mensual de Climatolog´ıa  and Calendario Meteorol´ogico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' These were first published in 1866, 1893, 1940 and  1943, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' A detailed explanation of the different sources can be found in  Carreras and Tafunell (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Currently, AEMET (Agencia Estatal de Meterolog´ıa) is the agency responsible  for storing, managing and providing meteorological data to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Some of the  historical publications, such as the Bolet´ın Diario and Calendario Meteorol´ogico can  Climate change heterogeneity  12  be found in digital format in their respective archives for whose use it is necessary  to use some kind of Optical Character Recognition (OCR) software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='5  In 2015, AEMET developed AEMET OpenData, an Application Programming  Interface (API REST) that allows the dissemination and reuse of Spanish meteoro-  logical and climatological information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' To use it, the user needs to obtain an API  key to allow access to the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Then, either through the GUI or through  a programming language such as Java or Python, the user can request data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' More  information about the use of the API can be found on their webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='6  In this paper, we are concerned with Spanish daily station data, specifically  temperature data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Each station records the minimum, maximum and average tem-  perature as well as the amount of precipitation, measured as liters per square meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The data period ranges from 1920 to 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' However, in 1920 there were only 13  provinces (out of 52) who had stations available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It was not until 1965 that all the  52 provinces had at least one working station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Moreover, it is important to keep in  mind that the number of stations has increased substantially from only 14 stations in  1920 to more than 250 in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' With this information in mind, we select the longest  span of time that guarantees a wide sample of stations so that all the geographical  areas of peninsular Spain are represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For this reason, we decided to work with  station data from 1950 to 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' There are 30 stations whose geographical distri-  bution is displayed in the map in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The original daily data are converted  into monthly data, so that we finally work with a total of 30x12 station-month units  corresponding to peninsular Spain and, consequently, we have 360 observations each  year with which to construct the annual distributional characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='2  The Globe  In the case of the Globe, we use the database of the Climate Research Unit (CRU)  that offers monthly and yearly data of land and sea temperatures in both hemi-  spheres from 1850 to the present, collected from different stations around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='7  Each station temperature is converted to an anomaly, taking 1961-1990 as the base  5http : //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='aemet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='es/es/conocermas/recursosenlinea/calendarios?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='n = todos and https :  //repositorio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='aemet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='es/handle/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='11765/6290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  6https : //opendata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='aemet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='es/centrodedescargas/inicio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The use of AEMET data is regulated  in the following resolution https : //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='boe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='es/boe/dias/2016/01/05/pdfs/BOE − A − 2016 −  111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  7We  use  CRUTEM  version  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0,  which  can  be  downloaded  from  (https://crudata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='uea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='uk/cru/data/temperature/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  A recent revision of the methodology  can be found in Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  13  period, and each grid-box value, on a five-degree grid, is the mean of all the station  anomalies within that grid box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This database (in particular, the annual temper-  ature of the Northern Hemisphere) has become one of the most widely used to  illustrate GW from records of thermometer readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' These records form the blade  of the well-known “hockey stick” graph, frequently used by academics and other  institutions, such as, the IPCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In this paper, we prefer to base our analysis on raw  station data, as in GG2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The database provides data from 1850 to nowadays, although due to the high  variability at the beginning of the period it is customary in the literature to begin  in 1880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In this work, we have selected the stations that are permanently present  in the period 1950-2019 according to the concept of the station-month unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In this  way, the results are comparable with those obtained for Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Although there are  10,633 stations on record, the effective number fluctuates each year and there are  only 2,192 stations with data for all the years in the sample period, which yields  19,284 station-month units each year (see this geographical distribution in the map  in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='8 In summary, we analyze raw global data (stations instead of grids)  for the period 1950 to 2019, compute station-month units that remain all the time  and with these build the annual distributional characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  4  Empirical strategy  In this section we apply our three-step quantitative methodology to show the ex-  istent climate heterogeneity between Spain and the Globe as well as within Spain,  between Madrid and Barcelona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Because all our definitions are written in a test-  ing format, it is straightforward to empirically apply them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' First, we test for the  existence of warming by testing the existence of a trend in a given distributional  characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' How common are the trends of the different characteristics (revealed  by a co-trending test) determine the warming typology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Second, the strength of  the warming process is tested by testing the hypothesis of warming acceleration  and warming amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' And third, independently of the warming typology, we  determine how the warming process of Spain compares with that of the Globe as  a whole (we do the same for Madrid and Barcelona).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This is done by testing for  warming dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  8In the CRU data there are 115 Spanish stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' However, after removing stations not present  for the whole 1880 to 2019 period, only Madrid-Retiro, Valladolid and Soria remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Since 1950,  applying the same criteria, only 30 remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  14  (a) Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Selected stations, AEMET data 1950-2019  (b) The Globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Selected stations, CRU data 1950-2019  Figure 1  Geographical distribution of stations  The results are presented according to the following steps: first, we apply our  trend test (see Definition 4) to determine the existence of local or global warming  and test for any possible warming acceleration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' second, we test different co-trending  45  18d:w 135°W 90  45°  S  90Climate change heterogeneity  15  hypotheses to determine the type of warming of each area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' thirdly, we test the  warming amplification hypothesis for different quantiles with respect to the mean  (of Spain as well as of the Globe): H0 : β1 = 1 versus Ha : β1 > 1 in (17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' and  finally, we compare the CC of different regions, for Spain and the Globe, and within  Spain, between Madrid and Barcelona, with our warming dominance test (see 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='1  Local warming: Spain  The cross-sectional analysis is approached under two assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' First, choosing a  sufficiently long and representative period of the geographical diversity of the Span-  ish Iberian Peninsula, 1950-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Second, we work with month-station units from  daily observations to construct the annual observations of the time series object from  the data supplied by the stations, following a methodology similar to that carried  out for the whole planet in GG2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='10 The study comprises the steps described in  the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The density of the data and the evolution of characteristics  are displayed, respectively in Figures 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  We find positive and significant trends in the mean, max, min and all the quan-  tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Therefore from definition 1, we conclude there exists a clear local warming  (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The recursive evolution for the periods 1950-2019 and 1970-2019 shows a clear  increase in the trends of the mean, some dispersion measures and higher quantiles  (see the last column of Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' More precisely, there is a significant trend acceler-  ation in most of the distributional characteristics except the lower quantiles (below  q20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' These quantiles, q05 and q10, remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The co-trending tests for the full sample 1950-2019 show a similar evolution of  the trend for all the quantiles with a constant iqr (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This indicates  that in this period the warming process of Spain can be considered a W1 type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  More recently, 1970-2019, the co-trending tests (see Table 3) indicate the upper  quantiles grow faster than the lower ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This, together with a positive trend in  the dispersion measured by the iqr shows that Spain has evolved from a W1 to a  9Before testing for the presence of trends in the distributional characteristics of the data, we  test for the existence of unit roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' To do so, we use the well-known Augmented Dickey-Fuller test  (ADF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Dickey and Fuller, 1979), where the number of lags is selected in accordance with the SBIC  criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The results, available from the authors on request, show that the null hypothesis of a  unit root is rejected for all the characteristics considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  10The results with daily averages are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The decision to work with monthly data  instead of daily in the cross-sectional approach has been based on its compatibility with the data  available for the Globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  16  W3 warming type process  Finally, no evidence of “inner” amplification during the period 1950-2019 is found  in the lower quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Regarding the upper quantiles, we found both “inner” and  “outer” amplification in the second period, which supports the previous finding of  a transition from type W1 to type W3 (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Summing up, with our proposed tests for the evolution of the trend of the whole  temperature distribution, we conclude that Spain has evolved from a W1 type to a  much more dangerous W3 type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The results of acceleration and dynamic amplifi-  cation reinforce the finding of this transition to type W3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Figure 2  Spain annual temperature density calculated with monthly data across stations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='07Climate change heterogeneity  17  Figure 3  Characteristics of temperature data in Spain with stations selected since 1950  (monthly data across stations, AEMET, 1950-2019)  30  15  mean  max  13  25  1950  1970  1990  2010  1950  1970  1990  2010  0  6  min  std  1950  1970  1990  2010  1950  1970  1990  2010  35  range  30  iqr  8  25  1950  1970  1990  2010  1950  1970  1990  2010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='0275  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0471  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='6949  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0040)  q90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0321  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0548  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='2441  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0007)  q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0335  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0526  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='3568  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0005)  Note:  OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:  Ct = α + βt + ut, for two  different time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T, Ct =  α2 + β2t + ut, t = s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' We show the value of  the t-statistic and its HAC p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table 2  Co-trending analysis (Spain monthly data across stations, AEMET, 1950-2019)  Joint hypothesis tests  Wald test  p-value  All quantiles (q05, q10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=',q90, q95)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='235  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='211  Lower quantiles (q05, q10, q20, q30)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='310  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='958  Medium quantiles (q40, q50, q60)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='438  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='803  Upper quantiles (q70, q80, q90, q95)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='515  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='679  Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='771  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='993  Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='331  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='215  Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='705  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='111  Spacing hypothesis  Trend-coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  p-value  q50-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='786  q95-q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  q95-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='096  q75-q25 (iqr)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='179  Note: Annual distributional characteristics (quantiles) of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The top panel shows the  Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In the bottom panel, the TT is applied to the difference between two representative quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  19  Table 3  Co-trending analysis (Spain monthly data across stations, AEMET, 1970-2019)  Joint hypothesis tests  Wald test  p-value  All quantiles (q05, q10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=',q90, q95)  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='879  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Lower quantiles (q05, q10, q20, q30)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='121  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='373  Medium quantiles (q40, q50, q60)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='314  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='518  Upper quantiles (q70, q80, q90, q95)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='719  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='633  Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='771  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='047  Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='675  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='099  Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='892  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Spacing hypothesis  Trend-coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  p-value  q50-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='029  q95-q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='050  q55-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='032  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='002  q75-q25 (iqr)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='003  Note: Annual distributional characteristics (quantiles) of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The top panel shows the  Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In the bottom panel, the TT is applied to the difference between two representative quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table 4  Amplification hypothesis (Spain monthly data, AEMET 1950-2019  periods/variables  1950-2019  1970-2019  1950-2019  1970-2019  Inner  Outer  q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='39  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='866)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='998)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='990)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='996)  q10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='52  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='899)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='994)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='992)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='986)  q20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='81  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='816)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='890)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='993)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='899)  q30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='87  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='935)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='929)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='008)  Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1  in the regression: Cit = βi0 + βi1meant + ϵit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' mean refers to the average of the Spanish Global  temperature distribution for the “inner” and “outer”cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='2  Global warming: the Globe  In this section, we carry out a similar analysis to that described in the previous  subsection for Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Figures 4 and 5 show the time evolution of the Global temper-  ature densities and their different distributional characteristics from 1950 to 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The data in both figures are obtained from stations that report data throughout the  sample period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table 5 shows a positive trend in the mean as well as in all the quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This  indicates the clear existence of Global warming, more pronounced (larger trend) in  the lower part of the distribution (a negative trend in the dispersion measures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The  warming process suffers an acceleration in all the quantiles above q30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  From the co-trending analysis (see Tables 6 and 7) we can determine the type  of warming process characterizing the whole Globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Table 6 indicates that in the  period 1950-2019 the Globe experimented a W2 warming type (the lower part of  the temperature distribution grows faster than the middle and upper part, implying  iqr and std have a negative trend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Similar results are maintained for the period  1970-2019 (in this case only the dispersion measure std has a negative trend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The asymmetric amplification results shown in Table 8 reinforce the W2 typology  for the whole Globe: an increase of one degree in the global mean temperature  increases the lower quantiles by more than one degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This does not occur with  the upper part of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Notice that this amplification goes beyond the  standard Artic amplification (q05) affecting also q10, q20 and q30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Summing up, the results from our different proposed tests for the evolution of  the trend of the whole temperature distribution indicate that the Globe can be  cataloged as a undergoing type W2 warming process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  This warming type may  have more serious consequences for ice melting, sea level increases, permafrost, CO2  migration, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' than the other types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  21  Figure 4  Global annual temperature density calculated with monthly data across stations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='005 ~  0  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='6962  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='5404  1970  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='3846  1960  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='2288  1950  temperature in degrees Celsius (month-station units)2010  2000  1990  1980  vears0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='035Climate change heterogeneity  22  Figure 5  Characteristics of temperature data in the Globe (monthly data across stations, CRU,  1950-2019)  12  40  11  38  10  mean  max  1950196019701980199020002010  1950196019701980  199020002010  13  44  46  48  50  52  std  11  1950196019701980199020002010  1950196019701980199020002010  8  90  85  16  iqr  range  80  19501960 1970198019902000 2010  1950196019701980199020002010  G  kur  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='9  skw  195019601970:1980199020002010  195019601970198019902000:2010  20  10  0  10  1950196019701980199020002010  q5  q10  q20  q30  q40  q50  q60  q70  q80  q90  q95Climate change heterogeneity  23  Table 5  Trend acceleration hypothesis (CRU monthly data across stations, 1950-2019)  Trend test by periods  Acceleration test  names/periods  1950-2019  1970-2019  1950-2019, 1970-2019  mean  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='0410)  Note:  OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:  Ct = α + βt + ut, for two  different time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T, Ct =  α2 + β2t + ut, t = s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' We show the value of  the t-statistic and its HAC p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table 6  Co-trending analysis (CRU montly data, 1950-2019)  Joint hypothesis tests  Wald test  p-value  All quantiles (q05, q10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=',q90, q95)  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='005  Lower quantiles (q05, q10, q20, q30)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='023  Medium quantiles (q40, q50, q60)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='962  Upper quantiles (q70, q80, q90, q95)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='777  Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='007  Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='916  Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='001  Spacing hypothesis  Trend-coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  p-value  q50-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='022  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  q95-q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='193  q95-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='026  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  q75-q25 (iqr)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='043  Note: Annual distributional characteristics (quantiles) of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The top panel shows the  Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In the bottom panel, the TT is applied to the difference between two representative quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  24  Table 7  Co-trending analysis (CRU montly data, 1970-2019)  Joint hypothesis tests  Wald test  p-value  All quantiles (q05, q10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=',q90, q95)  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='478  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='047  Lower quantiles (q05, q10, q20, q30)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='523  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='137  Medium quantiles (q40, q50, q60)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='569  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='752  Upper quantiles (q70, q80, q90, q95)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='667  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='446  Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='606  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='268  Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='714  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='348  Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='520  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='043  Spacing hypothesis  Trend-coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  p-value  q50-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='047  q95-q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='462  q95-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='048  q75-q25 (iqr)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='418  Note: Annual distributional characteristics (quantiles) of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The top panel shows the  Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In the bottom panel, the TT is applied to the difference between two representative quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table 8  Amplification hypotheses (CRU monthly data across stations, 1950-2019)  periods/variables  1950-2019  1970-2019  q05  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='83  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  q10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='79  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='73  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='001)  q20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='41  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='37  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  q30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='00  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='089)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='502)  q40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='91  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='999)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='973)  q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='81  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='997)  q60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='85  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='999)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='973)  q70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='85  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='988)  q80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='78  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  q90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='70  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='64  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000)  Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1  in the regression: Cit = βi0 +βi1meant +ϵit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' mean refers to the average of the Global temperature  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='3  Micro-local warming: Madrid and Barcelona  The existence of warming heterogeneity implies that in order to design more ef-  ficient mitigation policies, they have to be developed at different levels: global,  country, region etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' How local we need to go will depend on the existing degree of  micro-warming heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In this subsection, we go to the smallest level, cli-  mate station level .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' We analyze, within Spain, the warming process in two weather  stations corresponding to two cities: Madrid (Retiro station) and Barcelona (Fabra  station).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  11 Obviously, the data provided by these stations is not cross-sectional  data but directly pure time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Our methodology can be easily applied to  higher frequency time series, in this case daily data, to compute the distributional  characteristics (see Figures A1 and A2)12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The results are shown in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' These two stations, Madrid-Retiro and  Barcelona-Fabra clearly experience two different types of warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  First, there  is evidence of micro-local warming, understood as the presence of significant and  positive trends, in all the important temperature distributional characteristics of  both stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The acceleration phenomenon is also clearly detected, in other words,  the warming increases as time passes (see Tables A1 and A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Secondly, from the  co-trending tests (Tables A2-A3 and A6-A7), it can be concluded that the warming  process of Madrid-Retiro is type W3 while for Barcelona-Fabra it is type W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In  both cases the warming typology is stable through both sample periods (1950-2019  and 1970-2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Thirdly, as expected, Madrid-Retiro presents “inner” and “outer”  amplification for the upper quantiles, while Barcelona-Fabra does so only for the  center part of its temperature distribution (see Tables A4 and A8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Summing up, even within Spain we find evidence of warming heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  While Madrid (Continental Mediterranean climate) has a similar pattern as that  of peninsular Spain (1970-2019) W3, Barcelona (Mediterranean coastline climate)  maintains a W1 typology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Thus there are two different warming processes which  require mitigation policies at the country as well as the very local level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  11From Madrid and Barcelona there is data since 1920’s, nevertheless we began the study in 1950  for consistency with the previous analysis of Spain and the Globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  12See the application to Central England in GG2020 and in Gadea and Gonzalo (2022) to Madrid,  Zaragoza and Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  26  5  Comparing results  The goal of this section is to show the existence of climate heterogeneity by com-  paring the results obtained from applying our three-step methodology to different  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' These results are summarized in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It is clear that there is distribu-  tional warming in all the analyzed areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' but this warming follows different patterns  and sometimes the warming type is not even stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In the case of Spain, it depends  on the period under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Figure 6 captures graphically the different trend  behavior and intensity of the distributional characteristics by regions (Spain and the  Globe and Madrid and Barcelona).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='13 The graphical results in this figure coincide  with the results of the warming typology tests shown in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The middle of Table 10 shows that warming acceleration is detected in all the  locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This acceleration is more general in Spain than in the Globe (see also the  heatmap in Figure 7) and in Barcelona than in Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Apart from these differences,  the acceleration shares certain similarities across regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This is not the case for  the warming amplification that is clearly asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Spain suffers an amplification  in the upper quantiles while the Globe does so in the lower ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Notice that the  latter amplification goes beyond the standard results found in the literature for the  Arctic region (q05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' We detect amplification also for the regions corresponding to  the quantiles q10-q30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In the case of Madrid and Barcelona, Madrid suffers a wider  warming amplification than Barcelona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The results of the first two steps of our methodology are obtained region by region  (Spain, the Globe, Madrid and Barcelona).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It is the last step, via the warming  dominance test (see the numerical results in Table 9) where we compare directly  one region with another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Warming in Spain dominates that of the Globe in all the  quantiles except the lower q05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='14 This would support the idea held in European  institutions and gathered in international reports on the greater intensity of climate  13The analysis of other characteristics such as the third and fourth order moments can contribute  to the temperature distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In the case of Spain, the kurtosis is always negative with a mean  value of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='8 and a significant negative trend, which means that we are dealing with a platykurtic  distribution with tails less thick than Normal, a shape that is accelerating over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' However, it  is ot possible to draw conclusions about symmetry given its high variability over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Conversely,  the temperature distribution in the Globe is clearly leptokurtic with an average kurtosis of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='9  and a negative but not significant trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The global temperature observations are therefore more  concentrated around the mean and their tails are thicker than in a Normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The  skewness is clearly negative although a decreasing and significant trend points to a reduction of the  negative skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  14A more detailed analysis of the warming process suffered in the Artic region can be found in  Gadea and Gonzalo (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  27  change in the Iberian Peninsula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Warming in Madrid dominates that of Barcelona  in the upper quantiles, while the reverse is the case in the lower quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This  latter result coincides with the idea that regions close to the sea have milder upper  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Further research (beyond the scope of this paper) will go in the direction of  finding the possible causes behind the warming types W1, W2, and W3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Following  the literature, on diurnal temperature asymmetry (Diurnal Temperature Range =  DTR = Tmax − Tmin) we can suggest as possible causes for W2 the cloud coverage  (Karl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' 1993) and the planetary boundary layer (see Davy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For  W3, the process of desertification (see Karl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Summarizing, in this section we describe, measure and test the existence of  warming heterogeneity in different regions of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It is important to note  that these extensive results can not be obtained by the standard analysis of the  average temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table 9  Warming dominance  Spain-Globe  Madrid-Barcelona  Quantile  β  t-ratio  β  t-ratio  q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='018  (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='770)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='013  (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='730)  q10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='010  (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='504)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='013  (-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='215)  q20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='004  (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='950)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='012  (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='988)  q30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='001  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='180)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='013  (-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='164)  q40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='002  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='788)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='009  (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='909)  q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='003  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='025)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='003  (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='701)  q60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='006  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='933)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='001  (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='219)  q70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='009  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='266)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='006  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='252)  q80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='012  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='203)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='016  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='331)  q90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='017  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='862)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='010  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='869)  q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='019  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='930)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='014  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='993)  Note: The slopes (t-statistic) of the following regression  qτt(A) − qτt(B) = ατ + βτt + uτt  In the first column A=Spain, B=Globe and in the second A=Madrid, B=Barcelona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  28  Table 10  Summary of results  Cross analysis  Sample  Period  Type  Acceleration  Amplification  Dominance  Inner  Outer  Spain  1950-2019  W1  [mean, std, iqr, rank,  [q70, q80, q95]  [q90, q95]  [q60,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  q20,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  1970-2019  W3  [q50,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q80]  [q60,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  The Globe  1950-2019  W2  [mean  [q05,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q30]  [q05]  q40,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  1970-2019  W2  [q05,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q20]  Time analysis  Sample  Period  Type  Acceleration  Amplification  Dominance  Madrid, Retiro Station  1950-2019  W3  [mean, std, rank,  [q50,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  [ q40,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  [q80,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  q40, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  1970-2019  W3  [q50,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  [q40,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  Barcelona, Fabra Station  1950-2019  W1  [mean, [q30,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q90]  [q05,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q40]  q20,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q95]  1970-2019  W1  [q60, q70]  [q30,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', q70]  Note: For Spain and the Globe we build characteristics from station-months units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For Madrid and Barcelona we use daily  frequency time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' A significance level of 10% is considered for all tests and characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='01  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='06  mean  max  min  std  iqr  rank  kur  skw  q5  q10  q20  q30  q40  q50  q60  q70  q80  q90  q95  Globe-montly-1950  Spain-montly-1950  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='01  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='06  mean  max  min  std  iqr  rank  kur  skw  q5  q10  q20  q30  q40  q50  q60  q70  q80  q90  q95  Spain-montly-1950  Madrid-daily-1950  Barcelona-daily-1950  Note: The bars represent the intensity of the trends found in each characteristic measured through  the value of the β-coefficient estimated in the regression Ct = α + βt + ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Figure 6  Trend evolution of different temperature distributional characteristics  Climate change heterogeneity  30  1950-2019  1955-2019  1960-2019  1965-2019  1970-2019  1975-2019  1980-2019  1985-2019  1990-2019  1995-2019  2000-2019  mean  max  min  std  iqr  rank  kur  skw  q5  q10  q20  q30  q40  q50  q60  q70  q80  q90  q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='1  (a) Globe  Spain  1950-2019  1955-2019  1960-2019  1965-2019  1970-2019  1975-2019  1980-2019  1985-2019  1990-2019  1995-2019  2000-2019  mean  max  min  std  iqr  rank  kur  skw  q5  q10  q20  q30  q40  q50  q60  q70  q80  q90  q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='1  (b) Spain  Note: The color scale on the right side of the figure shows the intensity of the trend, based on the  value of the β-coefficient estimated in the regression Ct = α + βt + ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Figure 7  Comparing heatmaps  Climate change heterogeneity  31  6  Conclusions  The existence of Global Warming is very well documented in all the scientific reports  published by the IPCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In the last one, the AR6 report (2022), special attention is  dedicated to climate change heterogeneity (regional climate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Our paper presents a  new quantitative methodology, based on the evolution of the trend of the whole tem-  perature distribution and not only on the average, to characterize, to measure and  to test the existence of such warming heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' It is found that the local warm-  ing experienced by Spain (one of most climatically diverse areas) is very different  from that of the Globe as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' In Spain, the upper-temperature quantiles tend  to increase more than the lower ones, while in the Globe just the opposite occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In both cases the warming process is accelerating over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Both regions suffer an  amplification effect of an asymmetric nature: there is warming amplification in the  lower quantiles of the Globe temperature (beyond the standard well-known results  of the Arctic zone) and in the upper ones of Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Overall, warming in Spain domi-  nates that of the Globe in all the quantiles except the lower q05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' This places Spain in  a very difficult warming situation compared to the Globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Such a situation requires  stronger mitigation-adaptation policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For this reason, future climate agreements  should take into consideration the whole temperature distribution and not only the  average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Any time a novel methodology is proposed, new research issues emerge for future  investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Among those which have been left out of this paper (some are part  of our current research agenda), three points stand out as important:  There is a clear need for a new non-uniform causal-effect climate change anal-  ysis beyond the standard causality in mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In order to improve efficiency, mitigation-adaptation policies should be de-  signed containing a common global component and an idiosyncratic regional  element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  The relation between warming heterogeneity and public awareness of climate  change deserves to be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  32  References  [1] Berenguer-Rico, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='  Journal of Econometrics 178, 331-341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Econometric 24, 72-87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=', Fundacion BBVA / BBVA Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='L, Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Quasi-likelihood ratio tests for cointe-  gration, cobreaking, and cotrending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Econometric Reviews 38(8),881-898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Time series  analysis of global temperature distributions: identifying and estimating per-  sistent features in temperature anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Working Paper 15-13, University of  Missouri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Should the Carbon Price Be the Same in  All Countries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Journal of Public Economy Theory 18(5), 709-725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Diurnal  asymmetry to the observed global warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=', 37: 79-93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Temperature  Dynamics, in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Chudek, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Hsiao and A Timmermann (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' ), Essays in Honor  of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Hashem Pesaran (Advances in Econometrics, Volume 43), 9-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Working paper at arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='06303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  [11] Durlauf, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', Phillips, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Trends versus random walks in time series  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Econometrica 56, 1333-1354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=', Kl¨uppelberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Modelling Extremal Events  for Insurance and Finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Springer-Verlag, Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  33  [13] Gadea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Trends in distributional characteristics: Exis-  tence of global warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' Polar Warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' mimeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  [15] Gadea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' A Tale of three cities: Climate heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  SERIES 13(1-2) (Special Issue in honour of Juan Jos´e Dolado) 475-511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Statistical Analysis of Stationary Time  Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' New York: Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Huang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Waterfield, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Zhou (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=')].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='  Contribution of Working Group II to the Sixth Assessment Report of the In-  tergovernmental Panel on Climate Change [H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Rama (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Climate Change 2022: Mitigation of Climate Change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content=' Cambridge University  Press, Cambridge, UK and New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='  Climate change heterogeneity  36  7  Appendix: Climate change of Madrid and Barcelona  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='1  Madrid-Retiro  Figure A1  Characteristics of temperature data in Madrid-Retiro (AEMET daily data, 1950-2019)  16  32  30  mean  28  max  1950  1970  1990  2010 2019  1950  1970  1990  2010 2019  5  8  wmw  min  std  5  6  1950  1970  1990  2010 2019  1950  1970  1990  2010 2019  15  35  rank  w  30  10  igr  25  1950  1970  1990  2010 2019  1950  1970  1990  2010 2019  kur  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='5  0  1950  1970  1990  2010 2019  1950  1970  1990  2010 2019  30  20  10  0  1950  1970  1990  2010 2019  q5  q10  q20  q30  q40  q50  q60  q70  q80  q90  q95Climate change heterogeneity  37  Table A1  Trend acceleration hypothesis (Madrid, daily data, AEMET, 1950-2019)  Trend test by periods  Acceleration test  names/periods  1950-2019  1970-2019  1950-2019, 1970-2019  mean  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0326  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='0383)  Note:  OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:  Ct = α + βt + ut, for two  different time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T, Ct =  α2 + β2t + ut, t = s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' We show the value of  the t-statistic and its HAC p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table A2  Co-trending analysis (Madrid-Retiro daily data, AEMET 1950-2019)  Joint hypothesis tests  Wald test  p-value  All quantiles (q05, q10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=',q90, q95)  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='046  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Lower quantiles (q05, q10, q20, q30)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='360  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='715  Medium quantiles (q40, q50, q60)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='036  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='361  Upper quantiles (q70, q80, q90, q95)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='944  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='268  Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='707  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='349  Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='822  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='967  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Spacing hypothesis  Trend-coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  p-value  q50-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='505  q95-q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  q05-q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  q75-q25 (iqr)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='027  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Note: Annual distributional characteristics (quantiles) of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The top panel shows the  Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In the bottom panel, the TT is applied to the difference between two representative quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  38  Table A3  Co-trending analysis (Madrid-Retiro daily data, AEMET, 1970-2019)  Joint hypothesis tests  Wald test  p-value  All quantiles (q05, q10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=',q90, q95)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='371  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Lower quantiles (q05, q10, q20, q30)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='424  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='935  Medium quantiles (q40, q50, q60)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='111  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='017  Upper quantiles (q70, q80, q90, q95)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='214  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='360  Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='687  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='851  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='004  Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='094  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Spacing hypothesis  Trend-coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  p-value  q50-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='036  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='004  q95-q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='051  q05-q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  q75-q25 (iqr)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='000  Note: Annual distributional characteristics (quantiles) of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The top panel shows the  Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In the bottom panel, the TT is applied to the difference between two representative quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table A4  Amplification hypothesis (Madrid daily data, AEMET 1950-2019)  periods/variables  1950-2019  1970-2019  1950-2019  1970-2019  Inner  Outer  q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='0915  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='0012)  q70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0330  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0583  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='9241  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0020)  q80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0357  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0551  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='4081  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0087)  q90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0394  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0567  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0957  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0190)  q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0390  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0525  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='3435  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0000)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='0907)  Note:  OLS estimates and HAC p-values in parenthesis of the tβ=0 test from regression:  Ct = α + βt + ut, for two  different time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' For the acceleration hypothesis we run the system: Ct = α1 + β1t + ut, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T, Ct =  α2 + β2t + ut, t = s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=', T, and test the null hypothesis β2 = β1 against the alternativeβ2 > β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' We show the value of  the t-statistic and its HAC p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table A6  Co-trending analysis (Barcelona-Fabra daily data, AEMET, 1950-2019)  Joint hypothesis tests  Wald test  p-value  All quantiles (q05, q10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=',q90, q95)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='368  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='971  Lower quantiles (q05, q10, q20, q30)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='036  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='792  Medium quantiles (q40, q50, q60)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='073  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='964  Upper quantiles (q70, q80, q90, q95)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='784  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='853  Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='171  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='978  Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='901  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='929  Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='969  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='888  Spacing hypothesis  Trend-coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  p-value  q50-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='528  q95-q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='233  q05-q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='856  q75-q25 (iqr)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='442  Note: Annual distributional characteristics (quantiles) of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The top panel shows the  Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In the bottom panel, the TT is applied to the difference between two representative quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Climate change heterogeneity  41  Table A7  Co-trending analysis (Barcelona-Fabra daily data, AEMET, 1970-2019)  Joint hypothesis tests  Wald test  p-value  All quantiles (q05, q10,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=',q90, q95)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='165  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='215  Lower quantiles (q05, q10, q20, q30)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='904  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='593  Medium quantiles (q40, q50, q60)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='267  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='531  Upper quantiles (q70, q80, q90, q95)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='384  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='943  Lower-Medium quantiles (q05, q10, q20, q30, q40, q50, q60)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='120  Medium-Upper quantiles (q40, q50, q60, q70, q80, q90, q95)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='642  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='949  Lower-Upper quantiles (q05, q10, q20,q30, q70, q80, q90, q95 )  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='693  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='207  Spacing hypothesis  Trend-coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  p-value  q50-q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='192  q95-q50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='821  q05-q95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='241  q75-q25 (iqr)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='189  Note: Annual distributional characteristics (quantiles) of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content=' The top panel shows the  Wald test of the null hypothesis of equality of trend coefficients for a given set of characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  In the bottom panel, the TT is applied to the difference between two representative quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='  Table A8  Amplification hypothesis (Barcelona daily data, AEMET 1950-2019)  periods/variables  1950-2019  1970-2019  1950-2019  1970-2019  Inner  Outer  q05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='87  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='523)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='918)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='225)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='720)  q10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='79  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='94  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='824)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='980)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='263)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='668)  q20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
+page_content='04  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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+page_content='298)  Note: OLS estimates and HAC p-values of the t-statistic of testing H0 : βi = 1 versus Ha : βi > 1 in  the regression: Cit = βi0 + βi1meant + ϵit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfxgLI/content/2301.02648v1.pdf'}
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diff --git a/B9AzT4oBgHgl3EQfh_2U/content/tmp_files/2301.01493v1.pdf.txt b/B9AzT4oBgHgl3EQfh_2U/content/tmp_files/2301.01493v1.pdf.txt
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+ 
+This is the ‘accepted manuscript’ version of our paper:  
+ 
+ 
+ 
+ 
+Therese Weissbach, Tobias Kluge, Stéphane Affolter, Markus 
+C. Leuenberger, Hubert Vonhof, Dana F.C. Riechelmann, Jens 
+Fohlmeister, Marie-Christin Juhl, Benedikt Hemmer, Yao Wu, 
+Sophie F. Warken, Martina Schmidt, Norbert Frank, Werner 
+Aeschbach, 2023, Constraints for precise and accurate fluid 
+inclusion stable isotope analysis using water-vapour saturated 
+CRDS techniques, Chemical Geology 617, 121268. 
+https://doi.org/10.1016/j.chemgeo.2022.121268 
+ 
+ 
+Please contact the corresponding author, if you want to discuss 
+the content. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+© 2023. This manuscript version is made available under the 
+CC-BY-NC-ND 4.0 license 
+http://creativecommons.org/licenses/by-nc-nd/4.0/  
+
+Constraints for precise and accurate fluid inclusion stable isotope 
+analysis using water-vapour saturated CRDS techniques  
+Therese Weissbach1,2, Tobias Kluge1,2,3,4,*, Stéphane Affolter5, Markus C. Leuenberger6, Hubert 
+Vonhof7, Dana F.C. Riechelmann8, Jens Fohlmeister9, 10, Marie-Christin Juhl2, Benedikt Hemmer2,  Yao 
+Wu2, Sophie F. Warken2,11, Martina Schmidt2, Norbert Frank2, Werner Aeschbach2, 3 
+1Heidelberg Graduate School of Fundamental Physics, Heidelberg University, Im Neuenheimer Feld 
+226, 69120 Heidelberg, Germany 
+2Institute of Environmental Physics, Heidelberg University, Im Neuenheimer Feld 229, 69120 
+Heidelberg, Germany 
+3Heidelberg Center for the Environment, Heidelberg University, Im Neuenheimer Feld 229, 69120 
+Heidelberg, Germany 
+4now at: Institute of Applied Geosciences, Karlsruhe Institute of Technology, Adenauerring 20b, 
+76131 Karlsruhe, Germany 
+5Department of Environmental Sciences, University of Basel, Bernoullistrassse 30/32, 4056 Basel, 
+Switzerland 
+6Climate and Environmental Physics Division, Physics Institute and Oeschger Centre for Climate 
+Change Research, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland 
+7Climate Geochemistry Department, Max Planck Institute for Chemistry, Hahn-Meitner-Weg 1, 55128 
+Mainz, Germany 
+8Institute for Geosciences, Johannes Gutenberg University Mainz, Johann-Joachim-Becher-Weg 21, 
+55128 Mainz, Germany 
+9Federal Office for Radiation Protection, Köpenicker Allee 120-130, 10318 Berlin, Germany 
+10GFZ German Research Centre for Geosciences, Section ‘Climate Dynamics and Landscape 
+Development’, Telegrafenberg, 14473 Potsdam, Germany 
+11Institute of Earth Sciences, Heidelberg University, Im Neuenheimer Feld 234, 69120 Heidelberg, 
+Germany 
+*corresponding author: tobias.kluge@kit.edu 
+ 
+Journal: Chemical Geology 
+
+Highlights: 
+ 
+Laser-based fluid inclusion analysis with water-vapour purged extraction enables high 
+precision (δ2H≤±1.5‰, δ18O ≤±0.5‰) 
+ 
+Biasing effects (memory, adsorption, amount) in fluid inclusion isotope analysis are negligible 
+for ≥1 µl water/g calcite 
+ 
+Isotopic interference is negligible for sample isotope ratios within 10‰(δ18O) and 50‰(δ2H) 
+of the water vapour background 
+ 
+Reconstructed temperatures of a 20th century  stalagmite trace the recent warming of 1 °C in 
+Central Europe 
+
+Abstract 
+ 
+Hydrogen (δ2H) and oxygen (δ18O) isotopes of water extracted from speleothem fluid 
+inclusions are important proxies used for paleoclimate reconstruction. In our study we use a cavity 
+ring-down laser spectroscopy system for analysis and modified the approach of Affolter et al. (2014) 
+for sample extraction. The method is based on crushing of small sub-gram speleothem samples in a 
+heated and continuously water-vapour purged extraction line. The following points were identified:  
+ 
+Injection of reference water shows a precision (1σ) of 0.4-0.5 ‰ for δ18O values and 1.1-1.9 ‰ 
+for δ2H values for water amounts of 0.1-0.5 µl, which improves with increasing water amount to 0.1-
+0.3 ‰ and 0.2-0.7 ‰, respectively, above 1 µl. The accuracy of measurements of water injections and 
+water-filled glass capillaries crushed in the system is better than 0.08 ‰ for δ18O and 0.3 ‰ for δ2H 
+values. The reproducibility (1σ) based on replicate analysis of speleothem fluid inclusion samples with 
+water amounts > 0.2 µl is 0.5 ‰ for δ18O and 1.2 ‰ for δ2H values, respectively. Isotopic differences 
+between the water vapour background of the extraction system and the fluid inclusions have no 
+significant impact on the measured fluid inclusion isotope values if they are within 10 ‰ for δ18O and 
+50 ‰ for δ2H values of the background. Tests of potential adsorption effects with inclusion free spar 
+calcite confirm that the isotope values are unaffected by adsorption for water contents of about 1 µl 
+(fluid inclusion) water per g of carbonate or above.  
+Fluid inclusion analysis on three different modern to late Holocene speleothems from caves in 
+northwest Germany resulted in δ18O and δ2H values that follow the relationship as defined by the 
+meteoric water line and that correspond to the local drip water. Yet, due to potential isotope exchange 
+reactions for oxygen atoms, hydrogen isotope measurements are preferentially to be used for 
+temperature reconstructions. We demonstrate this in a case study with a Romanian stalagmite, for 
+which we reconstruct the 20th century warming with an amplitude of approximately 1 °C, with a 
+precision for each data point of better than ±0.5 °C. 
+ 
+Keywords: laser spectroscopy, water isotopes, cavity-ring-down measurement, speleothems, 
+paleoclimate, small samples 
+
+1. Introduction 
+Speleothem fluid inclusions can provide direct insight into past climatic conditions as they are 
+a unique archive for the original drip water and the corresponding meteoric water (e.g., Griffiths et al., 
+2010; Affolter et al., 2014; Labuhn et al., 2015; Warken et al., 2022). Fluid inclusion water isotope ratios 
+(δ18O and δ2H values) are increasingly usedas proxies in hydrology and paleoclimate studies (e.g., 
+McGarry et al., 2004; Demény et al., 2017; Millo et al., 2017; Affolter et al., 2019; Wilcox et al., 2020; 
+Matthews et al., 2021). Two physically different measurement principles, laser spectroscopy (mainly 
+cavity ring-down spectroscopy - CRDS) and isotope ratio mass spectrometry (IRMS), allow determining 
+the isotopic composition of speleothem fluid inclusion water (CRDS: Arienzo et al., 2013; Affolter et 
+al., 2014; Uemura et al., 2016; Dassié et al., 2018; IRMS: Dennis et al., 2001; Vonhof et al., 2006; 
+Dublyansky and Spötl, 2009). Although CRDS and IRMS systems yield comparable results (de Graaf et 
+al., 2020) challenges remain for both methods regarding precise and reproducible analysis of small 
+water amounts. Often only a single measurement attempt is possible due to low growth rates of the 
+speleothems (often 10-100 µm/a) or intended high resolution. Water contents in natural speleothems 
+range from ~ 0 up to several 10 µl per g (McDermott et al., 2006). The necessary water sample amount 
+(depending on the setup 0.05-0.2 µl, e.g., Dublyanksy and Spötl, 2009; Uemura et al., 2016) limits the 
+temporal resolution and restricts analytical repetition.  
+Fluid inclusion water for isotope analysis is released either by crushing (e.g., Schwarcz et al., 
+1976; Dennis et al., 2001; Vonhof et al., 2006; Dublyansky and Spötl, 2009; Demény et al., 2013) or 
+thermal decrepitation (e.g., Yonge, 1982; McGarry et al., 2004; Verheyden et al., 2008). Thermal 
+decrepitation has the disadvantage that structurally-bound water with a very low δ2H value may be 
+released during extraction, resulting in large isotopic shifts of up to 30 ‰ in comparison to parent cave 
+drip water (Yonge, 1982; Matthews et al., 2000; McGarry et al., 2004; Verheyden et al., 2008). This 
+analytical artefact can be largely avoided by crushing the sample mechanically. For fluid inclusion 
+analysis using IRMS (Schwarcz et al., 1976; Harmon et al., 1978; 1979), water was extracted by crushing 
+the sample under vacuum conditions and then subsequently converted to water vapour followed by 
+conversion into directly measurable gases such as H2 for H isotopic analysis. Fluid inclusion δ18O values 
+were initially not measured but calculated from measured δ2H values via the relationship between 
+δ18O and δ2H values of the meteoric water line (e.g., Schwarcz et al., 1976). In recent extraction systems 
+the oxygen in fluid inclusion water is converted to CO gas during a high temperature reaction with 
+glassy carbon which is then used for analysis of δ18O values in an IRMS (e.g., Dublyansky and Spötl, 
+2009). The first combined method for oxygen and hydrogen measurements with an off-line crushing 
+method and dual-inlet IRMS was developed by Dennis et al. (2001). It achieved good precision of ± 0.4 
+‰ for δ18O and ± 3 ‰ for δ2H values, but required a comparatively large sample with water amounts 
+of 1-3 μl (see also Matthews et al., 2000). A reduction in sample amount down to 0.1 μl, which 
+
+corresponds to 0.1 g of calcite for samples that contain 1 µl of water per gram, was achieved by Vonhof 
+et al. (2006) by combining off-line preparation and continuous-flow mass spectrometry. This technique 
+enables a faster analysis of 0.1 - 0.2 μl sized samples with a precision of ± 0.5 ‰ for δ18O and ± 1.5 ‰ 
+for δ2H values (Vonhof et al., 2006; Dublyansky and Spötl, 2009; de Graaf et al., 2020). 
+Laser spectroscopy is less expensive and represents a reliable, precise and easy technique to 
+directly measure stable water isotopes (Brand et al., 2009; Gupta et al., 2009). The first application 
+using CRDS to measure fluid inclusions in speleothems was developed by Arienzo et al. (2013). They 
+used a CRDS analyser with a stainless-steel line heated to 115 °C that was constantly flushed with dry 
+nitrogen as a carrier gas. It achieved comparable precisions as the traditional IRMS technique, with ± 
+0.5 ‰ for δ18O and ± 2.0 ‰ for δ2H values. The development of another analysis system using off-axis 
+integrated cavity output spectroscopy (OA-ICOS) achieved similar precision (Czuppon et al., 2014). The 
+latest analytical systems are able to measure released water volumes in the nano-litre range (50 to 
+260 nl) with a precision of ± 0.3 ‰ for δ18O and ± 1.6 ‰ for δ2H values using the CRDS technique 
+(Uemura et al., 2016). 
+The above discussed fluid inclusion extraction lines of Arienzo et al. (2013), Czuppon et al. 
+(2014), and Uemura et al. (2016) are working with dry carrier gas and low water vapour concentrations 
+in the analyser cavity, which may influence the stable isotope measurements by adsorption and cause 
+memory effects. The measured isotopic signal needs to be corrected for, e.g., the isotopic dependence 
+on the water vapour concentration (Uemura et al., 2016) or the memory effect (e.g., van Geldern and 
+Barth, 2012). The memory effect in the analyser cavity is due to limitations during the removal of all 
+gas between two measurements preventing the full desorption of water molecules from the cavity 
+walls. A standard technique to deal with memory effects in liquid water analysis is the repeated 
+injection of the same water sample. The measured signal converges exponentially towards the actual 
+sample signal (e.g., van Geldern and Barth, 2012). However, multiple crushing steps on the same 
+sample are typically not feasible for fluid inclusion measurements of speleothems since the amount of 
+water of these samples is often too low to split it in several aliquots (sub-μl range). The adsorption 
+issue was addressed by Affolter et al. (2014) with an extraction line that is continuously purged with a 
+moist gas providing a water vapour background with constant and known δ18O and δ2H values. The 
+extraction line with a “wet” N2 gas allows reproducible and precise measurement of released fluid 
+inclusion water. The continuous heating of the system enables the instantaneous evaporation of the 
+water released from inclusions followed by spectroscopic analysis of the resulting mixture of 
+background and sample water vapour. The main advantage of the water-vapour flushing is that this 
+procedure avoids additional corrections of the measured water stable isotopes related to memory 
+effects. The achievable standard deviations of the measurements with this analytical system are 
+smaller than 0.4 ‰ for δ18O and 1.5 ‰ δ2H values, which is comparable with the traditional IRMS 
+
+technique (Vonhof et al., 2006; Dublyansky and Spötl, 2009) and CRDS setups working with a dry carrier 
+gas (Arienzo et al., 2013; Czuppon et al., 2014). 
+One important application of isotope fluid inclusion studies is the reconstruction of 
+paleotemperatures using the oxygen isotope fractionation between water and calcite. The 
+temperature reconstruction was initially based on the measurement of fluid inclusion δ2H values from 
+which the fluid inclusion δ18O values were calculated using the δ2H-δ18O relationship of the global 
+meteoric water line (Craig, 1961). Combined with the δ18O value of the calcite, temperatures were 
+calculated from the oxygen isotope fractionation between the carbonate mineral and water. This 
+indirect approach achieved a reported precision of about ± 2 °C in the early studies of Schwarcz et al. 
+(1976). In the decades prior to 2010, direct temperature calculation from measured fluid inclusion δ18O 
+values has been rare due to severe challenges in its analysis with standard mass spectrometric 
+approaches. Therefore, mostly the indirect way of first converting measured δ2H values into fluid 
+inclusion water δ18O values was used (e.g., Matthews et al., 2000). More recent studies provided 
+comparable temperature precision based on inclusion water δ18O values: ± 1.3 °C (van Breukelen et 
+al., 2008), ± 0.9-2.1 °C (Meckler et al., 2015); ± 2.7 °C (Arienzo et al., 2015), and ± 0.6-3.1 °C (Uemura 
+et al., 2016). The uncertainty of the indirect paleotemperature reconstruction from δ2H variations in 
+speleothem fluid inclusions is similar: ± 1.5 °C (Zhang et al., 2008) and ± 0.9-2.5 °C (Meckler et al., 
+2015). More recently, a better precision has been achieved when using the rainfall δ2H/T relation in 
+mid-latitudes: ± 0.2-0.5 °C (Affolter et al., 2019). 
+In this study, we systematically assess the measurement method of stable water isotopes in 
+speleothem fluid inclusion analysis using CRDS, including in particular the effect of sample amount 
+(water amount per analysis), water adsorption on freshly crushed calcite surfaces, influence of the 
+isotope values of the water vapour background on the sample signal, as well as the external 
+reproducibility of speleothem fluid inclusion samples (using adjacent aliquots along growth layers). In 
+addition, we present a recent case study allowing to determine paleo-temperature trends in the 1°C 
+range. 
+ 
+2. Methods and site description 
+2.1 Water extraction from fluid inclusions  
+ 
+At the Institute of Environmental Physics (Heidelberg) water from speleothem fluid inclusions 
+is extracted within a system that is constantly purged by an artificially prepared moist gas, leading to 
+a water vapour background with known δ18O and δ2H values (Fig. 1). In the extraction line this stable 
+water vapour background is generated by mixing water of a known isotopic composition into a dry 
+nitrogen gas flow (300 ml/min). A peristaltic pump (Ismatec -REGLO Digital, Wertheim, Germany) 
+continuously supplies small amounts of water (1 μl/min) to the line through a T-injection port with a 
+
+septum (Fig. 1 A). A constant temperature of 120 °C ensures a complete and immediate evaporation. 
+Instant water evaporation is induced in a fused silica capillary, which slightly touches the heated base 
+of the port. A two-litre mixing cavity placed after the T-injection port generates a stable water vapour 
+background and compensates fluctuations caused by the peristaltic pump cycles. The nitrogen flow is 
+controlled by a mass flow controller (Analyt MTC, model GFC-17, Müllheim, Germany) and creates a 
+constant overpressure of 0.5 bar. The flow rate is 40 ml/min into the CRDS analyser (L2130-i, Picarro, 
+Santa Clara, USA). The surplus gas stream is vented through a purge capillary before the crusher unit. 
+With this setup the water vapour concentration in the CRDS cavity ranges between 6000 and 8000 
+ppmV, but the cavity can also be adjusted to higher or lower water vapour concentrations if needed. 
+We have chosen a range between 6000 and8000 ppmV to allow for the detection and analysis of small 
+fluid inclusion water amounts (sufficiently high ratio of water vapour from the sample relative to the 
+background) but also to provide a background water vapour concentration that prevents memory 
+effects. 
+The sample (speleothem fragment or glass capillary)is inserted in a copper tube (Fig. 1 B) which 
+is connected to the extraction and measurement system. Due to limitation of the copper tube with 
+respect to length and diameter, compact mineral pieces with rectangular dimensions of 6 mm x 6 mm 
+x 10-20 mm are preferred. The copper tube including the sample is purged for at least 30 min until 
+water vapour concentration and isotope values reach a constant signal. The stability of the water 
+vapour background is verified by monitoring the standard deviation of the water vapour concentration. 
+When the standard deviation of the water vapour concentration remained less than 20 ppmV for 30 
+minutes, the speleothem sample is crushed by compressing the copper tube from the outside with a 
+hydraulic press at 200-300 bar. This compression in the heated system leads to the release and 
+immediate evaporation of inclusion water and a sudden pressure increase. This pressure increase 
+could cause gas flow not only towards the CRDS but also in direction of the purge capillary and may 
+provoke sample gas loss. Therefore, a reflux valve is installed between the 2l mixing cavity and the 
+crushing unit to prevent a backflow and loss of the sample. In general, a very good crushing efficiency 
+has been achieved with an average grain size of 37 μm after the crushing (Weißbach, 2020). A 
+reproducibility test related to the crushing procedure showed similar particle size distributions for five 
+different speleothem samples investigated with laser diffraction (Analysette-22 Micro Tec) after 
+crushing with the hydraulic system.  
+An injection port is situated next to the copper tube, allowing to mimic a water release from a 
+mineral sample and helps to evaluate and control accuracy and precision of the δ18O and δ2H values 
+from small (inclusion) water samples. An additional small mixing cavity (400 ml) directly after the 
+crushing unit prolongs the generated sample signal from an usually few seconds lasting peak to a well 
+
+measurable signal with a duration of several minutes. After the mixing cavity the gas proceeds to the 
+L2130-i isotope and gas concentration analyser (Picarro). 
+ 
+Figure 1: Fluid inclusion line for extraction and measurement of stable oxygen and hydrogen isotopes of fluid 
+inclusions in speleothems. Water with a known isotope composition is mixed into a nitrogen gas flow to create a 
+stable water vapour background (position A). The purge capillary reduces the background vapour flow from 280 
+to 40 ml/min as required for the CRDS analyser (L2130-i, Picarro). The two mixing cavities provide a smoothing of 
+the background vapour signal and a dispersion of the measurement signal. The speleothem sample or the glass 
+capillary is placed in a copper tube and installed at position B inside the heated oven. In order to prevent the 
+backflow of the water vapour from the freshly crushed sample, a reflux valve is installed. The flow directions are 
+indicated as blue arrows. [black/white for figures in print, colour online] 
+ 
+2.2  Analysis and data evaluation 
+The water vapour concentration and the δ18O and δ2H values are determined with the L2130-i 
+isotope and gas concentration analyser of Picarro. The L2130-i analyser is based on wavelength-
+scanned CRDS in the spectral range from 7183.5 to 7184 cm-1 and uses a multi-pass cell that creates a 
+
+N2
+tankgas
+injection
+glass
+port
+capillary
+peristalticpump
+CRDS
+background
+heatingtape
+40
+water
+ml/min
+(120°C)
+purge capillary
+~280
+ml/min
+refluxvalve
+mixing cavity -21
+filter
+injection
+port
+oven
+mixingcavity-400ml
+loadedcopper
+(120C)
+tubelong effective absorption path length of about 12 km (Aemisegger et al., 2012). A stable cavity 
+temperature of 80°C ± 0.002 °C is maintained). The cavity pressure is set to 66.66 hPa (50 Torr). The 
+water isotopologue lines pertaining to 18O and 2H are measured simultaneously over a 0.8 s interval. 
+In our setup the L2130i allows isotope measurements in the water vapour concentration range 
+between 1 000 and 50 000 ppmV.  
+In constant flow mode the actual measured signal is composed of a background signal and a peak 
+signal after crushing and must be integrated over a corresponding time interval (Fig. 2). The evaluation 
+routine follows the approach of Affolter et al. (2014) but was extended by the correction of a potential 
+level change in the water vapour background (Weißbach, 2020). The import of the data and the 
+evaluation was carried out with the assistance of the Python script IsoFluid (https: 
+//github.com/bhemmer/IsoFluid and http://doi.org/10.5281/zenodo.5911265). IsoFluid determines 
+the sample or calibration standard peak start and end based on a slope criterion that compares the 
+start and end slope with the background slope of the water vapour concentration in user-defined time 
+intervals. The sample isotope value (δ18Osample) is calculated by subtraction of the background signal 
+(index back) from the measured signal of the mixed gas signal (index mix) and follows the approach of 
+Affolter et al. (2014): 
+𝛿��𝑂������ =
+∫
+𝛿��𝑂������(𝑡) ∗ 𝐻�𝑂������(𝑡) ∙ 𝑑𝑡
+��
+��
+∫
+𝐻�𝑂������(𝑡)
+��
+��
+∙ 𝑑𝑡
+=
+∫
+𝛿��𝑂���(𝑡) ∗ 𝐻�𝑂���(𝑡) ∙ 𝑑𝑡 − ∫
+𝛿��𝑂����(𝑡) ∗ 𝐻�𝑂����(𝑡) ∙ 𝑑𝑡
+��
+��
+��
+��
+∫
+𝐻�𝑂���(𝑡)
+��
+��
+∙ 𝑑𝑡 − ∫
+𝐻�𝑂����(𝑡)
+��
+��
+∙ 𝑑𝑡
+ 
+ 
+(Eq.1) 
+H2O refers to the related water amount. The calculation for δ2H values is correspondent.  
+All uncertainties are reported at the 1σ level. 
+
+ 
+Figure 2: Water vapour signal in the CRDS before, during, and after speleothem sample crushing. Background 
+intervals  (orange) are used to determine the value of the water background during a sample analysis (blue 
+shaded). [black/white for figures in print, colour online] 
+ 
+2.3 Water vapour background calibration 
+For the calibration of the oxygen and hydrogen isotope signal of the water vapour background 
+five independently measured in-house reference waters were used (Table 1). Water from 
+Willersinnweiher water (WW) was not used for calibration but for precision and accuracy assessment. 
+ 
+Table 1: In - house reference waters for isotopic calibration of the fluid inclusions CRDS system, Isotope 
+values were independently measured at the Institute of Environmental Physics (IUP) at Heidelberg 
+University.  Uncertainties are given as 1σ errors. 
+Water type 
+Code 
+δ18O values 
+(‰ VSMOW) 
+δ2H values 
+(‰ VSMOW) 
+Artificially evaporated water 
+AE 
+3.8 ± 0.3 
+-21.79 ± 1.8 
+Ocean water 
+Kona 
+-0.05 ± 0.08 
+0.5 ± 0.7 
+Lake 
+Water 
+- 
+Willersinnweiher 
+(S. 
+Germany) 
+WW 
+-0.32 ± 0.11 
+-18.8 ± 0.4 
+De-ionized local tap water 
+VE 
+-8.57 ± 0.08 
+-61.0 ± 0.7 
+Alpine Water 
+VCL 
+-13.04 ± 0.08 
+-98.3 ± 0.7 
+
+8500
+samplepeak
+water vapour concentration [ppmV]
+8000
+background interval
+background interval
+7500
+peak -
+start
+peak-end
+linearregression
+7000
+11:15
+11:30
+11:45
+12:00
+12:15
+12:30
+time [hour]Alpine Water - Colle Gniffeti ice core 
+CC 
+-15.13 ± 0.08 
+-110.6 ± 0.7 
+North Greenland water-surface snow 
+NG 
+-26.54 ± 0.08 
+-212.1 ± 0.7 
+ 
+ 
+ 
+ 
+ 
+The isotope values of these five reference waters were determined independently with a Los 
+Gatos Research LGR1 analyser. These reference waters cover a range of −26.5 up to -0.05 ‰ in δ18O 
+values and −212.1 up to 0.5 ‰ in δ2H values (both VSMOW) (Table 1), which includes the relevant 
+range for speleothem samples. Five different isotope background values were realised by using the 
+corresponding reference water as supply, injected with the peristaltic pump into the system. Once a 
+sufficiently stable water vapour concentration was achieved in the preparation line (standard deviation 
+below 20 ppmV for 30 min) the isotope signal was averaged over 60 minutes, which results in a 
+standard deviation of 0.2 ‰ and 0.7 ‰ for the δ18O and δ2H background values, respectively. Figure 3 
+shows the CRDS-measured isotope value against the reference value (Table 1). The value of the water 
+vapour background was constantly monitored via repeated measurements of the in-house reference 
+waters, and has remained constant over several years. 
+ 
+Figure 3: The calibration of δ18O and δ2H values of the water vapour background results from a linear regression 
+(red lines). The calibration equation is y = (0.994 ± 0.007 · x + 2.30 ± 0.14) ‰ for δ18O values (R²=0.999) and y = 
+(0.980 ± 0.003 · x−7.77 ± 0.38) ‰ for δ2H values(R²=0.999). Both calibrations remained constant over several 
+years. Isotope data on the reference waters used for water vapour background calibration are given in Table 1. 
+The residuals from the linear regression indicate that calibrated values are within 0.08 ‰ (δ18O) and 0.35 ‰ 
+(δ2H) of the expected reference value. Furthermore, the residuals show a random distribution. Uncertainties on 
+the 1σ level in the calibration graphs are smaller than the symbol size. [black/white for figures in print, colour 
+online] 
+
+residuals (%o)
+0.08
+8180
+(0%)
+0.30
+2H
+0.04
+0.15
+S
+8
+0.00
+dual
+0.00
+-0.15
+-0.04
+0
+-0.08
+-0.30
+0
+-30
+-25
+-20
+-15
+-10
+-5
+0
+-250
+-200
+-150
+-100
+-50
+0
+50
+50
+0
+Kona
+1:1
+(% VSMOW)
+1:1
+(%。 VSMOW)
+0
+-5
+Kona
+VE
+-10
+-50
+VCL
+VE
+-15
+CC
+-100
+VCL
+linearregression
+pected
+CC
+-20
+-150
+exp
+-25
+NG
+linearregression
+-200
+NG
+-30
+-250
+-30
+-25
+-20
+-15
+-10
+-5
+0
+-250
+-200
+-150
+-100
+-50
+0
+50
+8180.
+measured (%。 VSMOW)2.4  Water amount calibration  
+A precise water amount calibration is necessary for determining the exact amount of released 
+water from the crushed speleothem calcite. The released water amount is a major parameter for the 
+calculation of the fluid inclusion isotope value and is determined via water vapour signal integration 
+(see Eq.1). Isotope values could also be calculated with the time-integrated water vapour mixing ratio 
+alone, however, knowledge of the released water amounts is recommended for uncertainty 
+assessment (amount dependence) and for assessment of speleothem growth conditions (fluid 
+inclusion water yield). Typically, volume calibrations are carried out by injecting water in the μl range 
+with syringes (here: SGE 1BR-7RAX and 5BR-7RAX and  Hamilton 70001KH and 75N), however, the 
+variability of the calculated water amount only using syringe injections is significant and can be as high 
+as 10 % (inset in Fig. 4, Weißbach, 2020).  
+Here we present a water amount calibration method with glass capillaries that follows the 
+approach of Kluge et al. (2008). The glass capillaries (borosilicate, Hirschmann) can be filled with 0.1-
+5.0 μl water at varying isotopic composition. They can be closed airtight by melting both ends. The size 
+of the filled capillary can be adjusted to the size of the crushing cell down to a minimum length of 
+approximately 1 cm. The exact volume is determined by scanning the capillary with a high-resolution 
+office scanner and comparison with the pre-marked 1 μl labels on the capillary. The volume uncertainty 
+of the glass capillary water amount is ± 0.025 μl and was determined by five repetitions of the manual 
+evaluation of a scan. The accuracy is given by the uncertainty of the pre-marked 1 µl labels (± 0.003 
+µl). Water-filled capillaries were analysed weekly to monitor the stability of the water amount 
+calibration (Fig. 4). The uncertainty of the water amount determination from the calibration is 
+approximately ± 0.02 µl at a water volume of 1 µl and ± 0.04 µl at 2.5 µl using the 1σ uncertainty of 
+the linear regression. In general, we rarely observed outliers in the water amount calibration when 
+using glass capillaries. 
+
+ 
+Figure 4: The time-integrated measured volume signal in ppmV*s as given by the Picarro analyser is plotted 
+against the water volume of  the glass capillaries. The resulting linear regression y = (5.9 × 10−7 ·x−0.011) μl 
+(R²=0.999) is used to determine the released amount of water from speleothem samples. In total 45 capillaries 
+were measured for calibration spanning a water amount range from 0.2 up to 4.3 μl. The upper inset shows a 
+glass capillary filled with about 0.5 µl of water (in the middle of the capillary). The lower inset shows the 
+comparison of water injections with different syringes (blue and red symbols) and the glass capillaries (black 
+circles). The uncertainties are given on the 1σ level. [black/white for figures in print, colour online] 
+ 
+2.5 Site and sample description  
+Hüttenbläserschacht Cave (Germany) 
+For direct comparison with rainwater δ18O values and measured drip water isotope values, we 
+selected a suite of modern and late Holocene samples from the Hüttenbläserschacht Cave, located 
+only a few 100 meters west of the well-monitored Bunker Cave in northwest Germany (e.g., 
+Riechelmann et al., 2011). Both caves are situated in the upper Middle Devonian limestone in Iserlohn 
+(Sauerland). Hüttenbläserschacht Cave hosts pool spar calcites that are expected to provide fluid 
+inclusion isotope values close to drip water as they grow under the water table of the pools. Pool spars 
+from this cave have already been investigated by Kluge et al. (2013) using clumped isotope ∆47 and 
+calcite δ18O values for calculation of the (drip) water δ18O value. Calcite was actively precipitating in 
+the pools (e.g., abundant calcite rafts) at the time of pool spar removal. 230Th-U disequilibrium dating 
+
+8.0x106
+label
+watercolumn
+I area (ppmV*s)
+6.0x106
+integrated signal
+4.0x106
+3x106
+linearregression
+oglass capillary
+syringes
+Hamilton
+2x106
+SGE
+2.0x106
+1x106
+0
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0
+1
+2
+3
+4
+5
+water volume capillary (μl)at Heidelberg Academy of Sciences provided radiometric ages of one pool spar and one raft sample of 
+0.05 ± 0.27 ka BP and 0.36 ± 0.12 ka BP, respectively (Supplemental material S1), corroborating the 
+assumption that the pool spars and rafts are modern age. 
+Cloşani Cave (Romania) 
+For the second case study we selected a 20th century stalagmite (Stam 4) from Cloşani Cave, 
+Romania (e.g., Constantin and Lauritzen, 1999). A monitoring program from 2010 to 2012 and 2015 
+demonstrated a stable cave environment with an air temperature of 11.4 ± 0.5 °C and a relative 
+humidity close to 100 % (Warken et al., 2018). The isotopic composition of the drip water in direct 
+vicinity (1 m) of the former location of Stam 4 showed no seasonal cycle and was constant throughout 
+the monitoring. The mean dripwater δ18O value was −9.6 ± 0.2‰ and −66.3 ± 1.7‰ for δ2H values. 
+Stalagmite Stam 4 has a total length of 6 cm and an average growth rate of 510 μm per year, as 
+deduced from counting of layers related to annual cycles in the concentration of various elements 
+(Supplemental material S2). The speleothem grew actively until the removal in spring 2010 C.E. as drip 
+water was feeding the stalagmite. The recent growth of the stalagmite was further constrained by the 
+detection of the 20th century radiocarbon bomb spike, which was imprinted by the transport of the 
+atmospheric signal into the speleothem (see Supplemental material S3). Combined layer counting and 
+radiocarbon measurements suggest a growth period from 1910 to 2010 C.E. For the fluid inclusion 
+study, pieces were taken from the peripheral part of Stam 4 with a distance of approximately 1 to 1.5 
+cm from the actual growth axis (see Supplementary Fig. S2). 
+ 
+3. Results 
+3.1 Precision of isotope measurements 
+The precision of isotopic measurements (Fig. 5, Table 2) was quantified using the standard 
+deviation of repeated analyses of the reference waters injected via syringes (VE water; Table 1) and 
+independently cross-checked with water-filled glass capillaries using VE and WW reference waters. The 
+injected water amount using syringes) varied between 0.1 and 4.0 μl.  
+Using syringe injection method, a clear decrease of the standard deviation of these isotope 
+analyses with increasing water amount becomes apparent (Fig. 5, Table 2). The standard deviation 
+decreases strongest between 0.1 and 1 µL and reaches values between 0.1 and0.3 ‰ for δ18O and 
+between 0.2 and0.7 ‰ for δ2H values, for samples larger than 1 µl (Supplementary Table S4). For 
+smaller water amounts, i.e., of 0.5 µl and below, the isotope values of the injections show a 
+significantly larger scatter, leading to standard deviations between 0.4 and0.5 ‰ for δ18O and between 
+
+1.1 and1.9 ‰for δ2H values. These uncertainties are based on an exponential fit of the standard 
+deviation against the water volume using repeated measurements at a given water volume (Fig. 5).  
+For the determination of the precision, reference water sealed in glass capillaries was crushed in 
+the fluid inclusion system. Consistent with the results from the water injections, the precision for 
+isotopic analyses of water released from crushing glass capillaries is between 0.07 and0.10 ‰ for δ18O 
+values and between 0.3 and0.4 ‰ for δ2H values, for water amounts above 0.5 µl. Smaller water 
+amounts resulted in a significant increase in the uncertainty and are expressed through a lower 
+precision (Table 2).   
+
+Table 2: Measurement precision and accuracy in dependence of the water amount. The precision 
+(± 1σ) was determined from repeated injections of isotopically well-characterized water standards 
+using syringes. The accuracy (given at the 1σ level) was assessed by measurement of reference water 
+both from injections and by release from crushing of sealed glass capillaries and comparison with the 
+independently determined isotope values (Table 1). The error estimate for the accuracy assumes a 
+Gaussian distribution and includes the uncertainty of the expected value (VE water: ± 0.08 ‰ for δ18O, 
+± 0.7 ‰ for δ2H, WW water: ± 0.11 ‰ for δ18O, ± 0.4 ‰ for δ2H). n represents the number of analyses 
+in the investigated water volume range. The water isotope values are given relative to VSMOW. 
+Type 
+Reference 
+water 
+Water 
+volume (µl) 
+Precision (1σ) 
+δ18O value(‰) 
+                        
+δ2H 
+value 
+(‰) 
+Accuracy (1σ) 
+δ18O 
+value 
+(‰) 
+      
+ δ2H 
+value(‰) 
+n 
+Water 
+injection 
+VE 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0.1 
+0.54 
+1.8 
+ 
+ 
+6 
+ 
+ 
+0.2 
+0.34 
+1.6 
+ 
+ 
+6 
+ 
+ 
+0.3 
+0.49 
+1.6 
+ 
+ 
+6 
+ 
+ 
+0.4 
+0.53 
+1.1 
+ 
+ 
+6 
+ 
+ 
+0.5 
+0.58 
+1.4 
+ 
+ 
+6 
+ 
+ 
+1 
+0.17 
+0.4 
+ 
+ 
+16 
+ 
+ 
+2 
+0.21 
+0.4 
+ 
+ 
+11 
+ 
+ 
+3 
+0.16 
+0.4 
+ 
+ 
+11 
+ 
+ 
+4 
+0.10 
+0.1 
+ 
+ 
+9 
+ 
+ 
+Mean all 
+0.35 
+1.0 
+0.09 ± 0.19 
+0.3 ± 0.7 
+77 
+ 
+ 
+Mean < 1µl 
+0.50 
+1.5 
+0.12 ± 0.22 
+0.3 ± 0.7 
+30 
+ 
+ 
+Mean ≥ 1µl 
+0.16 
+0.3 
+0.08 ± 0.12 
+0.3 ± 0.7 
+47 
+Glass 
+capillary 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+WW 
+0.3-4.3 
+0.42 
+0.4 
+0.10 ± 0.44 
+0.1 ± 0.6 
+16 
+ 
+WW 
+> 0.5 
+0.07 
+0.3 
+-0.02 ± 0.13 
+0.1 ± 0.5 
+14 
+ 
+VE 
+0.3-4.1 
+0.30 
+1.3 
+0.14 ± 0.31 
+0.6 ± 1.5 
+17 
+ 
+VE 
+> 0.5 
+0.10 
+0.4 
+-0.03 ± 0.13 
+0.1 ± 0.8 
+9 
+ 
+
+ 
+Figure 5: Upper panels: precision (1σ) of the isotope measurements for varying amounts using the water injection 
+method. The red lines represent the least-square exponential fits to the data. The standard deviation decreases 
+with increasing water amount for both δ18O and δ2H values. . Lower panel: accuracy determination for varying 
+water amounts based on individual injections (open circles). The related mean values with their 1σ standard 
+deviation are shown as filled dots with error bars. The black horizontal lines represent the reference value for VE 
+- tap water (δ18O=- 8.57 ‰, δ2H =- 61.0 ‰) with its uncertainty band (grey shading, ± 0.08‰ for δ18O, ± 0.7 ‰ for 
+δ2H values). [black/white for figures in print, colour online] 
+ 
+ 
+ 
+ 
+
+2.0
+0.6
+18
+1α standard deviation (%o)
+0.5
+5
+ standard deviation (
+0.4
+1.0
+0.3
+0.2
+0.5
+0.1
+0.0
+0.0
+0
+1
+2
+3
+4
+0
+1
+2
+3
+4
+water amount (μl)
+water amount (μl)
+-57
+singlevalue
+8H [%。 VSMOW]
+-58
+meanvaluewithstdv
+8
+-60
+-61
+-62
+-63
+6180 [%VSMOW]
+-8.0
+-8.5
+-9.0
+9.5
+singlevalue
+meanvaluewithstdy
+-10.0
+3
+1
+2
+4
+volume [u]3.2 Accuracy of isotope analysis of micro-litre water amounts  
+The accuracy of the water δ18O and δ2H values was assessed for reference waters (Table 1) by the 
+injection with syringes and by crushing glass capillaries. The injected water amounts covered the 
+typical range of water extracted from inclusions (Fig. 5). The glass capillaries were filled with reference 
+water with similar water amounts between 0.3 and 4.3 µl and were crushed in the copper tube with 
+the same hydraulic press as the stalagmite samples. 
+Considering the water isotope mean values of all measurements performed with the glass 
+capillaries, the δ18O value deviated from the expected reference values (Table 1) by 0.10 ± 0.44 ‰ for 
+WW water (n=16) and by 0.14 ± 0.31 ‰ for VE water (n=17) (Table 2). For δ2H values the deviation 
+from the reference value was 0.1 ± 0.6 ‰ for WW water (n=16) and 0.6 ± 1.50 ‰ for VE water (n=17). 
+Considering only those measurements with water amounts above 0.5 µl reduces the uncertainty. For 
+this selection, the δ18O value of both reference waters deviates on average from the expected 
+reference value by -0.02 ± 0.13 ‰ for WW water (n=14) and -0.03 ± 0.13 ‰ for VE water (n=9). For 
+δ2H values the deviation from the reference value was 0.1 ± 0.5 ‰ for WW water (n=14) and 0.1 ± 0.8 
+‰ for VE water (n=9).  
+The accuracy as determined by crushing of water-filled glass capillaries is confirmed by the 
+injection-based data (Table 2). Overall, the δ18O value of the injected VE water deviated from the 
+expected value on average by 0.09 ± 0.19 ‰ (n=77), that of the δ2H value by 0.3 ± 0.7 ‰ (n=77). 
+ 
+3.3 Adsorption and/or desorption on the calcite surface 
+Adsorption on a calcite surface and, in particular, on freshly crushed carbonate with a large surface 
+to volume ratio provides the possibility to alter the isotope values of the fluid inclusion water (Dennis 
+et al., 2001). Therefore, an artificial fluid inclusion system (speleothem analogue) as described by 
+Dennis et al. (2001) has been prepared to quantify the influence of adsorption on the measured 
+isotopic signal in our setup. We measured water vapour released from a water-filled glass capillary in 
+direct contact with inclusion-free Iceland spar carbonate. The compact Iceland spar pieces (0.45-0.81 
+g) as well as the released water of the capillaries (1.4-3.7 µl water) represent a speleothem sample 
+with a water content of 2.2 up to 7.8 μl per g calcite. In total, we prepared and analysed five artificial 
+fluid inclusion - calcite systems in the range between 5.2 and7.8 µl/g and three at about 2.2 µl/g and 
+compared them to water-filled glass capillaries without additional calcite. The crushing of the compact 
+Iceland spar pieces provided fresh and fine-grained calcite for interaction and adsorption testing. The 
+measurements suggest that the adsorption of water molecules on the calcite surfaces does not affect 
+the measured isotopic signal in the investigated water/calcite ratio range (Fig. 6). Both measured 
+oxygen and hydrogen isotope values accurately match the expected value. With a standard deviation 
+
+of ± 0.05 ‰ for δ18O and ± 0.22 ‰ for δ2H values (high water/calcite ratio, n=5) and ± 0.15 ‰ for δ18O 
+and ± 0.31 ‰ for δ2H values (low water/calcite ratio, n=3) in both adsorption tests, a good 
+reproducibility of the individual measurements was achieved. We observed that after crushing of 
+Iceland spar (0.25 g) 0.023 μl water was adsorbed on the crushed calcite from the moist carrier gas 
+(Supplementary Fig. S3), which corresponds to a ratio of approximately 0.1 μl water per g calcite. Thus, 
+for low water contents of < 0.1 µl per g calcite an influence of adsorption on the released water amount 
+and the isotopic values probably cannot be excluded. Therefore, we rejected all fluid inclusion samples 
+with water amounts below 0.1 µl based on this observation (see Weißbach, 2020). 
+ 
+Figure 6: Isotopic values measured for the artificial inclusion calcite system, for which compact Iceland spar pieces 
+were crushed together with VE water - filled glass capillaries (triangles). Open circles indicate water - filled glass 
+capillaries (VE) without calcite addition, measured for comparison. An isotopic fractionation due to adsorption of 
+water molecules on the calcite surface is not detectable for the investigated water content range of 5.2-7.8 µl 
+water/g calcite (left side) and for 2.22±0.08 µl water/g calcite (right). Marginal differences in the isotope values 
+between left and right panel are within the expected variations in the reference water isotope values due to a 
+several year time lag between both experimental series. The uncertainties are displayed on the 1σ level. 
+[black/white for figures in print, colour online] 
+ 
+3.4 Isotopic effect of the water vapour background 
+
+5.2 - 7.8 μl/g
+~2.2 μl/g
+?H (% VSMOW)
+-58
+144
+-60
+444
+D
+-62
+(%。VSMOW)
+8.0
+丰
+-8.4
+0-8.8
+-9.2
+water plus
+only water
+waterplus
+only water
+iceland spar
+iceland sparThe potential influence of the isotope ratio of the water vapour background on that of the 
+measured sample could be relevant for speleothem samples whose isotopic composition strongly 
+differs from that of the water vapour background. For testing this potential effect, we injected our VE 
+water standard on four different water vapour backgrounds with different isotopic composition (Fig. 
+7). We used VE-water as injection fluid, because its isotopic composition is comparable to the majority 
+of fluid inclusion of speleothems from mid-latitudes. For each water vapour background 3.0 μl of VE 
+water were injected five times. For the background waters with the two most extreme isotope 
+compositions we additionally injected 1.0 µl of VE (n=6) to assess the robustness also for smaller water 
+amounts. .  
+If VE water is injected on VE background water vapour, the average isotope value corresponds to 
+the expected value within uncertainty. A deviation from the expected isotope value is notable for 
+injections on a different water vapour background. For example, VE injections on  a negative water 
+vapour background (NG, δ18O = -26.54 ± 0.08 ‰, δ2H = -212.1 ± 0.7 ‰, Table 1) yield a deviation of 
++0.40 ‰ for δ18O and +2.9 ‰ for δ2H values from the reference values. VE injections on a background, 
+which is based on lake water (WW)  with higher isotope values compared to VE water, yield deviations 
+of -0.15 ‰ for δ18O and -0.3 ‰ for δ2H values. Tests with injected water amounts of 1 µl corroborate 
+the observed trend (Fig.7). The standard deviation of repeated water injections is independent from 
+the isotopic composition of the water vapour background. The effect of the isotopic difference 
+between the samples and the background water vapour exceeds the measurement uncertainty only 
+for differences larger than 10 ‰ (δ18O). ). This experiment highlights that it is not necessary to correct 
+samples when using background water with an isotope composition close to the paleoclimate samples. 
+For speleothem measurements we used VE water as background water. 
+ 
+Figure 7: Deviation of the measured injection isotope value relative to the expected value. The deviation is related 
+to the difference between the isotope signal of the injection and that of the water vapour background. Single 
+injections are shown as open circles and the mean values as filled circles s. The green andblue lines indicate the 
+linear regression with all individual 3 µl measurements. An increasing deviation between the measured and 
+expected isotopic signal is observed for an increasing difference between injection isotope value and that of the 
+
+0.8
+oxygen
+hydrogen
+D
+3 μl
+1 μl
+0.6
+3 μl
+1 μl
+(0%)
+singleinjections
+O
+singleinjections
+-expected) (
+meanvaluewithuncertainty
+expected)(
+3
+mean value with uncertainty
+0.4
+linearregression
+Viation(measured
+0.2
+1
+0.0
+-0.2
+devi
+T:
+VE
+VE.
+injections
+injections
+-0.4
+uo
+on wWIAE
+VE
+cC
+CC
+NG
+AE
+ww
+VE
+NG
+-10
+-5
+0
+5
+10
+15
+20
+-50
+50
+100
+150
+200
+deviation (injection-background signal)(%o)
+deviation(injectionbackgroundsignal)(%o)water vapour background. The uncertainty of the expected value (black horizontal line) is shown as grey envelope. 
+Measurement uncertainties are given on the 1σ level. [black/white for figures in print, colour online] 
+ 
+3.5 Case applications 
+Case example 1: Modern – late Holocene sinter samples  
+The two fluid inclusion replicates of each modern or late Holocene sample from 
+Hüttenbläserschacht Cave reproduce very well and are within uncertainty of each other (Table 3). 
+Related standard deviations of the mean (0.2 and 1.6 ‰ for δ18O and δ2H values, respectively) are 
+comparable (δ18O values) or slightly larger (δ2H values)  than the measurement precision in this water 
+amount range (0.5-3.0 µl, Fig. 5). The mean fluid inclusion δ18O value of -7.6 ± 0.2 ‰ is identical to the 
+calculated drip-water value of Kluge et al. (2013) of -7.6 ± 0.3 ‰, independently confirming the former 
+finding. Drip water in Hüttenbläserschacht Cave was not monitored but should be close to the 
+neighbouring Bunker Cave and shares the same karst aquifer with comparable residence times of a 
+few years (e.g., Kluge et al., 2010). Fluid inclusion isotope values are close to the mean drip water 
+values from Bunker Cave of -7.9 ± 0.2 ‰ for δ18O and -53.3 ± 1.6 ‰ for δ2H values (Riechelmann et al., 
+2017). 
+Table 3: Measurement of fluid inclusions in three CaCO3 spar samples from Hüttenbläserschacht Cave 
+(Germany). Each sample was split in two to allow for a replication test. ‘Avg.’ refers to the average of the two 
+analyses. For comparison also the calculated pool water δ18O value of Kluge et al. (2013) is shown that uses an 
+independent temperature estimate and clumped isotope ∆47 for correction of kinetic isotope effects. The Bunker 
+Cave drip water is taken from Riechelmann et al. (2017). Uncertainties are given on the 1σ level. 
+ 
+ID 
+Sample weight 
+(g) 
+Water 
+amount     
+(µl) 
+Water 
+content 
+(µl/g) 
+δ18O  value     
+ (‰ VSMOW) 
+δ2H      value 
+ (‰ VSMOW) 
+Pond A 
+A-1 
+0.52 
+0.24 
+0.46 
+-7.5 ± 0.5 
+-53.2 ± 1.5 
+ 
+A-2 
+0.52 
+0.31 
+0.60 
+-7.9 ± 0.5 
+-51.9 ± 1.5 
+ 
+Avg. 
+- 
+- 
+- 
+-7.7 
+-52.5 
+Pond B 
+B-1 
+0.57 
+0.43 
+0.76 
+-7.6 ± 0.5 
+-49.7 ± 1.5 
+ 
+B-2 
+0.65 
+0.42 
+0.64 
+-7.8 ± 0.5 
+-48.6 ± 1.5 
+ 
+Avg. 
+- 
+- 
+- 
+-7.7 
+-49.1 
+Pond C (little 
+pond) 
+C-1 
+0.59 
+1.78 
+3.02 
+-7.3 ± 0.3 
+-51.1 ± 1.0 
+ 
+C-2 
+0.45 
+0.48 
+1.08 
+-7.7 ± 0.5 
+-51.0 ± 1.5 
+ 
+Avg. 
+ 
+ 
+ 
+-7.5 
+-51.0 
+Average all 
+ 
+ 
+ 
+ 
+-7.6 ± 0.2 
+-50.9 ± 1.6 
+
+Reconstructed 
+after Kluge et al. 
+(2013) 
+ 
+ 
+ 
+ 
+-7.6 ± 0.3 
+ 
+Drip 
+water 
+Bunker Cave  
+ 
+ 
+ 
+ 
+ 
+ 
+range 
+2006- 
+2013 
+ 
+ 
+ 
+ 
+-8.5 to -7.0 
+-48 to -58 
+mean 
+2006-
+2013 
+ 
+ 
+ 
+ 
+-7.9 ± 0.2 
+-53.3 ± 1.6 
+ 
+Case example 2: Speleothem sample from the 20th century – Stam 4 from Cloşani Cave 
+Comparison with current drip water and reproducibility assessment 
+We sampled calcite pieces at the outer surface of the stalagmite for comparison with current drip 
+water. It can be assumed that recent calcite precipitated there and accordingly, recent drip water is 
+enclosed in the fluid inclusions. The water yields during crushing were between 0.49 and 1.38 µl/g with 
+a mean of 0.93 ± 0.28 µl/g (one sample was excluded due to a low water amount of 0.18 µl) 
+(Supplementary Table S3). The mean value of 13 fluid inclusion measurements of samples from the 
+outer stalagmite layer is δ18O = −9.5 ± 0.5 ‰ and δ2H = −64.6 ± 1.2 ‰ (Supplementary Table S3). These 
+values agree within uncertainty with the mean of the related drip site CL3 of δ18O = −9.6 ± 0.2 ‰ and 
+δ2H = −66.3 ± 1.7 ‰ (Fig. 8). The 13 individual measurements reproduce with a standard deviation of 
+0.5 ‰ and 1.2 ‰ for δ18O and δ2H values, respectively, which is slightly higher than the analytical 
+uncertainty based on the standard deviation of repeated syringe injection for water amounts between 
+0.5-and 1.7 µl (0.2-0.4 ‰ for δ18O values, 0.4-1.1 ‰ for δ2H values, Fig. 5). The standard deviation for 
+the 13 individual speleothem analyses is also comparable to that of other CRDS systems and similar 
+water amount ranges, such as of Arienzo et al. (2013) with ± 0.5/2.0 ‰ for δ18O/δ2H values and Affolter 
+et al. (2014) with ± 0.5/1.5 ‰ for δ18O/δ2H values . The precision of the Stam4 sample analysis also 
+compares well with traditional IRMS measurement techniques which achieve a precision of ± 0.5 ‰ 
+for δ18O and ± 2.0 ‰ for δ2H values for water amounts > 0.2 µL (Dublyansky and Spötl, 2009). 
+ 
+
+ 
+Figure 8: Fluid inclusion water isotope ratios of samples from the outer stalagmite surface (light blue dots), with 
+corresponding mean value (dark blue). Drip water data from the same cave chamber where Stam 4 was removed 
+(light green triangles; drip site CL3) and its mean value (dark green) agree with the fluid inclusion results. Both drip 
+water and fluid inclusion data match the local meteoric water line (LMWL) of Cluj-Napoca of δ2H = 8.03 · δ18O + 
+11.29 ‰ (Cozma et al., 2017).  The uncertainties are given on the 1σ level. [black/white for figures in print, colour 
+online] 
+ 
+Fluid inclusion analysis of samples along the growth axis 
+We used the stalagmite pieces closest to the growth axis of Stam 4 for paleo-drip water and -
+temperature reconstruction (Fig. S2). Where possible, the reproducibility of the individual 
+measurements was tested with a second set of fluid inclusion samples, extracted  adjacent to the first 
+set of samples (Table 4). The second set had a larger distance from the growth axis than the first set. 
+The samples corresponding to the same growth period are grouped in levels, indicated by letters A-K 
+(Fig. 9). For sample level D, only the second sample is used because the first sample is close to the 
+applied water amount limit and contains only 0.18 μl. On average, the δ2H values of sample and 
+replicate are largely consistent (mean deviation: 0.1 ± 0.8 ‰). The same is observed for the inclusion 
+water δ18O value (mean deviation: 0.31 ± 0.51 ‰). In addition the water content of the different levels 
+appears characteristic. For level D and E with 5 replicates each, the water content varies only 0.1 µl/g 
+(excluding one sample each with low total water amount). For the other levels, a higher scatter has 
+
+LMWL
+FI-Stam4
+-60
+FImeanwithstdy
+dripwater (CL3)
+dripwatermeanwithstdv
+-62-
+8’H [% VSMOW]
+-64
+-66
+-68
+-70-
+-11.0
+-10.5
+-10.0
+-9.5
+-9.0
+-8.5
+8180 [%。 VSMOW]been observed, potentially due to a general heterogeneity of the speleothem inclusion distribution 
+(e.g., Muñoz-García et al., 2012). Generally, the water content was between 0.45 and 1.66 µl/g, 
+suggesting minimal or negligible influence of adsorption on the freshly crushed surface (Table 4). Fluid 
+inclusion δ18O values vary between -10.4 ‰ and -8.0 ‰ and, with one exception (level C), follow a 
+temporal trend towards higher values towards more recent times (Fig. 9, Table 4). 
+ 
+Table 4: Measurement results of fluid inclusion samples of stalagmite Stam-4 from Cloşani 
+Cave (Romania). Several samples were cut from individual layers that reflect contemporaneously 
+grown carbonate and allow for replication tests. The fractionation factor 18α(CaCO3-H2O) between 
+water and CaCO3 was calculated based on the difference between the calcite δ18O values (averaged 
+over the edge length of the fluid inclusion sample of typically 5 mm) and the fluid inclusion water δ18O 
+values. The temperature T18O,cc was determined using the 18α(CaCO3-H2O) - T relationship proposed by 
+Kim and O’Neil (1997). TH is related to the relative temperature change calculated using the δ2H-
+temperature relationship in rainfall (4.72‰/°C) and was referenced to top level K and the current cave 
+temperature. T18O, Fi refers to the temperature difference relative to sample level K with the modern 
+cave temperature as reference and was calculated using the δ18O-T relationship in rainfall (0.59‰/°C). 
+Samples in grey are not included in the interpretation and discussion as the water amount was 0.19 µl 
+or below. Samples closest to the growth axis (‘1’ closest, higher numbers are further away) were used 
+for temperature assessment based on the classical carbonate thermometer. Samples A1 and A2 were 
+the oldest samples and were excluded from the discussion as they belong to the stalagmite base with 
+unclear chronology. The age corresponds to the mean age of each sample level. Dft: distance from top. 
+ID 
+Dft 
+(mm) 
+Age    
+(year AD) 
+Sample 
+weight 
+(g) 
+Water 
+amount     
+(µl) 
+Water 
+content 
+(µl/g) 
+δ2H value     
+ (‰ 
+VSMOW) 
+δ18O value      
+(‰ 
+VSMOW) 
+18α 
+(CaCO3-
+H2O) (‰) 
+T18O,cc 
+(°C) 
+T18O,Fi (°C) 
+TH (°C) 
+A1 
+ 
+unknown 
+0.58 
+0.30 
+0.52 
+-65.4 ± 1.5 
+-9.5 ± 0.5 
+ 
+ 
+ 
+ 
+A2 
+ 
+ 
+0.49 
+0.29 
+0.59 
+-59.7 ± 1.5 
+-9.8 ± 0.5 
+ 
+ 
+ 
+ 
+B1 
+48.2 
+1928 
+0.42 
+0.40 
+0.95 
+-64.1 ± 1.5 
+-9.6 ± 0.5 
+31.8 ± 0.5 
+7.7 
+± 
+2.2 
+9.4 ± 0.2 
+10.4 ± 0.5 
+B2 
+ 
+ 
+0.49 
+0.37 
+0.75 
+-64.7 ± 1.5 
+-9.6 ± 0.5 
+ 
+ 
+ 
+ 
+B3 
+ 
+ 
+0.53 
+0.40 
+0.76 
+-64.8 ± 1.5 
+-8.9 ± 0.5 
+ 
+ 
+ 
+ 
+B4 
+ 
+ 
+0.52 
+0.29 
+0.56 
+-66.3 ± 1.5 
+-9.1 ± 0.5 
+ 
+ 
+ 
+ 
+B5 
+ 
+ 
+0.42 
+0.19 
+0.45 
+-68.5 ± 1.5 
+-9.4 ± 0.5 
+ 
+ 
+ 
+ 
+C1 
+43.9 
+1937 
+0.49 
+0.81 
+1.66 
+-57.6 ± 1.0 
+-8.0 ± 0.3 
+30.7 ± 0.5 
+12.7 
+± 
+2.3 
+12.1 ± 0.1 
+11.8 ± 0.4 
+C2 
+ 
+ 
+0.42 
+0.51 
+1.21 
+-59.6 ± 1.0 
+-8.5 ± 0.3 
+ 
+ 
+ 
+ 
+C3 
+ 
+ 
+0.54 
+0.44 
+0.81 
+-60.4 ± 1.5 
+-9.0 ± 0.5 
+ 
+ 
+ 
+ 
+D1 
+ 
+ 
+0.32 
+0.18 
+0.57 
+-63.8 ± 1.5 
+-8.5 ± 0.5 
+ 
+ 
+ 
+ 
+D2 
+39.0 
+1944 
+0.51 
+0.42 
+0.83 
+-63.8 ± 1.5 
+-10.0 ± 0.5 
+32.6 ± 0.5 
+4.3 
+± 
+2.1 
+8.7 ± 0.2 
+10.5 ± 0.5 
+D3 
+ 
+ 
+0.56 
+0.50 
+0.89 
+-63.7 ± 1.5 
+-10.4 ± 0.5 
+ 
+ 
+ 
+ 
+
+D4 
+ 
+ 
+0.55 
+0.54 
+0.98 
+-64.0 ± 1.5 
+-10.1 ± 0.5 
+ 
+ 
+ 
+ 
+D5 
+ 
+ 
+0.40 
+0.35 
+0.88 
+-61.7 ± 1.5 
+-9.0 ± 0.5 
+ 
+ 
+ 
+ 
+E1 
+34.5 
+1952 
+0.49 
+0.38 
+0.78 
+-62.9 ± 1.5 
+-10.3 ± 0.5 
+32.7 ± 0.5 
+3.5 
+± 
+2.1 
+8.2 ± 0.2 
+10.7 ± 0.5 
+E2 
+ 
+ 
+0.58 
+0.46 
+0.78 
+-62.5 ± 1.5 
+-9.4 ± 0.5 
+ 
+ 
+ 
+ 
+E3 
+ 
+ 
+0.54 
+0.44 
+0.82 
+-62.8 ± 1.5 
+-10.4 ± 0.5 
+ 
+ 
+ 
+ 
+E4 
+ 
+ 
+0.53 
+0.32 
+0.60 
+-63.1 ± 1.5 
+-10.0 ± 0.5 
+ 
+ 
+ 
+ 
+E5 
+ 
+ 
+0.25 
+0.14 
+0.55 
+-59.4 ± 1.5 
+-9.0 ± 0.5 
+ 
+ 
+ 
+ 
+F1 
+30.1 
+1960 
+0.47 
+0.35 
+0.75 
+-63.5 ± 1.5 
+-9.6 ± 0.5 
+31.9 ± 0.5 
+7.0 
+± 
+2.2 
+9.4 ± 0.2 
+10.5 ± 0.5 
+F2 
+ 
+ 
+0.58 
+0.82 
+1.4 
+-63.2 ± 1.0 
+-9.4 ± 0.3 
+ 
+ 
+ 
+ 
+F3 
+ 
+ 
+0.56 
+0.87 
+1.56 
+-59.8 ± 1.0 
+-8.3 ± 0.3 
+ 
+ 
+ 
+ 
+G1 
+26.2 
+1968 
+0.46 
+0.40 
+0.87 
+-62.0 ± 1.5 
+-9.0 ± 0.5 
+31.3 ± 0.5 
+9.7 
+± 
+2.2 
+10.4 ± 0.2 
+10.9 ± 0.5 
+G2 
+ 
+ 
+0.50 
+0.76 
+1.53 
+-61.8 ± 1.0 
+-8.9 ± 0.3 
+ 
+ 
+ 
+ 
+H1 
+22.1 
+1977 
+0.40 
+0.46 
+1.16 
+-61.6 ± 1.0 
+-9.1 ± 0.3 
+31.9 ± 0.5 
+7.2 
+± 
+2.2 
+10.2 ± 0.1 
+10.9 ± 0.4 
+H2 
+ 
+ 
+0.35 
+0.46 
+1.30 
+-61.0 ± 1.0 
+-8.8 ± 0.3 
+ 
+ 
+ 
+ 
+I 
+17.8 
+1990 
+0.39 
+0.28 
+0.72 
+-60.3 ± 1.5 
+-9.3 ± 0.5 
+32.0 ± 0.5 
+6.8 
+± 
+2.2 
+9.9 ± 0.2 
+11.2 ± 0.5 
+J 
+10.9 
+2004 
+0.50 
+0.43 
+0.86 
+-59.2 ± 1.5 
+-8.7 ± 0.5 
+31.3 ± 0.5 
+9.9 
+± 
+2.2 
+10.9 ± 0.2 
+11.4 ± 0.5 
+K 
+3.9 
+2008 
+0.46 
+0.43 
+0.94 
+-59.4 ± 1.5 
+-8.4 ± 0.5 
+30.7 ± 0.5 
+12.3 
+± 
+2.3 
+11.4 ± 0.2 
+11.4 ± 0.5 
+ 
+4. Discussion 
+4.1 Constraints for precise and accurate fluid inclusion isotope data 
+The presented setup allows for a good reproducibility with respect to isotope measurements 
+of pure water samples in the µl range, either injected via a syringe or by crushing of water-filled glass 
+capillaries in the copper tube (similar to speleothems samples; section 3.3). The achievable precision 
+is 0.4-0.5 ‰ for δ18O and 1.1-1.9 ‰ for δ2H analyses at extracted water amounts between 0.1 µl and 
+0.5 µl and decreases with increasing water amount to ± 0.1-0.3 ‰ for δ18O and ± 0.2-0.7 ‰ for δ2H 
+measurements at extracted water amounts >1 µl (Fig. 5). The improved precision with increasing water 
+amount is consistent with the observations of Dassié et al. (2018) who reported similar precision of 
+0.2-0.3 ‰ for δ18O and 0.6-2.6 ‰ for δ2H values for 0.2-1 µl as well as a strong increase of the 
+uncertainty at water amounts lower than 0.1 µL. Replicate analyses of calcite samples from the 
+outermost surface of a Romanian stalagmite corroborate the precision as determined by crushing of 
+water-filled glass capillaries and water injections.   
+Adsorption of water on freshly crushed surfaces appears negligible for water contents of about 
+1 µl water per g calcite or above Dennis et al. (2001) similarly observed a decreasing adsorption 
+influence at increasing H2O/CaCO3 ratios at room temperature. However, an adsorption effect could 
+
+be relevant if the water content in the crushed samples approaches 0.1 µl/g or is below this value. We 
+therefore recommend to use the water content as one parameter to check the robustness of the 
+analysis and to carefully assess or conservatively reject samples with water contents below 0.1 µl/g. 
+We observed a small dependence of the measured isotope value on the water vapour 
+background (Fig. 7).. After injection of a certain water amount, the δ18O value of the (hypothetically) 
+well mixed water vapour consisting of background and injection water is an amount-weighted mixture 
+of both δ18O values. For background water with relatively depleted values such as North Greenland 
+Water (NG, -δ18O =-26.5 ‰ and δ2H = -212.1 ‰) this would mean that the δ18O value of the VE water 
+with δ18O = -8.57 ‰ and δ2H = -61.0 ‰ is higher than the mixed water. For example, if the background 
+to injection volume is 1.8:3.0, the isotopic composition of the mixture is expected to be δ18O = -15.3 
+‰ and δ2H = -117.7 ‰.Given the short residence time of the water vapour in the mixing cavity before 
+the measurement in the CRDS, a full isotopic mixing is not reached. The kinetically slower molecules 
+containing an 18O atom remain preferentially in the gas stream compared  to the faster molecules 
+containing only 16O atoms that preferentially  take part in the mixing with the background water. Thus, 
+for this case example it is expected that the injection water isotopes are slightly higher relative to the 
+background and the mixed signal. Conversely, for a positive background as the WW water, the isotope 
+value of the VE injection is more negative relative to the isotope value of the hypothetical fully mixed 
+gas stream. Due to the kinetic behaviour of 18O, the injection stays more negative relative to the 
+expected value for this background. The adsorption effects and the influence of kinetic isotope 
+exchange are similar for the 1 µl and 3 µl injections (Fig. 7).  
+ For water amounts in the µl range this dependence on the vapour background isotope value 
+is relevant if the isotopic composition of the fluid inclusions is significantly different from the 
+background (> 10 ‰ for δ18O and > 50 ‰ for δ2H). Otherwise, the potential effect of the isotopic 
+difference to the background water vapour is within the analytical uncertainty of water samples 
+between 0.1 and 1.0 µl. The maximum expected deviations are < 0.25 ‰ for δ18O and < 1.0 ‰ for δ2H 
+values, if the sample is within the 10 ‰ range of the water vapour background for δ18O and 50 ‰ for 
+δ2H values. For water amounts larger than 1 µl the acceptable deviation between sample and 
+background water isotope values reduces in relation to the higher measurement precision at higher 
+water amounts (Fig. 5). 
+ 
+4.2 Paleotemperature calculation from Stam 4 using fluid inclusion isotopes 
+For calculation of the CaCO3-H2O isotope fractionation, we averaged the calcite δ18O values 
+which correspond to the growth period of the spatially larger fluid inclusion sample (Fig. S4). The 
+calculated fractionation factor 18α(CaCO3-H2O) between calcite and fluid inclusion water yields values 
+between 30.7 and 32.9 ‰. This range would correspond to temperatures between 3.5 ± 1.5 °C and 
+
+12.5 ± 1.5 °C using the 18α(CaCO3-H2O)-T relationship of Kim and O’Neil (1997) (Table 4). The calculated 
+absolute temperatures deviate slightly from these values depending on the used 18α(CaCO3-H2O)-T 
+relationship (e.g., Démeny et al., 2010; Tremaine et al., 2011). However, relative differences between 
+the coldest and warmest periods and the trend in the data set is largely independent of the selected 
+fractionation-temperature relationship as most experimental and empirical studies yield similar 
+18α(CaCO3-H2O)-T slopes. Following an apparent change of 2.2 ‰ in 18α(CaCO3-H2O) a temperature 
+change of about 9°C would formally correspond to the growth period of the stalagmite. This 
+temperature difference is much larger compared to that observed at local meteorological stations with 
+maximum and minimum mean annual air temperature differing by approximately 3°C. This discrepancy 
+suggests that the temperature trend related to 18α(CaCO3-H2O) in the stalagmite has been enhanced, 
+e.g., by stronger isotopic disequilibrium. As the measured fluid inclusion water isotopes correspond to 
+the meteoric water line (Fig. S5), post-depositional and other significantly altering effects are unlikely 
+for the water-filled inclusions. However, mineral formation in speleothems often takes place in a non-
+equilibrium regime (Deininger et al., 2021) and may also have influenced the calcite δ18O values of 
+Stam 4 due to a high growth rate and strong seasonal variations in prior calcite precipitation (PCP, 
+Warken et al., 2018).. We refrain from correcting the disequilibrium effect in calcite δ18O values due 
+to the related large and hardly quantifiable uncertainties and only focus on the fluid inclusion δ2H 
+values in the following. Note, that it may be possible in other cases to derive temperature variations 
+from the oxygen isotope fractionation between fluid and calcite if the degree of PCP is negligible or 
+constant and the length of drip interval has not changed significantly during growth. 
+Affolter et al. (2019) demonstrated that δ2H values and its temperature relationship in 
+rainwater of mid-latitudes can be used to deduce temperature changes throughout the Holocene. In 
+stalagmite Stam 4, a long-term trend towards higher δ2H values is observed from the oldest to 
+youngest fluid inclusion samples (Fig. 9). A significant increase for δ2H values of +4.8 ± 2.1 ‰ was 
+identified between sample level F and K and similarly between B and C (Fig. 9 C). This transfers into to 
+a temperature change of +1.0 ± 0.4 °C using the relationship between the isotopic composition of 
+precipitation and temperature for Central Europe of +0.59 ± 0.04 ‰/°C for oxygen and +4.72 ± 
+0.32‰/°C for hydrogen isotopes (Rozanski et al., 1992). Since stalagmite Stam 4 from Cloşani Cave 
+grew under continental climatic influence, the mean value for Central Europe seems to be the best 
+reference for the determination of the relative temperature change with the δ2H/T relationship. GNIP 
+(Global Network of Isotopes in Precipitation) stations and other weather stations in Hungary, Austria, 
+Slovakia and Poland with more than 10 years of isotope analysis show similar slopes of +3.9-5.4 ‰/°C 
+(Demény et al., 2021). Considering the observed range of the rainfall δ2H-temperature slopes in Central 
+and Eastern Europe by Gaussian error propagation, the uncertainty increases slightly to 0.5°C.  
+
+ With the confirmed recent growth of the stalagmite, the topmost stalagmite piece is assigned 
+to the year 2010 C.E.( year of stalagmite removal). Annual growth layers provide a possibility to assign 
+ages to all other sample depths (Supplementary Fig. S2). Temperature changes ΔT relative to the 
+reference level B is close to zero up to ca. 1960 C.E. (3 cm distance from top, level F), followed by an 
+increase of 1.0 ± 0.4 °C at the stalagmite top (Fig. 9F). The mean annual air temperature for the time 
+period from 1928 to 2008 C.E. at the meteorological station Drobeta/Turnu Severin, which is located 
+in the vicinity of the cave, shows a similar temperature increase of about 1 °C from 1980 until 2008 
+C.E. (Fig. 9G). This is consistent with the general trend in Romania, which experienced a 0.8 °C increase 
+for the period of 1901-2012 C.E. (Ministry of Environment and Climate Change, 2013). The 
+temperature change determined from δ2H values in the fluid inclusions corresponds well to the trend 
+and magnitude measured in the mean annual air temperature of the region (Fig. 9). Directly 
+interpreting the fluid inclusion δ18O values using the rainfall δ18O-T relationship for Central Europe of 
++0.59 ± 0.04 ‰/°C by Rozanski et al. (1992) also leads to a temperature increase, albeit with a higher 
+amplitude of 2.0 ± 1.1°C relative to level B, but within uncertainty consistent with the temperature 
+reconstruction using δ2H values (Table 4). 
+
+ 
+
+growthaxis[cm]
+5
+A
+B
+C
+D
+E
+Base
+level
+A
+B
+c
+D
+E
+F
+G
+H
+K
+a (% VSMOW)
+T
+-8
+B
+II
+-9
+-10
+-11°H (% VSMOW)
+-60
+65
+(0%)
+33
+D
+32
+30
+(82H) (%)
+E
+reference
+0
+TevelB
+2
+reference13
+(C)
+MAAT
+11
+10
+1920
+1940
+1960
+1980
+2000
+yearFigure 9: A) Stam 4 with assignment of sample pieces based on visual correlation with the growth axis. Layers B to 
+K were used for temperature reconstruction. The small inset shows alternation of fluid inclusion-rich and inclusion-
+poor layers. Winter layers yield very little inclusions, while summer layers include abundant air- and water-filled 
+inclusions. Width of the image is ca. 3 mm. B) Fluid inclusion δ18O values C) δ2H values D) Fractionation factor α 
+between calcite and inclusion water. E) Change in δ2H values relative to level B (lowest temperature). F) Inferred 
+temperature change relative to level B. A trend is visible for Δ(δ2H) as well as for ΔT from the stalagmite bottom 
+to the top. Using the δ2H/T relationship of 4.72 ± 0.32 ‰/T (Rozanski et al., 1992) a total increase of ΔT =1.0 ± 0.5 
+°C is observed within the growth period of the investigated stalagmite. G) Mean annual air temperature (MAAT) 
+of the Drobeta/Turnu Severin station in the cave region (thin black line, Klein Tank et al., 2002) for the last 100 
+years with a 10-year running mean (red line) . The 10 year-smoothing interval corresponds to the average age that 
+is covered by the fluid inclusion samples. The uncertainties are given on the 1σ level. [black/white for figures in 
+print, colour online] 
+ 
+4.3 Paleotemperature reconstruction using fluid inclusions  
+Our study supports the conclusion of previous publications (e.g., Affolter et al., 2014; Uemura 
+et al., 2016; de Graaf et al., 2020) that an accurate and precise determination of the isotope 
+composition of micro-litre water amounts is possible. Our setup is able to produce small errors, which 
+are in the same range as the precision in the previous fluid inclusion isotope studies(Dublyansky and 
+Spötl,2009; Arienzo et al., 2013); Affolter et al., 2014),; Uemura et al. 2016; Dassié et al.,2018). In these 
+studies a precision of 0.3-0.5 ‰ for δ18O and 0.7-1.9 ‰ for δ2H values in the water amount range of 
+0.1-1.0 µl, and 0.1-0.3 ‰ for δ18O and 0.2-0.7 ‰ for δ2H values at water amounts > 1 µl was 
+demonstrated. The analytical precision determines the currently achievable temperature precision.  
+In principle, three possible ways of temperature calculation from fluid inclusion isotopes exist: 
+a) from the temperature-dependent oxygen isotope fractionation between calcite and fluid inclusion 
+water (e.g., Arienzo et al., 2015; Labuhn et al., 2015), b) indirectly via transfer of the fluid inclusion δ2H 
+value to the corresponding water δ18O value using the δ18O-δ2H relationship of the meteoric water line 
+and then using the oxygen isotope fractionation between carbonate and water for temperature 
+calculation (Zhang et al., 2008; Meckler et al., 2015), and c) from the hydrogen isotopes using a locally 
+valid  δ2H-temperature relationship of the rainfall (e.g., Affolter et al., 2019). Of the three methods for 
+temperature reconstruction the first two (a and b) show the highest uncertainty of 0.6-3.1 °C (Van 
+Breukelen et al., 2008; Zhang et al., 2008; Arienzo et al., 2015; Meckler et al., 2015; Uemura et al., 
+2016). The highest achievable temperature precision in the case of the best analytical fluid inclusion 
+δ18O precision of 0.1-0.2 ‰ (and a calcite δ18O uncertainty <0.1 ‰) is 0.6-1.1 °C. Approaches a and b 
+are additionally affected by the potential influence of disequilibrium isotope fractionation during 
+carbonate mineral formation (e.g., Deininger et al., 2021), causing too high temperatures or an 
+unrealistically large temperature spread in case of significant changes of isotopic disequilibrium.  
+
+Furthermore, diagenetic exchange between host calcite and fluid inclusion water could further alter 
+the water δ18O value (Demeny et al., 2016; Uemura et al., 2020). The precision of the temperature 
+reconstruction directly from fluid inclusion δ2H values depends critically on the value of the rainfall 
+δ2H/T relationship and the availability of well-defined rainfall δ2H/T functions at the study site. For the 
+Central European region a value of 4.72 ± 0.32 ‰/°C of Rozanski et al. (1992) can be used and can yield  
+a temperature precision of 0.2 °C for released water amounts of ~0.5 µl if the  analytical precision is  
+~1.0 ‰ for δ2H measurements. The uncertainty of the rainfall δ2H/T function is negligible for our case 
+study but could be relevant in case of a reduced temperature dependence of the rainfall δ2H values. 
+At locations with a stronger temperature dependence of the rainfall δ2H value an even better precision 
+is possible, e.g., ± 0.13 °C for the average of Swiss stations, which show a slope of 7.44 ‰/°C (Rozanski 
+et al., 1992) and for the typical analytical uncertainty of our setup. 
+ The temperature resolution of this method is slightly reduced at lower latitudes (e.g., ± 0.55 
+°C at Hong Kong with a δ2H rainfall-temperature relationship of 2 ‰/°C; Rozanski et al., 1992). Note 
+that temperature estimates using fluid inclusion δ2H values and the rainfall δ2H/T relationship without 
+climatic reference points are relative, i.e., they record only temperature changes. With an anchor, e.g., 
+modern reference temperature and rainfall δ2H values, absolute temperatures can be also inferred 
+from fluid inclusion δ2H values. The application of the rainfall δ2H/T relationship for calculating 
+temperature changes from fluid inclusion δ2H values also requires the δ2H/T relationship to be 
+constrained for the past. Information on the rainfall isotope systematic and the δ2H/T relationship in 
+the past can be gained for example from groundwater studies (Darling, 2004) in combination with 
+noble gas temperatures (e.g., Kreuzer et al., 2009; Varsány et al., 2011, Túri et al., 2020). The 
+uncertainty of the δ2H/T relationship needs to be considered and likely decreases the achievable 
+precision for pre-Holocene speleothems as the uncertainty for the δ2H/T relationship increases when 
+applying the modern or Holocene relationship back in time. 
+Affolter et al. (2019) used the δ2H/T relationship for temperature reconstruction from fluid 
+inclusions throughout the Holocene and achieved a precision of 0.2-0.5 °C for a Swiss stalagmite. Our 
+analytical approach allows for the same temperature resolution and with measurements of stalagmite 
+Stam 4 from Romania confidently verified the recent 20th century warming. Both studies together 
+illustrate the potential of the inclusion-based methodology for tracing and reconstructing minor 
+temperature fluctuations of < 2 °C during the Holocene and, at sufficient temporal resolution (requiring 
+high stalagmite growth rates), also of sub-degree changes such as the recent anthropogenic warming 
+trend. 
+ 
+5. Summary and conclusion 
+
+Fluid inclusion isotope analysis using CRDS measurements after mechanical sample crushing 
+benefits from fluid extraction and measurement under a constant and controlled water vapour 
+background. The specific isotope and water volume calibration of the CRDS system remained valid for 
+several years. For assessing the fluid inclusion extraction and measurement performance we used 
+syringe injections and boro-silicate glass capillaries filled with reference water. We have shown that 
+out setup has no drift in the isotope values for smaller water amounts and that the memory effect for 
+this system is negligible when using an isotopically appropriate background water vapour. The water 
+vapour background should be chosen such that the isotope values of sample and background do not 
+deviate significantly (maximum 10 ‰ for δ18O and 50 ‰ for δ2H values). 
+Direct comparison of calcite powder-free and -filled extraction tubes proved that the adsorption of 
+water on the speleothem surface has no effect on the measured isotope signal if the water content is 
+larger than 1 µl water per g calcite. For samples with a water content below 0.1 µl/g calcite results 
+have to be checked as we observed a corresponding adsorption of the water vapour background on 
+freshly crushed calcite. Related to the above-mentioned constraints, the precision (1σ) of isotope 
+measurements for aliquots of water from speleothem fluid inclusions improves with increasing water 
+amount. It is 0.4-0.5 ‰ for δ18O and 1.1-1.9 ‰ for δ2H values for water samples between0.1 and0.5 
+µl, which is comparable to other CRDS systems and IRMS techniques. This value was further confirmed 
+by replicated measurements of adjacent samples of the Romanian stalagmite Stam 4 (standard 
+deviation of  0.5 and 1.2 ‰ for δ18O andδ2H values). For water amounts larger than1 µl the precision 
+improves to 0.1-0.3 ‰ for δ18O and 0.2-0.7 ‰ for δ2H. 
+Analysis of fluid inclusions of recent pool spars from a German cave shows good agreement 
+between drip water and fluid inclusion isotope values. Similarly, the δ18O and δ2H values of a Romanian 
+stalagmite, grown during the 20th century, reflect the isotopic composition of the modern drip water 
+within uncertainty. In the same case study, we observed a T-trend from δ18O values, which is 
+inconsistent with local weather records, suggesting a major influence disequilibrium and kinetic effects 
+on the speleothem calcite δ18O signal of Stam 4. The isotopic disequilibrium causes a significant 
+overestimation of the temperature changes calculated from the oxygen isotope fractionation between 
+calcite and water (in our case 9 °C difference instead of ca. 1°C). In contrast, hydrogen isotopes are not 
+involved in calcite precipitation and therefore provide a relatively undisturbed link to the stable 
+isotopic composition of drip and rain water. Using the δ2H-temperature relationship in rainfall we 
+obtained a temperature increase for Cloşani Cave of +1.0 ± 0.5 °C between 1960 and 2010, which is in 
+excellent agreement with the local temperature record. Thus, applying the local rainfall δ2H--
+temperature relation on fluid inclusion δ2H variations appears to be a reliable method to determine 
+mean annual air temperatures for mid-latitude speleothems. The achieved precision furthermore 
+highlights the potential of fluid inclusion isotope studies in speleothems for high resolution 
+
+paleoclimate reconstruction, given that the rainfall isotope relationship is significantly linked to 
+temperature and is available for the studied area and valid for past periods. 
+
+Acknowledgements 
+ 
+The project was funded by DFG Grant KL 2391/2-1 and supported by the Heidelberg 
+Graduate School for Physics in the context of grant GSC 129. We thank Sylvia Riechelmann and 
+Jasper Wassenburg for collection of Stam 4, Silviu Constantin and Mihai Terente for 
+monitoring, Christoph Spötl for drip water analysis at CL3, Regina Mertz-Kraus for LA-ICP-MS 
+element analysis and Sven Brömme for calcite δ18O and δ13C analysis on Stam 4. We thank the 
+editor Michael E. Böttcher and three anonymous reviewers for their very detailed comments 
+and suggestions that helped to improve the manuscript. 
+Data availability 
+ 
+Data of this study are summarized in Tables 1-4 and Supplementary Table S1-S3. Raw 
+data related to Figs. 3-6 are given in the Appendix of Weißbach (2020), available at 
+https://doi.org/10.11588/heidok.00028559 
+Competing interests statement 
+ 
+The authors declare the absence of competing interests. 
+
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+palaeoenvironmental research, pp. 185–225, Springer. 
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+geochemical proxy data of speleothems from this site.  Journal of Hydrology, 409, 682–695. 
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+system. Geochimica et Cosmochimica Acta, 75, 4929–4950. 
+Túri, M., Saadi, R., Marah, H., Temovski, M., Molnár, M., Palcsu, L., 2020. Paleotemperature 
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+Cosmochim. Acta, 172, 159–176. 
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+oxygen and hydrogen stable isotope measurements of water by isotope ratio infrared 
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+https://doi.org/10.4319/lom.2012.10.1024 
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+266-81. 
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+Lett., 499, 122–133. 
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+ Frank, N., 2021. Last glacial millennial-scale hydro-climate and temperature changes in 
+
+Puerto Rico constrained by speleothem fluid inclusion δ18O and δ2H values. Clim. Past, 18, 
+167-181., https://doi.org/10.5194/cp-18-167-2022, 2022.  
+Weißbach, T., 2020. Spectroscopic isotope ratio analysis on speleothem fluid inclusions - analytics and 
+paleoclimatic case studies. PhD thesis, Heidelberg University, 225p. 
+https://doi.org/10.11588/heidok.00028559 
+Wilcox, P., Honiat, C., Trüssel, M., Edwards, R. L., Spötl, C., 2020. Exceptional warmth and climate 
+instability occurred in the European Alps during the Last Interglacial period. Commun. Earth 
+Environ., 1: 57, https://doi.org/10.1038/s43247-020-00063-w 
+Yonge, C. J., 1982. Stable isotope studies of water extracted from speleothems. PhD thesis, McMaster 
+University. 
+Zhang, R., Schwarcz, H.P., Ford, D.C., Schroeder, F.S., Beddows, P.A., 2008. An absolute 
+paleotemperature record from 10 to 6 Ka inferred from fluid inclusion D/H ratios of a 
+stalagmite from Vancouver Island, British Columbia, Canada. Geochim. Cosmochim. Ac., 72, 
+1014-1026. 
+
+Supplementary Material 
+for  
+Constraints for precise and accurate fluid inclusion stable isotope 
+analysis using water-vapour saturated CRDS techniques  
+by 
+Therese Weissbach, Tobias Kluge, Stéphane Affolter, Markus C. Leuenberger, Hubert Vonhof, Dana 
+F.C. Riechelmann, Jens Fohlmeister, Marie-Christin Juhl, Benedikt Hemmer, Yao Wu, Sophie Warken, 
+Martina Schmidt, Norbert Frank, Werner Aeschbach 
+ 
+S1) Hüttenbläserschacht Cave – sample dating 
+Table S1: Pieces of speleothem samples (pool spars and rafts) collected at Hüttenbläserschacht Cave 
+were dated using the Th-U disequilibrium method at the Heidelberg Academy of Sciences. The 
+analytical procedure followed the methods described in Fohlmeister et al. (2012). 
+ 
+Sample 
+232Th 
+[ppb] 
+238U 
+[ppb] 
+230Th 
+[fg/g] 
+± 
+(234U/238U) 
+± 
+(230Th/238U) 
+± 
+Age 
+Age  
+  
+  
+  
+  
+  
+  
+  
+  
+  
+Corr. 
+uncor. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Hinterm 
+Ballsaal 
+13.0 382.1 
+58.8 6.7 
+1.1407 0.0030 
+0.0005 0.0011 
+0.04±0.26 
+0.90±0.08 
+Kristall-
+häutchen 
+3.1 321.9 
+34.3 2.5 
+1.2159 0.0053 
+0.0040 0.0005 0.3557±0.13 0.5862±0.04 
+ 
+S2) Closani Cave and Stam 4 annual layer counting 
+Cloşani Cave is located on the southern slope of the Carpathians at an altitude of 433 m above sea 
+level and developed in massive limestones of Upper Jurassic-Aptian age (Constantin and Lauritzen, 
+1999). The cave is overlain by about 30 m of rock overburden. A monitoring programme showed 
+microclimatic stability for the cave interior with a mean air temperature of 11.4 ± 0.5 °C and a relative 
+humidity close to 100% for 2010-2012 and 2015 (Warken et al., 2018). The cave air pCO2 pattern 
+follows a strong seasonal cycle with high values in late summer (up to 8000 ppmV) and lower values 
+during winter (2000 ppmV). Water infiltration occurs predominantly during winter time (October - 
+March) where 75 to 100% of the meteoric precipitation is available for infiltration. Warken et al. (2018) 
+showed that calcite precipitation is favoured during winter time and reduced in summer, as a result of 
+seasonally varying CO2 concentrations in the cave air and related equilibrium DIC concentrations. The 
+water isotopic composition of the drip water in direct vicinity (1 m) of the former location of Stam 4 
+shows no seasonal cycle and is constant with a mean value of −9.6 ± 0.2‰ for δ18O and −66.3 ± 1.7‰ 
+for δ2H. 
+
+The relatively small and fast-grown stalagmite Stam 4 was collected from the “laboratory passage” in 
+the cave in 2010. It has a total length of 6 cm and an average growth rate of 510 μm per year, as 
+deduced from counting of elemental layers. Both summer and winter layers are clearly detectable in 
+the thin sections, whereas winter layers show a compact structure with a lower number of inclusions 
+and the milky-white porous summer layers contain abundant air- and water-filled inclusions. This 
+layering in Stam 4 was induced by the strongly changing pCO2 in the cave air, resulting in a seasonal 
+change in growth rate and corresponding seasonal cycles in Sr and Ba in the stalagmite calcite. Similar 
+seasonal Sr and Ba pattern have also been observed e.g., by Treble et al. (2003), Mattey et al. (2010), 
+and Warken et al. (2018). The visible annual layers in stalagmite Stam 4 are not as pronounced as the 
+annual cycles in the measured high-resolution Ba concentration. Ba concentration was measured with 
+a LA-ICP-MS (Agilent 7500 ce with Laser UP-213, Institute of Geosciences Mainz) at 4.3 µm resolution. 
+The minima of this record were counted five times. These five counted layer series were cross-dated 
+to each other. Layers have to be counted at minimum three times, layers only counted once or twice 
+were deleted from the time series. For each layer a mean value of layer thickness was calculated from 
+the five layer thickness series to a master chronology. This layer thickness chronology results in a 
+growth of Stam 4 from 1910 to 2010, the year of sampling under an active drip site. 
+S3) Radiocarbon dating 
+Four samples were drilled with a hand-held dental burr (1 mm). Calcite powder was acidified in vacuum 
+with HCl. The emerging CO2 was combusted to C with H2 and an iron catalyst at 575°C (Fohlmeister et 
+al., 2011). Measurements were performed with a MICADAS AMS system (Synal et al., 2007) in the 
+Klaus-Tschira laboratory Mannheim. The results for the four samples show a typical speleothem 
+radiocarbon bomb spike (Tab. S1, Fig. S1), constraining recent growth of the speleothem. 
+Table S2: Radiocarbon measurement results. Radiocarbon results and errors are expressed in fraction 
+modern (fm). 
+MAMS lab nr. 
+depth [mm] 
+14C [fm] 
+14C error [fm] 
+14709 0.5 
+1.0416 
+0.0029 
+14710 18.7 
+1.0730 
+0.0029 
+14711 39 
+0.9265 
+0.0025 
+14712 55 
+0.9133 
+0.0024 
+ 
+
+ 
+Fig. S1: Radiocarbon measurements (black) over depth (bottom-axis), plotted to fit the atmospheric 
+radiocarbon anomaly (blue, top x-axis) in the mid to late 20th century.  
+ 
+
+Year [A.D.]
+1900
+1920
+1940
+1960
+1980
+2000
+180
+180
+160.
+160
+140-
+140
+120
+120
+100
+100
+80
+80
+1900
+1920
+1940
+1960
+1980
+2000
+distancefromtop[mm]Additional figures 
+ 
+Fig. S2: Age assignment of the fluid inclusion samples 
+A) Fluid inclusion sample pieces (labelled B to K) are shown on the left half of the stalagmite slab. The 
+red lines illustrate the assignment of the individual sample blocks to the growth axis. The visible 
+lamination was used as guideline for correlation. Sample A is related to the base and due to a disturbed 
+growth structure does not allow to assign any age. Due to the intrinsic uncertainties of this procedure 
+(for details see Weißbach, 2020) we associated age ranges to the individual fluid inclusion samples B 
+to K. 
+B) Age depth model with distance from top (dft) in cm. The chronology was established by layer 
+counting and additional 14C measurements (see S1 and S2). 
+ 
+
+2025
+B
+2010-
+1995-
+2
+1980-
+H
+[C.E.]
+GFED
+1965-
+3
+year
+1950
+C
+1935-
+E5
+B
+5
+growth axis
+1920
+Base
+1905-
+5
+4
+3
+2
+1
+0
+dft[cm] 
+Fig. S3: Water vapour adsorption by the artificial fluid inclusion system. Water vapour concentration 
+during crushing of 0.25 g Iceland spar. The decrease of the water vapour concentration indicates an 
+adsorption of water molecules on the freshly crushed calcite. Using the water amount calibration, it 
+corresponds to about 0.023 µl of water adsorption. The reference water vapour background is marked 
+in orange with interpolated linear fit as dashed line. The small inset shows examples of compact and 
+inclusion-free pieces of Iceland spar. 
+ 
+
+6920+
+6900
+6880-
+1 cm
+6860
+water
+background
+6840
+water
+background
+6820
+iceland spar
+6800-
+crush
+:35
+:40
+:45
+:50
+:55
+11:00
+:05
+:10
+time [hour] 
+Fig. S4: from top to bottom: relative temperature change derived from α(CaCO3-H2O) relative to 
+sample level B (orange dots); fractionation factor α(CaCO3-H2O) (grey squares); calcite δ18O values 
+(green triangles) corresponding to intervals with an edge length of 0.5 cm of the fluid inclusion sample 
+pieces, with smoothed higher-resolution data (green line); fluid inclusion δ18O (blue triangles). For a 
+better overview the depth (dft) errors of α(CaCO3-H2O) and the calculated temperature change are not 
+shown, but are the same as for the fluid inclusions δ18O. 
+ 
+
+level
+B
+C
+D
+E
+F
+G
+K
+6
+△(T) [K]
+3
+0
+reference levelB
+-3
+-6
+33.6
+32.9
+32.2
+31.5
+30.8
+30.1
+-7.5
+-8.0
+8180.
+-8.5
+-8
+-9
+-10
+180
+-11
+5
+4
+3
+2
+0
+dft [cm] 
+Fig. S5: Samples B-K of stalagmite Stam 4 with replicates from the same growth phases (Table 4) 
+displayed relative to the meteoric water line. The aliquots closest to the growth axis of the stalagmite 
+are shown as red circles. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+-45
+closest to axis
+-50
+replicates
+8H (% VSMOW)
+55
+60
+65
+-70
+-75
+-11
+-10
+-9
+-8
+-7
+9-
+-5
+s180 (% VSMOW)Additional Tables 
+Table S3: Fluid inclusion data from the outermost layer of stalagmite Stam 4. The distance to the 
+growth axis increases with higher Roman numbers. Arabic numbers indicate replicates with similar 
+distance from the growth axis. Samples in grey are not included in the interpretation and discussion as 
+the water amount was below 0.2 µl. 
+ 
+ID 
+Sample 
+weight (g) 
+Water     
+(µl) 
+Water content 
+(µl/g) 
+δ2H              (‰ 
+VSMOW) 
+δ18O               
+(‰ VSMOW) 
+I-1 
+0.69 
+0.30 
+0.52 
+-65.4 ± 1.5 
+-9.5 ± 0.5 
+II-1 
+0.61 
+0.40 
+0.95 
+-64.1 ± 1.5 
+-9.6 ± 0.5 
+II-2 
+0.30 
+0.37 
+0.75 
+-64.7 ± 1.5 
+-9.6 ± 0.5 
+II-3 
+0.37 
+0.40 
+0.76 
+-64.8 ± 1.5 
+-8.9 ± 0.5 
+II-4 
+0.38 
+0.29 
+0.56 
+-68.5 ± 1.5 
+-9.1 ± 0.5 
+III-1 
+0.38 
+0.81 
+1.66 
+-59.6 ± 1.5 
+-8.0 ± 0.5 
+III-2 
+0.40 
+0.51 
+1.21 
+-60.4 ± 1.5 
+-8.5 ± 0.5 
+III-3 
+0.44 
+0.44 
+0.81 
+-63.8 ± 1.5 
+-9.0 ± 0.5 
+IV-1 
+0.41 
+0.18 
+0.57 
+-63.8 ± 1.5 
+-8.5 ± 0.5 
+IV-2 
+0.47 
+0.42 
+0.83 
+-63.7 ± 1.5 
+-10.0 ± 0.5 
+V-1 
+0.23 
+0.38 
+0.78 
+-62.5 ± 1.5 
+-10.3 ± 0.5 
+V-2 
+0.58 
+0.46 
+0.78 
+-62.8 ± 1.5 
+-9.4 ± 0.5 
+VI 
+0.47 
+0.35 
+0.75 
+-63.2 ± 1.5 
+-9.6 ± 0.5 
+VII 
+0.46 
+0.43 
+0.94 
+-65.4 ± 1.5 
+-8.4 ± 0.5 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Table S4: Precision of fluid inclusion δ18O and δ2H measurements, interpolated from repeated water 
+injections and crushing of water-filled glass capillaries. The values refer to an exponential fit to the 
+standard deviation at various water amounts (Fig.5). The precision at 0.02-0.1 µl are extrapolated using 
+the exponential fit. 
+ 
+Water 
+amount 
+(µl) 
+Precision (1σ) 
+δ18O 
+(‰) 
+ 
+ δ2H 
+ (‰) 
+0.02 
+0.55 
+2.08 
+0.05 
+0.54 
+2.00 
+0.08 
+0.53 
+1.92 
+0.1 
+0.53 
+1.87 
+0.2 
+0.50 
+1.65 
+0.3 
+0.47 
+1.45 
+0.4 
+0.44 
+1.28 
+0.5 
+0.42 
+1.14 
+0.6 
+0.40 
+1.01 
+0.7 
+0.37 
+0.90 
+0.8 
+0.35 
+0.81 
+0.9 
+0.34 
+0.73 
+1.0 
+0.32 
+0.66 
+2.0 
+0.20 
+0.33 
+3.0 
+0.14 
+0.26 
+4.0 
+0.11 
+0.24 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Additional references: 
+Fohlmeister, J., Kromer, B., Mangini, A., 2011. The influence of soil organic matter age spectrum on  
+the reconstruction of atmospheric 14C levels via stalagmites. Radiocarbon 53(1), 99–115. 
+Fohlmeister, J., Schröder-Ritzrau, A., Scholz, D., Spötl, C., Riechelmann, D. F. C., Mudelsee, M., 
+Wackerbarth, A., Gerdes, A., Riechelmann, S., Immenhauser, A., Richter, D. K., and Mangini, 
+A., 2012. Bunker Cave stalagmites: an archive for central European Holocene climate 
+variability. Clim. Past, 8, 1751–1764, doi:10.5194/cp-8-1751-2012. 
+Mattey, D. P., Fairchild, I.J., Atkinson, T. C., Latin, J.-P., Ainsworth, M., Durell, R., 2010. Seasonal 
+microclimate control of calcite fabrics, stable isotopes and trace elements in modern 
+speleothem from St Michaels Cave, Gibraltar. Geological Society, London, Special 
+Publications, 336, 323-344. 
+Synal , H.-A., Stocker, M., Suter, M., 2007. MICADAS: a new compact radiocarbon AMS system. Nucl.  
+Instr. Meth. Phys. Res. B, 259, 7-13. 
+Treble, P., Shelley, J.M.G., Chappell, J., 2003. Comparison of high resolution sub-annual records of 
+trace elements in a modern (1911–1992) speleothem with instrumental climate data from 
+southwest Australia. Earth Planet. Sci. Lett.216, 141–153. 
+ 
+
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+page_content=' Warken, Martina Schmidt, Norbert Frank, Werner Aeschbach, 2023, Constraints for precise and accurate fluid inclusion stable isotope analysis using water-vapour saturated CRDS techniques, Chemical Geology 617, 121268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content='0 license http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content='0/  Constraints for precise and accurate fluid inclusion stable isotope analysis using water-vapour saturated CRDS techniques Therese Weissbach1,2, Tobias Kluge1,2,3,4,*, Stéphane Affolter5, Markus C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Leuenberger6, Hubert Vonhof7, Dana F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Riechelmann8, Jens Fohlmeister9, 10, Marie-Christin Juhl2, Benedikt Hemmer2,  Yao Wu2, Sophie F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content=' Martina Schmidt2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Norbert Frank2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Werner Aeschbach2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 3 1Heidelberg Graduate School of Fundamental Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Heidelberg University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Im Neuenheimer Feld 226,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 69120 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany 2Institute of Environmental Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Heidelberg University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Im Neuenheimer Feld 229,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 69120 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany 3Heidelberg Center for the Environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Heidelberg University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Im Neuenheimer Feld 229,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 69120 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany 4now at: Institute of Applied Geosciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Karlsruhe Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Adenauerring 20b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 76131 Karlsruhe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany 5Department of Environmental Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' University of Basel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Bernoullistrassse 30/32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 4056 Basel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Switzerland 6Climate and  Environmental Physics Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Physics Institute and Oeschger Centre for Climate Change Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' University of Bern,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Sidlerstrasse 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 3012 Bern,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Switzerland 7Climate Geochemistry Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Max Planck Institute for Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Hahn-Meitner-Weg 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 55128 Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany 8Institute for Geosciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Johannes Gutenberg University Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Johann-Joachim-Becher-Weg 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 55128 Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany 9Federal Office for Radiation Protection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Köpenicker Allee 120-130,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 10318 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany 10GFZ German Research Centre for Geosciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Section ‘Climate Dynamics and Landscape Development’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Telegrafenberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 14473 Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany 11Institute of Earth Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Heidelberg University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Im Neuenheimer Feld 234,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 69120 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany *corresponding author: tobias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='kluge@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='edu  Journal: Chemical Geology  Highlights: \uf0b7 Laser-based fluid inclusion analysis with water-vapour purged extraction enables high precision (δ2H≤±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5‰, δ18O ≤±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5‰) \uf0b7 Biasing effects (memory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' adsorption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' amount) in fluid inclusion isotope analysis are negligible for ≥1 µl water/g calcite \uf0b7 Isotopic interference is negligible for sample isotope ratios within 10‰(δ18O) and 50‰(δ2H) of the water vapour background \uf0b7 Reconstructed temperatures of a 20th century  stalagmite trace the recent warming of 1 °C in Central Europe  Abstract  Hydrogen (δ2H) and oxygen (δ18O) isotopes of water extracted from speleothem fluid inclusions are important proxies used for paleoclimate reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In our study we use a cavity ring-down laser spectroscopy system for analysis and modified the approach of Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2014) for sample extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The method is based on crushing of small sub-gram speleothem samples in a heated and continuously water-vapour purged extraction line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The following points were identified:  Injection of reference water shows a precision (1σ) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O values and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ‰ for δ2H values for water amounts of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 µl, which improves with increasing water amount to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰, respectively, above 1 µl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The accuracy of measurements of water injections and water-filled glass capillaries crushed in the system is better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 ‰ for δ18O and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ for δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The reproducibility (1σ) based on replicate analysis of speleothem fluid inclusion samples with water amounts > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 µl is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ for δ2H values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Isotopic differences between the water vapour background of the extraction system and the fluid inclusions have no significant impact on the measured fluid inclusion isotope values if they are within 10 ‰ for δ18O and 50 ‰ for δ2H values of the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Tests of potential adsorption effects with inclusion free spar calcite confirm that the isotope values are unaffected by adsorption for water contents of about 1 µl (fluid inclusion) water per g of carbonate or above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Fluid inclusion analysis on three different modern to late Holocene speleothems from caves in northwest Germany resulted in δ18O and δ2H values that follow the relationship as defined by the meteoric water line and that correspond to the local drip water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Yet, due to potential isotope exchange reactions for oxygen atoms, hydrogen isotope measurements are preferentially to be used for temperature reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  We demonstrate this in a case study with a Romanian stalagmite, for which we reconstruct the 20th century warming with an amplitude of approximately 1 °C, with a precision for each data point of better than ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Keywords: laser spectroscopy, water isotopes, cavity-ring-down measurement, speleothems, paleoclimate, small samples  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Introduction Speleothem fluid inclusions can provide direct insight into past climatic conditions as they are a unique archive for the original drip water and the corresponding meteoric water (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Griffiths et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Labuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Warken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Fluid inclusion water isotope ratios (δ18O and δ2H values) are increasingly usedas proxies in hydrology and paleoclimate studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', McGarry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Demény et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Millo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Wilcox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Two physically different measurement principles, laser spectroscopy (mainly cavity ring-down spectroscopy - CRDS) and isotope ratio mass spectrometry (IRMS), allow determining the isotopic composition of speleothem fluid inclusion water (CRDS: Arienzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Dassié et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' IRMS: Dennis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Vonhof et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Dublyansky and Spötl, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Although CRDS and IRMS systems yield comparable results (de Graaf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2020) challenges remain for both methods regarding precise and reproducible analysis of small water amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Often only a single measurement attempt is possible due to low growth rates of the speleothems (often 10-100 µm/a) or intended high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Water contents in natural speleothems range from ~ 0 up to several 10 µl per g (McDermott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  The necessary water sample amount (depending on the setup 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 µl, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Dublyanksy and Spötl, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2016) limits the temporal resolution and restricts analytical repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Fluid inclusion water for isotope analysis is released either by crushing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Schwarcz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Dennis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Vonhof et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Dublyansky and Spötl, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Demény et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2013) or thermal decrepitation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Yonge, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' McGarry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Verheyden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Thermal decrepitation has the disadvantage that structurally-bound water with a very low δ2H value may be released during extraction, resulting in large isotopic shifts of up to 30 ‰ in comparison to parent cave drip water (Yonge, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' McGarry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Verheyden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This analytical artefact can be largely avoided by crushing the sample mechanically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For fluid inclusion analysis using IRMS (Schwarcz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Harmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 1979), water was extracted by crushing the sample under vacuum conditions and then subsequently converted to water vapour followed by conversion into directly measurable gases such as H2 for H isotopic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Fluid inclusion δ18O values were initially not measured but calculated from measured δ2H values via the relationship between δ18O and δ2H values of the meteoric water line (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Schwarcz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  In recent extraction systems the oxygen in fluid inclusion water is converted to CO gas during a high temperature reaction with glassy carbon which is then used for analysis of δ18O values in an IRMS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Dublyansky and Spötl, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The first combined method for oxygen and hydrogen measurements with an off-line crushing method and dual-inlet IRMS was developed by Dennis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' It achieved good precision of ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ‰ for δ18O and ± 3 ‰ for δ2H values, but required a comparatively large sample with water amounts of 1-3 μl (see also Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A reduction in sample amount down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 μl, which  corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 g of calcite for samples that contain 1 µl of water per gram, was achieved by Vonhof et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2006) by combining off-line preparation and continuous-flow mass spectrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This technique enables a faster analysis of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 μl sized samples with a precision of ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O and ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ2H values (Vonhof et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Dublyansky and Spötl, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' de Graaf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Laser spectroscopy is less expensive and represents a reliable, precise and easy technique to directly measure stable water isotopes (Brand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The first application using CRDS to measure fluid inclusions in speleothems was developed by Arienzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' They used a CRDS analyser with a stainless-steel line heated to 115 °C that was constantly flushed with dry nitrogen as a carrier gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' It achieved comparable precisions as the traditional IRMS technique, with ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O and ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ‰ for δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The development of another analysis system using off-axis integrated cavity output spectroscopy (OA-ICOS) achieved similar precision (Czuppon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The latest analytical systems are able to measure released water volumes in the nano-litre range (50 to 260 nl) with a precision of ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ for δ18O and ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ‰ for δ2H values using the CRDS technique (Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The above discussed fluid inclusion extraction lines of Arienzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2013), Czuppon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2014), and Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  (2016) are working with dry carrier gas and low water vapour concentrations in the analyser cavity, which may influence the stable isotope measurements by adsorption and cause memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The measured isotopic signal needs to be corrected for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', the isotopic dependence on the water vapour concentration (Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2016) or the memory effect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', van Geldern and Barth, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The memory effect in the analyser cavity is due to limitations during the removal of all gas between two measurements preventing the full desorption of water molecules from the cavity walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A standard technique to deal with memory effects in liquid water analysis is the repeated injection of the same water sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The measured signal converges exponentially towards the actual sample signal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', van Geldern and Barth, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' However, multiple crushing steps on the same sample are typically not feasible for fluid inclusion measurements of speleothems since the amount of water of these samples is often too low to split it in several aliquots (sub-μl range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The adsorption issue was addressed by Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2014) with an extraction line that is continuously purged with a moist gas providing a water vapour background with constant and known δ18O and δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The extraction line with a “wet” N2 gas allows reproducible and precise measurement of released fluid inclusion water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  The continuous heating of the system enables the instantaneous evaporation of the water released from inclusions followed by spectroscopic analysis of the resulting mixture of background and sample water vapour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The main advantage of the water-vapour flushing is that this procedure avoids additional corrections of the measured water stable isotopes related to memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The achievable standard deviations of the measurements with this analytical system are smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ‰ for δ18O and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ δ2H values, which is comparable with the traditional IRMS  technique (Vonhof et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Dublyansky and Spötl, 2009) and CRDS setups working with a dry carrier gas (Arienzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Czuppon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' One important application of isotope fluid inclusion studies is the reconstruction of paleotemperatures using the oxygen isotope fractionation between water and calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The temperature reconstruction was initially based on the measurement of fluid inclusion δ2H values from which the fluid inclusion δ18O values were calculated using the δ2H-δ18O relationship of the global meteoric water line (Craig, 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Combined with the δ18O value of the calcite, temperatures were calculated from the oxygen isotope fractionation between the carbonate mineral and water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This indirect approach achieved a reported precision of about ± 2 °C in the early studies of Schwarcz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In the decades prior to 2010, direct temperature calculation from measured fluid inclusion δ18O values has been rare due to severe challenges in its analysis with standard mass spectrometric approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Therefore, mostly the indirect way of first converting measured δ2H values into fluid inclusion water δ18O values was used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' More recent studies provided comparable temperature precision based on inclusion water δ18O values: ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 °C (van Breukelen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2008), ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 °C (Meckler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 °C (Arienzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2015), and ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 °C (Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  The uncertainty of the indirect paleotemperature reconstruction from δ2H variations in speleothem fluid inclusions is similar: ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2008) and ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C (Meckler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' More recently, a better precision has been achieved when using the rainfall δ2H/T relation in mid-latitudes: ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C (Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In this study, we systematically assess the measurement method of stable water isotopes in speleothem fluid inclusion analysis using CRDS, including in particular the effect of sample amount (water amount per analysis), water adsorption on freshly crushed calcite surfaces, influence of the isotope values of the water vapour background on the sample signal, as well as the external reproducibility of speleothem fluid inclusion samples (using adjacent aliquots along growth layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In addition, we present a recent case study allowing to determine paleo-temperature trends in the 1°C range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Methods and site description 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 Water extraction from fluid inclusions  At the Institute of Environmental Physics (Heidelberg) water from speleothem fluid inclusions is extracted within a system that is constantly purged by an artificially prepared moist gas, leading to a water vapour background with known δ18O and δ2H values (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In the extraction line this stable water vapour background is generated by mixing water of a known isotopic composition into a dry nitrogen gas flow (300 ml/min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A peristaltic pump (Ismatec -REGLO Digital, Wertheim, Germany) continuously supplies small amounts of water (1 μl/min) to the line through a T-injection port with a  septum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 1 A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A constant temperature of 120 °C ensures a complete and immediate evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Instant water evaporation is induced in a fused silica capillary, which slightly touches the heated base of the port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A two-litre mixing cavity placed after the T-injection port generates a stable water vapour background and compensates fluctuations caused by the peristaltic pump cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The nitrogen flow is controlled by a mass flow controller (Analyt MTC, model GFC-17, Müllheim, Germany) and creates a constant overpressure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The flow rate is 40 ml/min into the CRDS analyser (L2130-i, Picarro, Santa Clara, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The surplus gas stream is vented through a purge capillary before the crusher unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' With this setup the water vapour concentration in the CRDS cavity ranges between 6000 and 8000 ppmV, but the cavity can also be adjusted to higher or lower water vapour concentrations if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' We have chosen a range between 6000 and8000 ppmV to allow for the detection and analysis of small fluid inclusion water amounts (sufficiently high ratio of water vapour from the sample relative to the background) but also to provide a background water vapour concentration that prevents memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The sample (speleothem fragment or glass capillary)is inserted in a copper tube (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 1 B) which is connected to the extraction and measurement system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Due to limitation of the copper tube with respect to length and diameter, compact mineral pieces with rectangular dimensions of 6 mm x 6 mm x 10-20 mm are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The copper tube including the sample is purged for at least 30 min until water vapour concentration and isotope values reach a constant signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The stability of the water vapour background is verified by monitoring the standard deviation of the water vapour concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' When the standard deviation of the water vapour concentration remained less than 20 ppmV for 30 minutes, the speleothem sample is crushed by compressing the copper tube from the outside with a hydraulic press at 200-300 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This compression in the heated system leads to the release and immediate evaporation of inclusion water and a sudden pressure increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This pressure increase could cause gas flow not only towards the CRDS but also in direction of the purge capillary and may provoke sample gas loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Therefore, a reflux valve is installed between the 2l mixing cavity and the crushing unit to prevent a backflow and loss of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In general, a very good crushing efficiency has been achieved with an average grain size of 37 μm after the crushing (Weißbach, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A reproducibility test related to the crushing procedure showed similar particle size distributions for five different speleothem samples investigated with laser diffraction (Analysette-22 Micro Tec) after crushing with the hydraulic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  An injection port is situated next to the copper tube, allowing to mimic a water release from a mineral sample and helps to evaluate and control accuracy and precision of the δ18O and δ2H values from small (inclusion) water samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' An additional small mixing cavity (400 ml) directly after the crushing unit prolongs the generated sample signal from an usually few seconds lasting peak to a well  measurable signal with a duration of several minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' After the mixing cavity the gas proceeds to the L2130-i isotope and gas concentration analyser (Picarro).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Figure 1: Fluid inclusion line for extraction and measurement of stable oxygen and hydrogen isotopes of fluid inclusions in speleothems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Water with a known isotope composition is mixed into a nitrogen gas flow to create a stable water vapour background (position A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The purge capillary reduces the background vapour flow from 280 to 40 ml/min as required for the CRDS analyser (L2130-i, Picarro).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The two mixing cavities provide a smoothing of the background vapour signal and a dispersion of the measurement signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The speleothem sample or the glass capillary is placed in a copper tube and installed at position B inside the heated oven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In order to prevent the backflow of the water vapour from the freshly crushed sample, a reflux valve is installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The flow directions are indicated as blue arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' [black/white for figures in print, colour online]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2  Analysis and data evaluation The water vapour concentration and the δ18O and δ2H values are determined with the L2130-i isotope and gas concentration analyser of Picarro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The L2130-i analyser is based on wavelength- scanned CRDS in the spectral range from 7183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 to 7184 cm-1 and uses a multi-pass cell that creates a  N2 tankgas injection glass port capillary peristalticpump CRDS background heatingtape 40 water ml/min (120°C) purge capillary ~280 ml/min refluxvalve mixing cavity -21 filter injection port oven mixingcavity-400ml loadedcopper (120C) tubelong effective absorption path length of about 12 km (Aemisegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A stable cavity temperature of 80°C ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='002 °C is maintained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The cavity pressure is set to 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='66 hPa (50 Torr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The water isotopologue lines pertaining to 18O and 2H are measured simultaneously over a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 s interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In our setup the L2130i allows isotope measurements in the water vapour concentration range between 1 000 and 50 000 ppmV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In constant flow mode the actual measured signal is composed of a background signal and a peak signal after crushing and must be integrated over a corresponding time interval (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The evaluation routine follows the approach of Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2014) but was extended by the correction of a potential level change in the water vapour background (Weißbach, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The import of the data and the evaluation was carried out with the assistance of the Python script IsoFluid (https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='com/bhemmer/IsoFluid and http://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5911265).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' IsoFluid determines the sample or calibration standard peak start and end based on a slope criterion that compares the start and end slope with the background slope of the water vapour concentration in user-defined time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  The sample isotope value (δ18Osample) is calculated by subtraction of the background signal (index back) from the measured signal of the mixed gas signal (index mix) and follows the approach of Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2014): 𝛿��𝑂������ = ∫ 𝛿��𝑂������(𝑡) ∗ 𝐻�𝑂������(𝑡) ∙ 𝑑𝑡 �� �� ∫ 𝐻�𝑂������(𝑡) �� �� ∙ 𝑑𝑡 = ∫ 𝛿��𝑂���(𝑡) ∗ 𝐻�𝑂���(𝑡) ∙ 𝑑𝑡 − ∫ 𝛿��𝑂����(𝑡) ∗ 𝐻�𝑂����(𝑡) ∙ 𝑑𝑡 �� �� �� �� ∫ 𝐻�𝑂���(𝑡) �� �� ∙ 𝑑𝑡 − ∫ 𝐻�𝑂����(𝑡) �� �� ∙ 𝑑𝑡  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1) H2O refers to the related water amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The calculation for δ2H values is correspondent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' All uncertainties are reported at the 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Figure 2: Water vapour signal in the CRDS before, during, and after speleothem sample crushing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Background intervals  (orange) are used to determine the value of the water background during a sample analysis (blue shaded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' [black/white for figures in print, colour online]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 Water vapour background calibration For the calibration of the oxygen and hydrogen isotope signal of the water vapour background five independently measured in-house reference waters were used (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Water from Willersinnweiher water (WW) was not used for calibration but for precision and accuracy assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Table 1: In - house reference waters for isotopic calibration of the fluid inclusions CRDS system, Isotope values were independently measured at the Institute of Environmental Physics (IUP) at Heidelberg University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Uncertainties are given as 1σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Water type Code δ18O values (‰ VSMOW) δ2H values (‰ VSMOW) Artificially evaporated water AE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 -21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='79 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 Ocean water Kona -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 Lake Water - Willersinnweiher (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Germany) WW -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='11 -18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 De-ionized local tap water VE -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 -61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 Alpine Water VCL -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 -98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7  8500  samplepeak  water vapour concentration [ppmV]  8000  background interval  background interval  7500  peak start  peak end  linearregression  7000  11:15  11:30  11:45  12:00  12:15  12:30  time [hour]Alpine Water Colle Gniffeti ice core  CC 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7  North Greenland water surface snow  NG 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7  The isotope values of these five reference waters were determined independently with a Los Gatos Research LGR1 analyser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' These reference waters cover a range of −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 up to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='05 ‰ in δ18O values and −212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ in δ2H values (both VSMOW) (Table 1), which includes the relevant range for speleothem samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Five different isotope background values were realised by using the corresponding reference water as supply, injected with the peristaltic pump into the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Once a sufficiently stable water vapour concentration was achieved in the preparation line (standard deviation below 20 ppmV for 30 min) the isotope signal was averaged over 60 minutes, which results in a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰ for the δ18O and δ2H background values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Figure 3 shows the CRDS-measured isotope value against the reference value (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The value of the water vapour background was constantly monitored via repeated measurements of the in-house reference waters, and has remained constant over several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Figure 3: The calibration of δ18O and δ2H values of the water vapour background results from a linear regression (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The calibration equation is y = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='994 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='007 · x + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='14) ‰ for δ18O values (R²=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='999) and y = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='980 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='003 · x−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='38) ‰ for δ2H values(R²=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Both calibrations remained constant over several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Isotope data on the reference waters used for water vapour background calibration are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The residuals from the linear regression indicate that calibrated values are within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 ‰ (δ18O) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='35 ‰ (δ2H) of the expected reference value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Furthermore, the residuals show a random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Uncertainties on the 1σ level in the calibration graphs are smaller than the symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' [black/white for figures in print, colour online]  residuals (%o) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 8180 (0%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='30 2H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='15 S 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='00 dual 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='15 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='04 0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='30 0 -30 -25 -20 -15 -10 -5 0 -250 -200 -150 -100 -50 0 50 50 0 Kona 1:1 (% VSMOW) 1:1 (%。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' VSMOW) 0 -5 Kona VE -10 -50 VCL VE -15 CC -100 VCL linearregression pected CC -20 -150 exp -25 NG linearregression -200 NG -30 -250 -30 -25 -20 -15 -10 -5 0 -250 -200 -150 -100 -50 0 50 8180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' measured (%。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' VSMOW)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4  Water amount calibration A precise water amount calibration is necessary for determining the exact amount of released water from the crushed speleothem calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The released water amount is a major parameter for the calculation of the fluid inclusion isotope value and is determined via water vapour signal integration (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Isotope values could also be calculated with the time-integrated water vapour mixing ratio alone, however, knowledge of the released water amounts is recommended for uncertainty assessment (amount dependence) and for assessment of speleothem growth conditions (fluid inclusion water yield).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Typically, volume calibrations are carried out by injecting water in the μl range with syringes (here: SGE 1BR-7RAX and 5BR-7RAX and  Hamilton 70001KH and 75N), however, the variability of the calculated water amount only using syringe injections is significant and can be as high as 10 % (inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 4, Weißbach, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Here we present a water amount calibration method with glass capillaries that follows the approach of Kluge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  The glass capillaries (borosilicate, Hirschmann) can be filled with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 μl water at varying isotopic composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' They can be closed airtight by melting both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The size of the filled capillary can be adjusted to the size of the crushing cell down to a minimum length of approximately 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The exact volume is determined by scanning the capillary with a high-resolution office scanner and comparison with the pre-marked 1 μl labels on the capillary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The volume uncertainty of the glass capillary water amount is ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='025 μl and was determined by five repetitions of the manual evaluation of a scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The accuracy is given by the uncertainty of the pre-marked 1 µl labels (± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='003 µl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Water-filled capillaries were analysed weekly to monitor the stability of the water amount calibration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The uncertainty of the water amount determination from the calibration is approximately ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='02 µl at a water volume of 1 µl and ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='04 µl at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 µl using the 1σ uncertainty of the linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In general, we rarely observed outliers in the water amount calibration when using glass capillaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Figure 4: The time-integrated measured volume signal in ppmV*s as given by the Picarro analyser is plotted against the water volume of  the glass capillaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The resulting linear regression y = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 × 10−7 ·x−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='011) μl (R²=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='999) is used to determine the released amount of water from speleothem samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In total 45 capillaries were measured for calibration spanning a water amount range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 μl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The upper inset shows a glass capillary filled with about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 µl of water (in the middle of the capillary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The lower inset shows the comparison of water injections with different syringes (blue and red symbols) and the glass capillaries (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The uncertainties are given on the 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' [black/white for figures in print, colour online]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 Site and sample description Hüttenbläserschacht Cave (Germany) For direct comparison with rainwater δ18O values and measured drip water isotope values, we selected a suite of modern and late Holocene samples from the Hüttenbläserschacht Cave, located only a few 100 meters west of the well-monitored Bunker Cave in northwest Germany (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Riechelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Both caves are situated in the upper Middle Devonian limestone in Iserlohn (Sauerland).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Hüttenbläserschacht Cave hosts pool spar calcites that are expected to provide fluid inclusion isotope values close to drip water as they grow under the water table of the pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Pool spars from this cave have already been investigated by Kluge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2013) using clumped isotope ∆47 and calcite δ18O values for calculation of the (drip) water δ18O value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Calcite was actively precipitating in the pools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', abundant calcite rafts) at the time of pool spar removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 230Th-U disequilibrium dating  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0x106 label watercolumn I area (ppmV*s) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0x106 integrated signal 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0x106 3x106 linearregression oglass capillary syringes Hamilton 2x106 SGE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0x106 1x106 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 0 1 2 3 4 5 water volume capillary (μl)at Heidelberg Academy of Sciences provided radiometric ages of one pool spar and one raft sample of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='27 ka BP and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='12 ka BP, respectively (Supplemental material S1), corroborating the assumption that the pool spars and rafts are modern age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Cloşani Cave (Romania) For the second case study we selected a 20th century stalagmite (Stam 4) from Cloşani Cave, Romania (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Constantin and Lauritzen, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A monitoring program from 2010 to 2012 and 2015 demonstrated a stable cave environment with an air temperature of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C and a relative humidity close to 100 % (Warken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The isotopic composition of the drip water in direct vicinity (1 m) of the former location of Stam 4 showed no seasonal cycle and was constant throughout the monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The mean dripwater δ18O value was −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2‰ and −66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7‰ for δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Stalagmite Stam 4 has a total length of 6 cm and an average growth rate of 510 μm per year, as deduced from counting of layers related to annual cycles in the concentration of various elements (Supplemental material S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The speleothem grew actively until the removal in spring 2010 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' as drip water was feeding the stalagmite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  The recent growth of the stalagmite was further constrained by the detection of the 20th century radiocarbon bomb spike, which was imprinted by the transport of the atmospheric signal into the speleothem (see Supplemental material S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Combined layer counting and radiocarbon measurements suggest a growth period from 1910 to 2010 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For the fluid inclusion study, pieces were taken from the peripheral part of Stam 4 with a distance of approximately 1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 cm from the actual growth axis (see Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 Precision of isotope measurements The precision of isotopic measurements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 5, Table 2) was quantified using the standard deviation of repeated analyses of the reference waters injected via syringes (VE water;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Table 1) and independently cross-checked with water-filled glass capillaries using VE and WW reference waters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The injected water amount using syringes) varied between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 μl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Using syringe injection method, a clear decrease of the standard deviation of these isotope analyses with increasing water amount becomes apparent (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 5, Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The standard deviation decreases strongest between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 and 1 µL and reaches values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 and0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ for δ18O and between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 and0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰ for δ2H values, for samples larger than 1 µl (Supplementary Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For smaller water amounts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 µl and below, the isotope values of the injections show a significantly larger scatter, leading to standard deviations between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 and0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O and between  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 and1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ‰for δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' These uncertainties are based on an exponential fit of the standard deviation against the water volume using repeated measurements at a given water volume (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For the determination of the precision, reference water sealed in glass capillaries was crushed in the fluid inclusion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Consistent with the results from the water injections, the precision for isotopic analyses of water released from crushing glass capillaries is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='07 and0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='10 ‰ for δ18O values and between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 and0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ‰ for δ2H values, for water amounts above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 µl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Smaller water amounts resulted in a significant increase in the uncertainty and are expressed through a lower precision (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Table 2: Measurement precision and accuracy in dependence of the water amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The precision (± 1σ) was determined from repeated injections of isotopically well-characterized water standards using syringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The accuracy (given at the 1σ level) was assessed by measurement of reference water both from injections and by release from crushing of sealed glass capillaries and comparison with the independently determined isotope values (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The error estimate for the accuracy assumes a Gaussian distribution and includes the uncertainty of the expected value (VE water: ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 ‰ for δ18O, ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰ for δ2H, WW water: ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='11 ‰ for δ18O, ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ‰ for δ2H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' n represents the number of analyses in the investigated water volume range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The water isotope values are given relative to VSMOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Type Reference water Water volume (µl) Precision (1σ) δ18O value(‰)  δ2H  value  (‰)  Accuracy (1σ)  δ18O  value  (‰)  δ2H  value(‰)  n  Water  injection  VE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='54  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='34  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4  6  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4  16  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4  11  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4  11  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1  9  Mean all  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7  77  Mean < 1µl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7  30  Mean ≥ 1µl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7  47  Glass  capillary  WW  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6  16  WW  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  14  VE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  17  VE  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8  9  Figure 5: Upper panels: precision (1σ) of the isotope measurements for varying amounts using the water injection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The red lines represent the least-square exponential fits to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The standard deviation decreases with increasing water amount for both δ18O and δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Lower panel: accuracy determination for varying water amounts based on individual injections (open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The related mean values with their 1σ standard deviation are shown as filled dots with error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The black horizontal lines represent the reference value for VE - tap water (δ18O=- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='57 ‰, δ2H =- 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ‰) with its uncertainty band (grey shading, ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08‰ for δ18O, ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰ for δ2H values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' [black/white for figures in print, colour online]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 18 1α standard deviation (%o) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 5 standard deviation ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 0 1 2 3 4 0 1 2 3 4 water amount (μl) water amount (μl) -57 singlevalue 8H [%。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' VSMOW] -58 meanvaluewithstdv 8 -60 -61 -62 -63 6180 [%VSMOW] -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 singlevalue meanvaluewithstdy -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 3 1 2 4 volume [u]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 Accuracy of isotope analysis of micro-litre water amounts The accuracy of the water δ18O and δ2H values was assessed for reference waters (Table 1) by the injection with syringes and by crushing glass capillaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The injected water amounts covered the typical range of water extracted from inclusions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The glass capillaries were filled with reference water with similar water amounts between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 µl and were crushed in the copper tube with the same hydraulic press as the stalagmite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Considering the water isotope mean values of all measurements performed with the glass capillaries, the δ18O value deviated from the expected reference values (Table 1) by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='44 ‰ for WW water (n=16) and by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='31 ‰ for VE water (n=17) (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For δ2H values the deviation from the reference value was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ‰ for WW water (n=16) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='50 ‰ for VE water (n=17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Considering only those measurements with water amounts above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 µl reduces the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For this selection, the δ18O value of both reference waters deviates on average from the expected reference value by -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='13 ‰ for WW water (n=14) and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='13 ‰ for VE water (n=9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  For δ2H values the deviation from the reference value was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for WW water (n=14) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ‰ for VE water (n=9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The accuracy as determined by crushing of water-filled glass capillaries is confirmed by the injection-based data (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Overall, the δ18O value of the injected VE water deviated from the expected value on average by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='19 ‰ (n=77), that of the δ2H value by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰ (n=77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 Adsorption and/or desorption on the calcite surface Adsorption on a calcite surface and, in particular, on freshly crushed carbonate with a large surface to volume ratio provides the possibility to alter the isotope values of the fluid inclusion water (Dennis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Therefore, an artificial fluid inclusion system (speleothem analogue) as described by Dennis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2001) has been prepared to quantify the influence of adsorption on the measured isotopic signal in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' We measured water vapour released from a water-filled glass capillary in direct contact with inclusion-free Iceland spar carbonate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The compact Iceland spar pieces (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='45-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='81 g) as well as the released water of the capillaries (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 µl water) represent a speleothem sample with a water content of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 up to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 μl per g calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In total, we prepared and analysed five artificial fluid inclusion - calcite systems in the range between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 and7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 µl/g and three at about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 µl/g and compared them to water-filled glass capillaries without additional calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The crushing of the compact Iceland spar pieces provided fresh and fine-grained calcite for interaction and adsorption testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The measurements suggest that the adsorption of water molecules on the calcite surfaces does not affect the measured isotopic signal in the investigated water/calcite ratio range (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Both measured oxygen and hydrogen isotope values accurately match the expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' With a standard deviation  of ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='05 ‰ for δ18O and ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='22 ‰ for δ2H values (high water/calcite ratio, n=5) and ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='15 ‰ for δ18O and ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='31 ‰ for δ2H values (low water/calcite ratio, n=3) in both adsorption tests, a good reproducibility of the individual measurements was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' We observed that after crushing of Iceland spar (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='25 g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='023 μl water was adsorbed on the crushed calcite from the moist carrier gas (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S3), which corresponds to a ratio of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 μl water per g calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Thus, for low water contents of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 µl per g calcite an influence of adsorption on the released water amount and the isotopic values probably cannot be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Therefore, we rejected all fluid inclusion samples with water amounts below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 µl based on this observation (see Weißbach, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Figure 6: Isotopic values measured for the artificial inclusion calcite system, for which compact Iceland spar pieces were crushed together with VE water - filled glass capillaries (triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Open circles indicate water - filled glass capillaries (VE) without calcite addition, measured for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' An isotopic fractionation due to adsorption of water molecules on the calcite surface is not detectable for the investigated water content range of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 µl water/g calcite (left side) and for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 µl water/g calcite (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Marginal differences in the isotope values between left and right panel are within the expected variations in the reference water isotope values due to a several year time lag between both experimental series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The uncertainties are displayed on the 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' [black/white for figures in print, colour online]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 Isotopic effect of the water vapour background  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 μl/g ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 μl/g ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='H (% VSMOW) -58 144 -60 444 D -62 (%。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='VSMOW) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 丰 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 0-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 water plus only water waterplus only water iceland spar iceland sparThe potential influence of the isotope ratio of the water vapour background on that of the measured sample could be relevant for speleothem samples whose isotopic composition strongly differs from that of the water vapour background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For testing this potential effect, we injected our VE water standard on four different water vapour backgrounds with different isotopic composition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' We used VE-water as injection fluid, because its isotopic composition is comparable to the majority of fluid inclusion of speleothems from mid-latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For each water vapour background 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 μl of VE water were injected five times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For the background waters with the two most extreme isotope compositions we additionally injected 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 µl of VE (n=6) to assess the robustness also for smaller water amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' If VE water is injected on VE background water vapour, the average isotope value corresponds to the expected value within uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A deviation from the expected isotope value is notable for injections on a different water vapour background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For example, VE injections on  a negative water vapour background (NG, δ18O = -26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 ‰, δ2H = -212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰, Table 1) yield a deviation of +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='40 ‰ for δ18O and +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ‰ for δ2H values from the reference values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  VE injections on a background, which is based on lake water (WW)  with higher isotope values compared to VE water, yield deviations of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='15 ‰ for δ18O and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ for δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Tests with injected water amounts of 1 µl corroborate the observed trend (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The standard deviation of repeated water injections is independent from the isotopic composition of the water vapour background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The effect of the isotopic difference between the samples and the background water vapour exceeds the measurement uncertainty only for differences larger than 10 ‰ (δ18O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This experiment highlights that it is not necessary to correct samples when using background water with an isotope composition close to the paleoclimate samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For speleothem measurements we used VE water as background water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Figure 7: Deviation of the measured injection isotope value relative to the expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The deviation is related to the difference between the isotope signal of the injection and that of the water vapour background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Single injections are shown as open circles and the mean values as filled circles s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The green andblue lines indicate the linear regression with all individual 3 µl measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' An increasing deviation between the measured and expected isotopic signal is observed for an increasing difference between injection isotope value and that of the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 oxygen hydrogen D 3 μl 1 μl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 3 μl 1 μl (0%) singleinjections O singleinjections -expected) ( meanvaluewithuncertainty expected)( 3 mean value with uncertainty 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 linearregression Viation(measured 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 devi T: VE VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' injections injections -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 uo on wWIAE VE cC CC NG AE ww VE NG -10 -5 0 5 10 15 20 -50 50 100 150 200 deviation (injection-background signal)(%o) deviation(injectionbackgroundsignal)(%o)water vapour background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The uncertainty of the expected value (black horizontal line) is shown as grey envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Measurement uncertainties are given on the 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' [black/white for figures in print, colour online]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 Case applications Case example 1: Modern – late Holocene sinter samples The two fluid inclusion replicates of each modern or late Holocene sample from Hüttenbläserschacht Cave reproduce very well and are within uncertainty of each other (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Related standard deviations of the mean (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ‰ for δ18O and δ2H values, respectively) are comparable (δ18O values) or slightly larger (δ2H values)  than the measurement precision in this water amount range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 µl, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The mean fluid inclusion δ18O value of -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ is identical to the calculated drip-water value of Kluge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2013) of -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰, independently confirming the former finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Drip water in Hüttenbläserschacht Cave was not monitored but should be close to the neighbouring Bunker Cave and shares the same karst aquifer with comparable residence times of a few years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Kluge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Fluid inclusion isotope values are close to the mean drip water values from Bunker Cave of -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ for δ18O and -53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ‰ for δ2H values (Riechelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Table 3: Measurement of fluid inclusions in three CaCO3 spar samples from Hüttenbläserschacht Cave (Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Each sample was split in two to allow for a replication test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' ‘Avg.’ refers to the average of the two analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For comparison also the calculated pool water δ18O value of Kluge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2013) is shown that uses an independent temperature estimate and clumped isotope ∆47 for correction of kinetic isotope effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  The Bunker Cave drip water is taken from Riechelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Uncertainties are given on the 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  ID Sample weight (g) Water amount (µl) Water content (µl/g) δ18O  value (‰ VSMOW) δ2H      value (‰ VSMOW) Pond A A-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='46 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  A 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  Pond B  B 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='76 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  B 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='64 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1  Pond C (little  pond)  C 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='59  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='78  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0  C 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='48  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 Average all  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6  Reconstructed  after Kluge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  (2013)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3  Drip  water  Bunker Cave  range  2006 2013  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 to  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0  48 to  58 mean 2006  2013  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6  Case example 2: Speleothem sample from the 20th century – Stam 4 from Cloşani Cave Comparison with current drip water and reproducibility assessment We sampled calcite pieces at the outer surface of the stalagmite for comparison with current drip water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' It can be assumed that recent calcite precipitated there and accordingly, recent drip water is enclosed in the fluid inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The water yields during crushing were between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='49 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='38 µl/g with a mean of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='28 µl/g (one sample was excluded due to a low water amount of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='18 µl) (Supplementary Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The mean value of 13 fluid inclusion measurements of samples from the outer stalagmite layer is δ18O = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ and δ2H = −64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ (Supplementary Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' These values agree within uncertainty with the mean of the related drip site CL3 of δ18O = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ and δ2H = −66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The 13 individual measurements reproduce with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ for δ18O and δ2H values, respectively, which is slightly higher than the analytical uncertainty based on the standard deviation of repeated syringe injection for water amounts between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5-and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 µl (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ‰ for δ18O values, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ‰ for δ2H values, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The standard deviation for the 13 individual speleothem analyses is also comparable to that of other CRDS systems and similar water amount ranges, such as of Arienzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2013) with ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ‰ for δ18O/δ2H values and Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  (2014) with ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O/δ2H values .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The precision of the Stam4 sample analysis also compares well with traditional IRMS measurement techniques which achieve a precision of ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O and ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ‰ for δ2H values for water amounts > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 µL (Dublyansky and Spötl, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Figure 8: Fluid inclusion water isotope ratios of samples from the outer stalagmite surface (light blue dots), with corresponding mean value (dark blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Drip water data from the same cave chamber where Stam 4 was removed (light green triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' drip site CL3) and its mean value (dark green) agree with the fluid inclusion results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Both drip water and fluid inclusion data match the local meteoric water line (LMWL) of Cluj-Napoca of δ2H = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='03 · δ18O + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='29 ‰ (Cozma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The uncertainties are given on the 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' [black/white for figures in print, colour online]  Fluid inclusion analysis of samples along the growth axis We used the stalagmite pieces closest to the growth axis of Stam 4 for paleo-drip water and - temperature reconstruction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Where possible, the reproducibility of the individual measurements was tested with a second set of fluid inclusion samples, extracted  adjacent to the first set of samples (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The second set had a larger distance from the growth axis than the first set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The samples corresponding to the same growth period are grouped in levels, indicated by letters A-K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For sample level D, only the second sample is used because the first sample is close to the applied water amount limit and contains only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='18 μl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' On average, the δ2H values of sample and replicate are largely consistent (mean deviation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ‰).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The same is observed for the inclusion water δ18O value (mean deviation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='51 ‰).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In addition the water content of the different levels appears characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For level D and E with 5 replicates each, the water content varies only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 µl/g (excluding one sample each with low total water amount).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For the other levels, a higher scatter has  LMWL FI-Stam4 -60 FImeanwithstdy dripwater (CL3) dripwatermeanwithstdv -62- 8’H [% VSMOW] -64 -66 -68 -70- -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 8180 [%。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' VSMOW]been observed, potentially due to a general heterogeneity of the speleothem inclusion distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Muñoz-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Generally, the water content was between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='45 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='66 µl/g, suggesting minimal or negligible influence of adsorption on the freshly crushed surface (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Fluid inclusion δ18O values vary between -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ‰ and -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ‰ and, with one exception (level C), follow a temporal trend towards higher values towards more recent times (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 9, Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Table 4: Measurement results of fluid inclusion samples of stalagmite Stam-4 from Cloşani Cave (Romania).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Several samples were cut from individual layers that reflect contemporaneously grown carbonate and allow for replication tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The fractionation factor 18α(CaCO3-H2O) between water and CaCO3 was calculated based on the difference between the calcite δ18O values (averaged over the edge length of the fluid inclusion sample of typically 5 mm) and the fluid inclusion water δ18O values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The temperature T18O,cc was determined using the 18α(CaCO3-H2O) - T relationship proposed by Kim and O’Neil (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' TH is related to the relative temperature change calculated using the δ2H- temperature relationship in rainfall (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='72‰/°C) and was referenced to top level K and the current cave temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' T18O, Fi refers to the temperature difference relative to sample level K with the modern cave temperature as reference and was calculated using the δ18O-T relationship in rainfall (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='59‰/°C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Samples in grey are not included in the interpretation and discussion as the water amount was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='19 µl or below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Samples closest to the growth axis (‘1’ closest, higher numbers are further away) were used for temperature assessment based on the classical carbonate thermometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Samples A1 and A2 were the oldest samples and were excluded from the discussion as they belong to the stalagmite base with unclear chronology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The age corresponds to the mean age of each sample level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Dft: distance from top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  ID Dft (mm) Age (year AD) Sample weight (g) Water amount (µl) Water content (µl/g) δ2H value (‰ VSMOW) δ18O value (‰ VSMOW) 18α (CaCO3- H2O) (‰) T18O,cc (°C) T18O,Fi (°C) TH (°C) A1  unknown  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='52 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='72 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8  ±  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  J  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9  2004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='86 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9  ±  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  K  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9  2008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='94 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3  ±  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 Constraints for precise and accurate fluid inclusion isotope data The presented setup allows for a good reproducibility with respect to isotope measurements of pure water samples in the µl range, either injected via a syringe or by crushing of water-filled glass capillaries in the copper tube (similar to speleothems samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The achievable precision is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ‰ for δ2H analyses at extracted water amounts between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 µl and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 µl and decreases with increasing water amount to ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ for δ18O and ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰ for δ2H measurements at extracted water amounts >1 µl (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The improved precision with increasing water amount is consistent with the observations of Dassié et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2018) who reported similar precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ for δ18O and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ‰ for δ2H values for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-1 µl as well as a strong increase of the uncertainty at water amounts lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 µL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Replicate analyses of calcite samples from the outermost surface of a Romanian stalagmite corroborate the precision as determined by crushing of water-filled glass capillaries and water injections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Adsorption of water on freshly crushed surfaces appears negligible for water contents of about 1 µl water per g calcite or above Dennis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2001) similarly observed a decreasing adsorption influence at increasing H2O/CaCO3 ratios at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' However, an adsorption effect could  be relevant if the water content in the crushed samples approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 µl/g or is below this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' We therefore recommend to use the water content as one parameter to check the robustness of the analysis and to carefully assess or conservatively reject samples with water contents below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 µl/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' We observed a small dependence of the measured isotope value on the water vapour background (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='. After injection of a certain water amount, the δ18O value of the (hypothetically) well mixed water vapour consisting of background and injection water is an amount-weighted mixture of both δ18O values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For background water with relatively depleted values such as North Greenland Water (NG, -δ18O =-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ and δ2H = -212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ‰) this would mean that the δ18O value of the VE water with δ18O = -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='57 ‰ and δ2H = -61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ‰ is higher than the mixed water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For example, if the background to injection volume is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0, the isotopic composition of the mixture is expected to be δ18O = -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ and δ2H = -117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='Given the short residence time of the water vapour in the mixing cavity before the measurement in the CRDS, a full isotopic mixing is not reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The kinetically slower molecules containing an 18O atom remain preferentially in the gas stream compared  to the faster molecules containing only 16O atoms that preferentially  take part in the mixing with the background water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Thus, for this case example it is expected that the injection water isotopes are slightly higher relative to the background and the mixed signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Conversely, for a positive background as the WW water, the isotope value of the VE injection is more negative relative to the isotope value of the hypothetical fully mixed gas stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Due to the kinetic behaviour of 18O, the injection stays more negative relative to the expected value for this background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The adsorption effects and the influence of kinetic isotope exchange are similar for the 1 µl and 3 µl injections (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For water amounts in the µl range this dependence on the vapour background isotope value is relevant if the isotopic composition of the fluid inclusions is significantly different from the background (> 10 ‰ for δ18O and > 50 ‰ for δ2H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Otherwise, the potential effect of the isotopic difference to the background water vapour is within the analytical uncertainty of water samples between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 µl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The maximum expected deviations are < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='25 ‰ for δ18O and < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ‰ for δ2H values, if the sample is within the 10 ‰ range of the water vapour background for δ18O and 50 ‰ for δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For water amounts larger than 1 µl the acceptable deviation between sample and background water isotope values reduces in relation to the higher measurement precision at higher water amounts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 Paleotemperature calculation from Stam 4 using fluid inclusion isotopes For calculation of the CaCO3-H2O isotope fractionation, we averaged the calcite δ18O values which correspond to the growth period of the spatially larger fluid inclusion sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The calculated fractionation factor 18α(CaCO3-H2O) between calcite and fluid inclusion water yields values between 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ‰.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This range would correspond to temperatures between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C and  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C using the 18α(CaCO3-H2O)-T relationship of Kim and O’Neil (1997) (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The calculated absolute temperatures deviate slightly from these values depending on the used 18α(CaCO3-H2O)-T relationship (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Démeny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Tremaine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' However, relative differences between the coldest and warmest periods and the trend in the data set is largely independent of the selected fractionation-temperature relationship as most experimental and empirical studies yield similar 18α(CaCO3-H2O)-T slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Following an apparent change of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ in 18α(CaCO3-H2O) a temperature change of about 9°C would formally correspond to the growth period of the stalagmite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This temperature difference is much larger compared to that observed at local meteorological stations with maximum and minimum mean annual air temperature differing by approximately 3°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This discrepancy suggests that the temperature trend related to 18α(CaCO3-H2O) in the stalagmite has been enhanced, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', by stronger isotopic disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' As the measured fluid inclusion water isotopes correspond to the meteoric water line (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S5), post-depositional and other significantly altering effects are unlikely for the water-filled inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  However, mineral formation in speleothems often takes place in a non- equilibrium regime (Deininger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2021) and may also have influenced the calcite δ18O values of Stam 4 due to a high growth rate and strong seasonal variations in prior calcite precipitation (PCP, Warken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='. We refrain from correcting the disequilibrium effect in calcite δ18O values due to the related large and hardly quantifiable uncertainties and only focus on the fluid inclusion δ2H values in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Note, that it may be possible in other cases to derive temperature variations from the oxygen isotope fractionation between fluid and calcite if the degree of PCP is negligible or constant and the length of drip interval has not changed significantly during growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2019) demonstrated that δ2H values and its temperature relationship in rainwater of mid-latitudes can be used to deduce temperature changes throughout the Holocene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In stalagmite Stam 4, a long-term trend towards higher δ2H values is observed from the oldest to youngest fluid inclusion samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A significant increase for δ2H values of +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ‰ was identified between sample level F and K and similarly between B and C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 9 C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This transfers into to a temperature change of +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 °C using the relationship between the isotopic composition of precipitation and temperature for Central Europe of +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='04 ‰/°C for oxygen and +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='32‰/°C for hydrogen isotopes (Rozanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Since stalagmite Stam 4 from Cloşani Cave grew under continental climatic influence, the mean value for Central Europe seems to be the best reference for the determination of the relative temperature change with the δ2H/T relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' GNIP (Global Network of Isotopes in Precipitation) stations and other weather stations in Hungary, Austria, Slovakia and Poland with more than 10 years of isotope analysis show similar slopes of +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ‰/°C (Demény et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Considering the observed range of the rainfall δ2H-temperature slopes in Central and Eastern Europe by Gaussian error propagation, the uncertainty increases slightly to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  With the confirmed recent growth of the stalagmite, the topmost stalagmite piece is assigned to the year 2010 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' ( year of stalagmite removal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Annual growth layers provide a possibility to assign ages to all other sample depths (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Temperature changes ΔT relative to the reference level B is close to zero up to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 1960 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (3 cm distance from top, level F), followed by an increase of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 °C at the stalagmite top (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 9F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The mean annual air temperature for the time period from 1928 to 2008 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' at the meteorological station Drobeta/Turnu Severin, which is located in the vicinity of the cave, shows a similar temperature increase of about 1 °C from 1980 until 2008 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 9G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This is consistent with the general trend in Romania, which experienced a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 °C increase for the period of 1901-2012 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (Ministry of Environment and Climate Change, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The temperature change determined from δ2H values in the fluid inclusions corresponds well to the trend and magnitude measured in the mean annual air temperature of the region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Directly interpreting the fluid inclusion δ18O values using the rainfall δ18O-T relationship for Central Europe of +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='04 ‰/°C by Rozanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (1992) also leads to a temperature increase, albeit with a higher amplitude of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1°C relative to level B, but within uncertainty consistent with the temperature reconstruction using δ2H values (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  growthaxis[cm] 5 A B C D E Base level A B c D E F G H K a (% VSMOW) T -8 B II -9 -10 -11°H (% VSMOW) -60 65 (0%) 33 D 32 30 (82H) (%) E reference 0 TevelB 2 reference13 (C) MAAT 11 10 1920 1940 1960 1980 2000 yearFigure 9: A) Stam 4 with assignment of sample pieces based on visual correlation with the growth axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Layers B to K were used for temperature reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The small inset shows alternation of fluid inclusion-rich and inclusion- poor layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Winter layers yield very little inclusions, while summer layers include abundant air- and water-filled inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Width of the image is ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' B) Fluid inclusion δ18O values C) δ2H values D) Fractionation factor α between calcite and inclusion water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' E) Change in δ2H values relative to level B (lowest temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' F) Inferred temperature change relative to level B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A trend is visible for Δ(δ2H) as well as for ΔT from the stalagmite bottom to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Using the δ2H/T relationship of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='32 ‰/T (Rozanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 1992) a total increase of ΔT =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C is observed within the growth period of the investigated stalagmite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' G) Mean annual air temperature (MAAT) of the Drobeta/Turnu Severin station in the cave region (thin black line, Klein Tank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2002) for the last 100 years with a 10-year running mean (red line) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The 10 year-smoothing interval corresponds to the average age that is covered by the fluid inclusion samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The uncertainties are given on the 1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  [black/white for figures in print, colour online]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 Paleotemperature reconstruction using fluid inclusions Our study supports the conclusion of previous publications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' de Graaf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2020) that an accurate and precise determination of the isotope composition of micro-litre water amounts is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Our setup is able to produce small errors, which are in the same range as the precision in the previous fluid inclusion isotope studies(Dublyansky and Spötl,2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Arienzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2014),;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Dassié et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=',2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In these studies a precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ‰ for δ2H values in the water amount range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 µl, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ for δ18O and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰ for δ2H values at water amounts > 1 µl was demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The analytical precision determines the currently achievable temperature precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  In principle, three possible ways of temperature calculation from fluid inclusion isotopes exist: a) from the temperature-dependent oxygen isotope fractionation between calcite and fluid inclusion water (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Arienzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Labuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2015), b) indirectly via transfer of the fluid inclusion δ2H value to the corresponding water δ18O value using the δ18O-δ2H relationship of the meteoric water line and then using the oxygen isotope fractionation between carbonate and water for temperature calculation (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Meckler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2015), and c) from the hydrogen isotopes using a locally valid  δ2H-temperature relationship of the rainfall (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Of the three methods for temperature reconstruction the first two (a and b) show the highest uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 °C (Van Breukelen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Arienzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Meckler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The highest achievable temperature precision in the case of the best analytical fluid inclusion δ18O precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ (and a calcite δ18O uncertainty <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ‰) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Approaches a and b are additionally affected by the potential influence of disequilibrium isotope fractionation during carbonate mineral formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Deininger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2021), causing too high temperatures or an unrealistically large temperature spread in case of significant changes of isotopic disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Furthermore, diagenetic exchange between host calcite and fluid inclusion water could further alter the water δ18O value (Demeny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Uemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The precision of the temperature reconstruction directly from fluid inclusion δ2H values depends critically on the value of the rainfall δ2H/T relationship and the availability of well-defined rainfall δ2H/T functions at the study site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For the Central European region a value of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='32 ‰/°C of Rozanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (1992) can be used and can yield a temperature precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 °C for released water amounts of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 µl if the  analytical precision is ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ‰ for δ2H measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The uncertainty of the rainfall δ2H/T function is negligible for our case study but could be relevant in case of a reduced temperature dependence of the rainfall δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' At locations with a stronger temperature dependence of the rainfall δ2H value an even better precision is possible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='13 °C for the average of Swiss stations, which show a slope of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='44 ‰/°C (Rozanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 1992) and for the typical analytical uncertainty of our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The temperature resolution of this method is slightly reduced at lower latitudes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='55 °C at Hong Kong with a δ2H rainfall-temperature relationship of 2 ‰/°C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Rozanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Note that temperature estimates using fluid inclusion δ2H values and the rainfall δ2H/T relationship without climatic reference points are relative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', they record only temperature changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  With an anchor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', modern reference temperature and rainfall δ2H values, absolute temperatures can be also inferred from fluid inclusion δ2H values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The application of the rainfall δ2H/T relationship for calculating temperature changes from fluid inclusion δ2H values also requires the δ2H/T relationship to be constrained for the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Information on the rainfall isotope systematic and the δ2H/T relationship in the past can be gained for example from groundwater studies (Darling, 2004) in combination with noble gas temperatures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Kreuzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Varsány et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2011, Túri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The uncertainty of the δ2H/T relationship needs to be considered and likely decreases the achievable precision for pre-Holocene speleothems as the uncertainty for the δ2H/T relationship increases when applying the modern or Holocene relationship back in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Affolter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2019) used the δ2H/T relationship for temperature reconstruction from fluid inclusions throughout the Holocene and achieved a precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C for a Swiss stalagmite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Our analytical approach allows for the same temperature resolution and with measurements of stalagmite Stam 4 from Romania confidently verified the recent 20th century warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Both studies together illustrate the potential of the inclusion-based methodology for tracing and reconstructing minor temperature fluctuations of < 2 °C during the Holocene and, at sufficient temporal resolution (requiring high stalagmite growth rates), also of sub-degree changes such as the recent anthropogenic warming trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Summary and conclusion  Fluid inclusion isotope analysis using CRDS measurements after mechanical sample crushing benefits from fluid extraction and measurement under a constant and controlled water vapour background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The specific isotope and water volume calibration of the CRDS system remained valid for several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For assessing the fluid inclusion extraction and measurement performance we used syringe injections and boro-silicate glass capillaries filled with reference water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' We have shown that out setup has no drift in the isotope values for smaller water amounts and that the memory effect for this system is negligible when using an isotopically appropriate background water vapour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The water vapour background should be chosen such that the isotope values of sample and background do not deviate significantly (maximum 10 ‰ for δ18O and 50 ‰ for δ2H values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Direct comparison of calcite powder-free and -filled extraction tubes proved that the adsorption of water on the speleothem surface has no effect on the measured isotope signal if the water content is larger than 1 µl water per g calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For samples with a water content below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 µl/g calcite results have to be checked as we observed a corresponding adsorption of the water vapour background on freshly crushed calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Related to the above-mentioned constraints, the precision (1σ) of isotope measurements for aliquots of water from speleothem fluid inclusions improves with increasing water amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  It is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ‰ for δ18O and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ‰ for δ2H values for water samples between0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 and0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 µl, which is comparable to other CRDS systems and IRMS techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This value was further confirmed by replicated measurements of adjacent samples of the Romanian stalagmite Stam 4 (standard deviation of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 ‰ for δ18O andδ2H values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For water amounts larger than1 µl the precision improves to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ‰ for δ18O and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ‰ for δ2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Analysis of fluid inclusions of recent pool spars from a German cave shows good agreement between drip water and fluid inclusion isotope values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Similarly, the δ18O and δ2H values of a Romanian stalagmite, grown during the 20th century, reflect the isotopic composition of the modern drip water within uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In the same case study, we observed a T-trend from δ18O values, which is inconsistent with local weather records, suggesting a major influence disequilibrium and kinetic effects on the speleothem calcite δ18O signal of Stam 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The isotopic disequilibrium causes a significant overestimation of the temperature changes calculated from the oxygen isotope fractionation between calcite and water (in our case 9 °C difference instead of ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 1°C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' In contrast, hydrogen isotopes are not involved in calcite precipitation and therefore provide a relatively undisturbed link to the stable isotopic composition of drip and rain water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Using the δ2H-temperature relationship in rainfall we obtained a temperature increase for Cloşani Cave of +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C between 1960 and 2010, which is in excellent agreement with the local temperature record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Thus, applying the local rainfall δ2H-- temperature relation on fluid inclusion δ2H variations appears to be a reliable method to determine mean annual air temperatures for mid-latitude speleothems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The achieved precision furthermore highlights the potential of fluid inclusion isotope studies in speleothems for high resolution  paleoclimate reconstruction, given that the rainfall isotope relationship is significantly linked to temperature and is available for the studied area and valid for past periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Acknowledgements  The project was funded by DFG Grant KL 2391/2-1 and supported by the Heidelberg Graduate School for Physics in the context of grant GSC 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' We thank Sylvia Riechelmann and Jasper Wassenburg for collection of Stam 4, Silviu Constantin and Mihai Terente for monitoring, Christoph Spötl for drip water analysis at CL3, Regina Mertz-Kraus for LA-ICP-MS element analysis and Sven Brömme for calcite δ18O and δ13C analysis on Stam 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' We thank the editor Michael E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Böttcher and three anonymous reviewers for their very detailed comments and suggestions that helped to improve the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Data availability  Data of this study are summarized in Tables 1-4 and Supplementary Table S1-S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Raw data related to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' 3-6 are given in the Appendix of Weißbach (2020), available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='11588/heidok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='00028559 Competing interests statement  The authors declare the absence of competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' An absolute paleotemperature record from 10 to 6 Ka inferred from fluid inclusion D/H ratios of a stalagmite from Vancouver Island, British Columbia, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Geochim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Cosmochim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 72, 1014-1026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Supplementary Material for Constraints for precise and accurate fluid inclusion stable isotope analysis using water-vapour saturated CRDS techniques by Therese Weissbach, Tobias Kluge, Stéphane Affolter, Markus C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Leuenberger, Hubert Vonhof, Dana F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Riechelmann, Jens Fohlmeister, Marie-Christin Juhl, Benedikt Hemmer, Yao Wu, Sophie Warken, Martina Schmidt, Norbert Frank, Werner Aeschbach  S1) Hüttenbläserschacht Cave – sample dating Table S1: Pieces of speleothem samples (pool spars and rafts) collected at Hüttenbläserschacht Cave were dated using the Th-U disequilibrium method at the Heidelberg Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The analytical procedure followed the methods described in Fohlmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Sample  232Th  [ppb]  238U  [ppb]  230Th  [fg/g]  ±  (234U/238U)  ±  (230Th/238U)  ±  Age  Age  Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  uncor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Hinterm  Ballsaal  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1407 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='08  Kristall häutchen  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3557±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5862±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='04  S2) Closani Cave and Stam 4 annual layer counting Cloşani Cave is located on the southern slope of the Carpathians at an altitude of 433 m above sea level and developed in massive limestones of Upper Jurassic-Aptian age (Constantin and Lauritzen, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The cave is overlain by about 30 m of rock overburden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' A monitoring programme showed microclimatic stability for the cave interior with a mean air temperature of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 °C and a relative humidity close to 100% for 2010-2012 and 2015 (Warken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The cave air pCO2 pattern follows a strong seasonal cycle with high values in late summer (up to 8000 ppmV) and lower values during winter (2000 ppmV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Water infiltration occurs predominantly during winter time (October - March) where 75 to 100% of the meteoric precipitation is available for infiltration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Warken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2018) showed that calcite precipitation is favoured during winter time and reduced in summer, as a result of seasonally varying CO2 concentrations in the cave air and related equilibrium DIC concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The water isotopic composition of the drip water in direct vicinity (1 m) of the former location of Stam 4 shows no seasonal cycle and is constant with a mean value of −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2‰ for δ18O and −66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7‰ for δ2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  The relatively small and fast-grown stalagmite Stam 4 was collected from the “laboratory passage” in the cave in 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' It has a total length of 6 cm and an average growth rate of 510 μm per year, as deduced from counting of elemental layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Both summer and winter layers are clearly detectable in the thin sections, whereas winter layers show a compact structure with a lower number of inclusions and the milky-white porous summer layers contain abundant air- and water-filled inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' This layering in Stam 4 was induced by the strongly changing pCO2 in the cave air, resulting in a seasonal change in growth rate and corresponding seasonal cycles in Sr and Ba in the stalagmite calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Similar seasonal Sr and Ba pattern have also been observed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', by Treble et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2003), Mattey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2010), and Warken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The visible annual layers in stalagmite Stam 4 are not as pronounced as the annual cycles in the measured high-resolution Ba concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Ba concentration was measured with a LA-ICP-MS (Agilent 7500 ce with Laser UP-213, Institute of Geosciences Mainz) at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 µm resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The minima of this record were counted five times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' These five counted layer series were cross-dated to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Layers have to be counted at minimum three times, layers only counted once or twice were deleted from the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For each layer a mean value of layer thickness was calculated from the five layer thickness series to a master chronology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  This layer thickness chronology results in a growth of Stam 4 from 1910 to 2010, the year of sampling under an active drip site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S3) Radiocarbon dating Four samples were drilled with a hand-held dental burr (1 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Calcite powder was acidified in vacuum with HCl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The emerging CO2 was combusted to C with H2 and an iron catalyst at 575°C (Fohlmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Measurements were performed with a MICADAS AMS system (Synal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2007) in the Klaus-Tschira laboratory Mannheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The results for the four samples show a typical speleothem radiocarbon bomb spike (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S1), constraining recent growth of the speleothem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Table S2: Radiocarbon measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Radiocarbon results and errors are expressed in fraction modern (fm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' MAMS lab nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' depth [mm] 14C [fm] 14C error [fm] 14709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0416 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0029 14710 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0730 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0029 14711 39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0025 14712 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0024  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S1: Radiocarbon measurements (black) over depth (bottom-axis), plotted to fit the atmospheric radiocarbon anomaly (blue, top x-axis) in the mid to late 20th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  Year [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=']  1900  1920  1940  1960  1980  2000  180  180  160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  160  140 140  120  120  100  100  80  80  1900  1920  1940  1960  1980  2000  distancefromtop[mm]Additional figures  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S2: Age assignment of the fluid inclusion samples A) Fluid inclusion sample pieces (labelled B to K) are shown on the left half of the stalagmite slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The red lines illustrate the assignment of the individual sample blocks to the growth axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The visible lamination was used as guideline for correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Sample A is related to the base and due to a disturbed growth structure does not allow to assign any age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Due to the intrinsic uncertainties of this procedure (for details see Weißbach, 2020) we associated age ranges to the individual fluid inclusion samples B to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' B) Age depth model with distance from top (dft) in cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The chronology was established by layer counting and additional 14C measurements (see S1 and S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  2025 B 2010- 1995- 2 1980- H [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='] GFED 1965- 3 year 1950 C 1935- E5 B 5 growth axis 1920 Base 1905- 5 4 3 2 1 0 dft[cm] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S3: Water vapour adsorption by the artificial fluid inclusion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Water vapour concentration during crushing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='25 g Iceland spar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The decrease of the water vapour concentration indicates an adsorption of water molecules on the freshly crushed calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Using the water amount calibration, it corresponds to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='023 µl of water adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The reference water vapour background is marked in orange with interpolated linear fit as dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The small inset shows examples of compact and inclusion-free pieces of Iceland spar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  6920+ 6900 6880- 1 cm 6860 water background 6840 water background 6820 iceland spar 6800- crush :35 :40 :45 :50 :55 11:00 :05 :10 time [hour] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S4: from top to bottom: relative temperature change derived from α(CaCO3-H2O) relative to sample level B (orange dots);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' fractionation factor α(CaCO3-H2O) (grey squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' calcite δ18O values (green triangles) corresponding to intervals with an edge length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 cm of the fluid inclusion sample pieces, with smoothed higher-resolution data (green line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' fluid inclusion δ18O (blue triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' For a better overview the depth (dft) errors of α(CaCO3-H2O) and the calculated temperature change are not shown, but are the same as for the fluid inclusions δ18O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  level B C D E F G K 6 △(T) [K] 3 0 reference levelB -3 -6 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 8180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -8 -9 -10 180 -11 5 4 3 2 0 dft [cm] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' S5: Samples B-K of stalagmite Stam 4 with replicates from the same growth phases (Table 4) displayed relative to the meteoric water line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The aliquots closest to the growth axis of the stalagmite are shown as red circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  45 closest to axis  50 replicates 8H (% VSMOW) 55 60 65  70  75  11  10  9  8  7 9 5 s180 (% VSMOW)Additional Tables Table S3: Fluid inclusion data from the outermost layer of stalagmite Stam 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The distance to the growth axis increases with higher Roman numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Arabic numbers indicate replicates with similar distance from the growth axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Samples in grey are not included in the interpretation and discussion as the water amount was below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='2 µl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='  ID Sample weight (g) Water (µl) Water content (µl/g) δ2H              (‰ VSMOW) δ18O (‰ VSMOW) I-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='52 -65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 II-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='95 -64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 II-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='75 -64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 II-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='76 -64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 II-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='56 -68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 III-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='66 -59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 III-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='21 -60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 III-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='81 -63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 IV-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='57 -63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 IV-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='83 -63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 V-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='78 -62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 V-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='78 -62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content='94 -65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content='5  Table S4: Precision of fluid inclusion δ18O and δ2H measurements, interpolated from repeated water injections and crushing of water-filled glass capillaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The precision at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content='  Water  amount  (µl)  Precision (1σ)  δ18O  (‰)  δ2H  (‰)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content='24  Additional references: Fohlmeister, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Kromer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Mangini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' The influence of soil organic matter age spectrum on the reconstruction of atmospheric 14C levels via stalagmites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Radiocarbon 53(1), 99–115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Fohlmeister, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Schröder-Ritzrau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Scholz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Spötl, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Riechelmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Mudelsee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Wackerbarth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Gerdes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Riechelmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Immenhauser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Richter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', and Mangini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Bunker Cave stalagmites: an archive for central European Holocene climate variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Clim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Past, 8, 1751–1764, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='5194/cp-8-1751-2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Mattey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Fairchild, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Atkinson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Latin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Ainsworth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Durell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Seasonal microclimate control of calcite fabrics, stable isotopes and trace elements in modern speleothem from St Michaels Cave, Gibraltar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Geological Society, London, Special Publications, 336, 323-344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Synal , H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Stocker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Suter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' MICADAS: a new compact radiocarbon AMS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' B, 259, 7-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Treble, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', Shelley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+page_content=', Chappell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=', 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Comparison of high resolution sub-annual records of trace elements in a modern (1911–1992) speleothem with instrumental climate data from southwest Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Earth Planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
+page_content='216, 141–153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfh_2U/content/2301.01493v1.pdf'}
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+Tricking AI chips into Simulating the Human Brain:
+A Detailed Performance Analysis
+Lennart P. L. Landsmeer∗
+Quantum & Computer
+Engineering Department
+Delft University of Technology
+Delft, The Netherlands
+Dept. of Neuroscience
+Erasmus Medical Center
+Rotterdam, The Netherlands
+ORCID 0000-0003-0010-7249
+Max C.W. Engelen∗
+Dept. of Neuroscience
+Erasmus Medical Center
+Rotterdam, The Netherlands
+&
+Maxeler IoT Labs
+Delft, Netherlands
+ORCID 0000-0002-5762-1276
+Rene Miedema
+Quantum & Computer
+Engineering Department
+Delft University of Technology
+Delft, The Netherlands
+&
+Dept. of Neuroscience
+Erasmus Medical Center
+Rotterdam, The Netherlands
+ORCID 0000-0002-0447-1083
+Christos Strydis
+Quantum & Computer
+Engineering Department
+Delft University of Technology
+Delft, The Netherlands
+&
+Dept. of Neuroscience
+Erasmus Medical Center
+Rotterdam, The Netherlands
+ORCID 0000-0002-0935-9322
+Abstract—Challenging the Nvidia monopoly, dedicated AI-
+accelerator chips have begun emerging for tackling the compu-
+tational challenge that the inference and, especially, the training
+of modern deep neural networks (DNNs) poses to modern
+computers. The field has been ridden with studies assessing
+the performance of these contestants across various DNN model
+types. However, AI-experts are aware of the limitations of current
+DNNs and have been working towards the fourth AI wave
+which will, arguably, rely on more biologically inspired models,
+predominantly on spiking neural networks (SNNs). At the same
+time, GPUs have been heavily used for simulating such models
+in the field of computational neuroscience, yet AI-chips have not
+been tested on such workloads. The current paper aims at filling
+this important gap by evaluating multiple, cutting-edge AI-chips
+(Graphcore IPU, GroqChip, Nvidia GPU with Tensor Cores and
+Google TPU) on simulating a highly biologically detailed model of
+a brain region, the inferior olive (IO). This IO application stress-
+tests the different AI-platforms for highlighting architectural
+tradeoffs by varying its compute density, memory requirements
+and floating-point numerical accuracy. Our performance analysis
+reveals that the simulation problem maps extremely well onto the
+GPU and TPU architectures, which for networks of 125,000 cells
+leads to a 28x respectively 1,208x speedup over CPU runtimes.
+At this speed, the TPU sets a new record for largest real-time IO
+simulation. The GroqChip outperforms both platforms for small
+networks but, due to implementing some floating-point operations
+at reduced accuracy, is found not yet usable for brain simulation.
+Index Terms—AI accelerator, GPU, Brain simulation, com-
+puter architecture
+I. INTRODUCTION
+To date, GPUs have achieved spectacularly better perfor-
+mance in deep learning (DL) than CPUs [1]. Recently, novel,
+specialized AI hardware platforms have begun to emerge,
+holding the promise of accelerating training and inference
+even further. The workloads targeted mainly are artificial, and
+specifically, deep neural networks (DNNs), which have shown
+great potential in recent years. On the other hand, highly
+biologically plausible models such as conductance-based (e.g.,
+Hodgkin-Huxley) neurons have not attracted similar atten-
+tion from AI-chip manufacturers and analysts alike. This is
+strange, given that biological brains – the inspiration behind
+these DNNs – are modeled using equations built on similar
+elementary functions. What is more, high-detail models are
+touted as the next AI wave, which is intended to be more
+biologically inspired than its predecessors [2], [3]. Therefore,
+it makes sense both for neuroscientists and for AI researchers
+to reach for these AI accelerators and deploy them for brain
+simulations; yet no performance studies exist.
+In this work, we evaluate multiple, cutting-edge AI chips
+(Graphcore IPU [4], GroqChip [5], TensorRT-capable GPU [6]
+and Google TPU v3 [7]) on simulating a highly biologically
+detailed model of a brain region, the Inferior-Olivary nucleus
+(IO). Biologically detailed brain models, such as the IO,
+chiefly involve addition, multiplication, division and expo-
+nential operations, arranged as sparse computations. There
+is, thus, a large operation overlap with artificial networks.
+Therefore, new AI chips seem a good fit for these types
+of models. This IO application represents timely, relevant
+research and is constructed as an extended-Hodgkin-Huxley
+model. It is very suitable for stress-testing the different AI
+platforms and highlighting architectural tradeoffs by adjusting
+the compute density, memory requirements and numerical
+accuracy of the IO model. Evaluation is performed using the
+application encoded as a TensorFlow 2 [8] kernel, which in the
+case of the GroqChip, is necessarily compiled to its ONNX [9]
+equivalent. ONNX is an intermediary tool used to convert
+models between different machine-learning (ML) frameworks.
+In the analysis of the different accelerators, the exact same
+TensorFlow model is used, ensuring a fair comparison across
+the board. This is either used directly or ported to ONNX
+via the Python package tf2onnx. TensorFlow is a high-level
+API, requiring little to moderate intervention from the user
+and is therefore suitable for a wide user base. A schematic
+overview of this setup is presented in Fig. 1.
+While all accelerators in this study support chip-to-chip
+communication, this work constrains the application to single-
+chip performance comparisons; multi-chip is left as future
+1
+arXiv:2301.13637v1  [cs.LG]  31 Jan 2023
+
+IO-model 
+TensorFlow
+CPU
+GPU
+IPU
+ONNX 
+tf2onnx
+CPU
+GPU
+Groq
+TPU
+Fig. 1: Overview of the performance-analysis strategy followed in this work
+work. The contributions of this work are:
+• We take a deep dive into four cutting-edge AI architec-
+tures with a focus on biologically plausible spiking neural
+networks (SNNs).
+• We build the first ML-library-based, efficient implemen-
+tation of a detailed brain model, the Inferior Olive (IO).
+• We deploy the IO model onto the four AI platforms and
+benchmark their performance and numerical accuracy.
+• We demonstrate that modern ML libraries are seman-
+tically able to model classical problems in scientific
+computing, offering large performance gains and reduced
+development times while remaining hardware-agnostic.
+• Lastly, this work is the first to ever simulate a realistic
+mouse-sized IO model with real-time performance.
+The paper is organized as follows: Section II presents
+related works in the field. Section III introduces the IO
+model used as our benchmarking application, while Section IV
+briefly presents the four AI architectures under evaluation and
+attempts some performance predictions. Section V ensures
+experiment reproducibility by detailing the experiment param-
+eters and platform configurations used to acquire our results
+presented in Section VI. A general discussion of our findings
+is included in Section VII and conclusions in Section VIII.
+II. RELATED WORKS
+Models of biological neurons come in various levels of
+detail, ranging from population-level dynamics, from simpli-
+fied models of single neurons to highly detailed biophysically
+realistic neurons [10]. Coarse models of single neurons, no-
+tably leaky-integrate-and-fire (LIF) type models have seen a
+renewed interest in the DL-community (often referred here
+to as SNNs) as an alternative to artificial neural networks
+(ANNs) [11]. In contrast, computational neuroscience is usu-
+ally interested in biophysically accurate models that model the
+underlying biological processes in a way that makes it possible
+to gain insights about these processes. These conductance
+based models can be made more realistic by modeling of
+their 3-dimensional structure (the morphology) using multiple
+discretized compartments. Multi-compartmental conductance-
+based neurons are then simulated by explicit calculation of
+electrical currents flowing within, between and into discretized
+compartments [12].
+Due to the computional resources needed for large-scale
+conductance level brain simulations, computational neuro-
+science was an early adopter of general-purpose GPU (GP-
+GPU) in the HPC environment. Notable GPU-based ex-
+amples of large-scale, biologically detailed brain simulators
+include CoreNeuron [13], which enabled porting of exist-
+ing conductance-level NEURON [14] models to the GPU,
+and more recently, Arbor [15], a library-based approach to
+performance-portable, large-scale brain simulation. Their suc-
+cess shows that the computational problems of neuroscience
+map well to GP-GPU platforms and result in significant
+speedups for large-scale brain models. Still, even with hand-
+optimized CUDA code [16], the IO application (to be detailed
+in the next section) at biological sizes runs order-of-magnitude
+slower than the biological brain, hampering research.
+With respect to TensorFlow-based implementations of
+conductance-level models, there is PymoNNto [17], an attempt
+to bring the Brian [18] API of neural models to TensorFlow.
+While faster than the Brian simulator on a GTX1080 GPU,
+performance was not a primary goal and the architecture pro-
+hibits optimizations using TensorFlow’s JIT compiler backend,
+by scattering the computational definitions across the code-
+base. Although this shows that TensorFlow does express the
+right API surface for neural models, no efficient ML-library
+based conductance-level GP-GPU brain simulators exist.
+Simplified SNN models have readily available GP-GPU
+implementations of LIF and similar models as well. High-
+level ML-libraries like TensorFlow and PyTorch allowed for
+the hardware-agnostic implementation of their neural dy-
+namics, considerably lowering development efforts to build
+SNN simulators for GP-GPU simulation. For example, Nengo
+DL [19] allows for the GPU-based simulation of existing SNN
+models defined in the Nengo framework using TensorFlow.
+Beyond just simulating neural networks on the GPU, novel
+developments in surrogate gradients for event-based SNNs
+and automatic gradient calculation provided by ML-libraries
+allowed for the nearly simultaneous appearance of similar
+SNN deep-learning libraries Norse [20], snnTorch [21] and
+SpikingJelly [22]. BindsNET [23] is another, efficient SNN
+implementation in PyTorch with a focus on reinforcement
+learning. Again, these project show that not only ML-libraries
+have the expressive power and performance needed to run
+large-scale SNN models, also that this arguably can be devel-
+oped faster than hardware-specific low-level code. As these
+libraries had DL-oriented goals in mind, none of these imple-
+ments multi-compartmental, conductance-level neural models.
+Simplified SNN models also led to the development of
+specialized neuromorphic hardware to simulate them. Numer-
+ous publications show the benefits of using these chips for
+simplified-SNN simulation. For a short review of the various
+chips, we point the reader to [24]. While exciting with respect
+to low-power inference of SNN-based deep-learning models,
+these chips, due to their hardwired dynamics, lack the ability
+to simulate conductance level neural models.
+On AI chips that have the semantic power to capture
+more general HPC workloads, little has been published about
+both simplified and conductance-level SNNs. With respect
+to simplified SNN simulation, we find just one preprint tar-
+geting an AI chip, introducing an IPU-optimized version of
+2
+
+snnTorch [25]. Training throughput of a dense 3-layer LIF
+network on an image classification task is 3.4x higher on the
+IPU than on the A100. The reported performance benefits
+decrease if the network size is increased, with the A100
+apparently underutilized throughout the entire application.
+This shows the potential of using the IPU for simple SNN
+workloads, but the performance characteristics of other AI
+chips or more complex SNNs are not yet obvious.
+No works have been published targeting AI chips with
+conductance-level models or other biologically realistic brain
+simulation scenarios, neither using high-level ML libraries or
+hardware-specific SDKs. To the authors’ knowledge, this is the
+first work to implement an efficient, conductance-level, multi-
+compartmental neuron in an ML library and also the first to
+benchmark multiple AI chips on this workload class.
+III. THE INFERIOR-OLIVE APPLICATION
+The IO is a intrinsically oscillating brain region located
+in the brainstem, and is key to motor control and learn-
+ing [26]. The estimated neuron population for the mouse
+brain is approx. 104 neurons [27] and for humans between
+106 − 107 neurons [28]. These numbers will be referred to
+during hardware-performance evaluation (Section VI). In this
+work, we will capture in TensorFlow 2 the IO nucleus as
+an extended Hodgkin-Huxley (eHH) model, conductance-level
+brain model, first published in [29]. The model is a good
+example of the computational load of realistic brain models
+and, also, a good fit for our benchmarking purposes, since
+it captures complex neuron dynamics and fast interneural
+communication (in the form of gap junctions), as will be
+shown next.
+We restate the IO-neuron main equations in this section, but
+refer the reader to [29] for more details. In addition, we model
+connectivity based on the network described in [30].
+1) The cable model:
+Cm
+dV (i)
+dt
+= −
+�
+k∈Channels
+I(i)
+k
+−
+�
+i∈Compartments
+Ik,j
+−
+�
+i∈Gap junctions
+Igj,k,j − I(i)
+app
+(1)
+The eHH model describes the membrane that envelops the
+neurons as a capacitor. The cell internal voltage can thus be
+calculated by integrating currents flowing into and out of the
+cell (eq. 1). Here, Iapp is an optional term describing externally
+applied currents by the experimenter.
+2) Channel currents: Channels (CaL, h, KCa, Na, Kdr, K,
+CaH, Na, K) allow currents to flow through the cell membrane.
+They produce this current as function of internal state variables
+changing over time. In general, this current (eq. 2) results from
+the potential difference to an channel specific reversal potential
+E multiplied by the product of one or more internal gating
+variables, each optionally raised to an integer power (eq. 2).
+The gating variables follow an Ordinary Differential Equation
+(ODE), that brings them to a certain cell-voltage dependent
+steady state at a given speed (eq. 3). These latter equations
+Listing 1 Axonal sodium-channel current
+m inf = 1/(1+ t f . exp ( −(V axon+30) / 5 . 5 ) )
+h
+inf = 1/(1+ t f . exp ( ( V axon+60) / 5 . 8 ) )
+tau h = 1.5* t f . exp ( −(V axon+40) /33)
+dh dt = ( h inf −h ) / tau h
+I na
+= g Na * ( V axon−V Na) * m inf **3* h
+Listing 2 Sparse gap-junction current
+V d i f f
+=
+t f . gather ( V dend ,
+gj
+src ) \
+−
+t f . gather ( V dend ,
+g j
+t g t )
+I
+per
+gj = V d i f f
+*
+g gj
+*
+(0.2 + \
+0.8
+*
+t f . exp ( −0.01* V d i f f * V d i f f ) )
+I gapp
+=
+t f . tensor scatter nd add (
+t f . zeros
+like (V) ,
+t f . reshape ( gj
+tgt ,
+( −1 ,1) ) ,
+I
+per
+gj )
+are usually gaussian or sigmoidal functions of the voltage.
+For certain fast operating channels we set n(t) = n∞(V ) as
+a numerical stability optimization.
+Ii = ¯gi
+��
+k
+ni,k(t)mk
+�
+(V − Ei)
+(2)
+τn (V ) dn
+dt = n∞ (V ) − n (t)
+(3)
+3) Compartmental currents: A single IO cell consist of
+three separate compartments, the axon, soma and dendrite.
+Currents flowing between different compartments are modeled
+resistively as: Ii,j = gi,j (Vj − Vi)
+4) Gap-junction currents: Gap junctions are direct electri-
+cal connections between different IO cells and allow current
+to flow between them. They follow experimentally determined
+Connexin-36 protein dynamics:
+Igj = ggj∆V
+�
+0.2 + 0.8 exp
+�
+−∆V 2/100
+��
+(4)
+with ∆V the potential-difference between two connected cells.
+5) Topology: The real IO looks like a large, folded sheet
+with mostly local connectivity. As approximating this structure
+is not a focus of this paper, our model neurons are assumed
+to exist on a discrete 3-D grid with wrap-around connectivity
+(i.e., a hypertorus). This should exhibit the same non-local
+memory-access patterns as a more realistic model. Connec-
+tions are sampled as a function of inter-neuron distance r on
+a radially symmetric distribution: p(r) ∝ u(rmax − r)(e−r2 −
+e−r2
+max)n(r), where n(r) is the density of neurons in the
+volume shell around r. This distribution is sampled until we
+have 10 connections per neuron on average.
+6) TensorFlow Translation: The previous equations sum up
+to a total of 14 ODEs per neuron. This system of ODEs is
+translated to a series of TensorFlow operators in Python. By
+defining the model in TensorFlow instead of using platform-
+specific APIs, we make sure all platforms have equal op-
+timization opportunities. Furthermore, TensorFlow naturally
+translates to ONNX models, which is the only high-level
+API available for GroqChip. Straightforward translation to
+TensorFlow is achieved by storing all state in a large 2d-array
+and direct substitution of mathematical expressions by their
+3
+
+TensorFlow counterparts (see Listing 1). When certain model
+parameters need to be user-specified (e.g., gi or Iapp), these
+are passed to the TensorFlow kernel, which then needs to be
+recompiled before running again.
+Translating gap junctions to both TensorFlow and ONNX
+in a performant way requires expressing them as vector
+operations, as opposed to more traditional for-loop-based
+approaches [16]. With just 10 connections per IO neuron on
+average, cell-to-cell communication is sparse. The effective
+operation from a TensorFlow perspective is two sparse-matrix
+(SM) multiplications. As a novel contribution in computa-
+tional neuroscience, we model those as tf.gather and
+tf.tensor_scatter_nd_add operations (see Listing 2).
+Apart from being more specific and memory-efficient in
+describing SM multiplications, these functions have a direct
+mapping to ONNX operators as Gather and ScatterND since
+ONNX specification opset 11, contrary to SM multiplica-
+tions which currently are not possible in ONNX.
+At each timestep, ODEs are integrated using Forward-Euler
+to produce the next state array, resulting in a hardware-
+agnostic timestepping function. For TensorFlow backends, a
+JIT-compilable TensorFlow function is constructed that exe-
+cutes 40 timesteps at a ∆t of 0.025ms, resulting in a 1ms
+sampling accuracy. For ONNX backends, the timestep function
+is converted to an ONNX model and either the public onnx-
+runtime library or Groq compiler is used to compile this
+into executable code. This does not lead to the best possible
+performance by default, thus hardware-specific optimizations
+are discussed in Section V-B.
+IV. TARGET PLATFORMS
+Hardware platforms were selected from the top-performing
+AI accelerators in the MLCommons MLPerf training bench-
+mark v2.0 [31]. From this, the Intel Habana Gaudi was not
+available to us. The GroqChip was included as it was already
+available through academic channels. An overview of all AI
+chips is given in Tab. I and will be presented next. A modern,
+server-grade CPU is also included as a baseline for our
+subsequent performance and numerical-accuracy comparisons.
+1) Nvidia GPU [6]: These are well-established accelerators
+in the HPC world. With the introduction of the Tensor Cores
+in Nvidia GPUs, they also became well-known for their AI
+capabilities. Tensor Cores are capable of matrix multiplications
+in a very efficient manner. The current generation of tensor
+cores can support up to TensorFloat-32 (TF32) precision TF32
+is a floating point with float 32 dynamic range but float 16
+precision. There are multiple ways of interacting with them;
+e.g., via cuBLAS and TensorRT.
+2) GroqChip [5]: This is a deterministic Tensor Streaming
+Processor (TSP), resembling a modified systolic array archi-
+tecture. The chip layout is a conventional 2D mesh of cores,
+each with its own dedicated functionality. A column of these
+cores – all of the same type – is called a functional slice.
+Data travels horizontally, executing 320 SIMD-style lanes. A
+single instruction can control 16 lanes, effectively creating 20
+superlanes that can all be operated independently from each
+other. The functional slices consist of one vector processor
+(VXM), two matrix execution modules (MXM), switch ex-
+ecution modules (SXM) and memory modules (MEM). Each
+functional unit (core) accepts a set of instructions; for example,
+the MEM unit could receive the instruction to put a vector onto
+one of the data streams or store the results from the data stream
+in its available SRAM. As soon as data is loaded onto a data
+stream, it automatically ‘flows’ in the direction of the stream,
+which can be either EAST-bound or WEST-bound. When an
+addition needs to be performed, both inputs need to arrive
+at the same time as the add instruction at the corresponding
+VXM core. This design choice puts the burden of optimization
+on the software generating the instructions. This is either done
+by the Groq compiler automatically from an ONNX-graph
+input or manually controlled by a user through the exposed
+Groq-API, which has different levels of abstraction on top of
+the Groq-ISA. To support the creation of large-scale systems,
+the GroqChip has dedicated Chip-to-Chip modules that are
+capable of performing off-chip communication without losing
+their determinism [32]. For this work, we will mainly utilize
+the VXM and MEM units, The memory modules add up
+to a total of 220 MiB of on-chip SRAM. Each superlane
+implements a 4x4 mesh of vector ALUs capable of doing
+x16-SIMD. Each ALU has a 32-bit input operand but with
+the exception of additions and multiplications, instructions are
+done in a reduced-precision FP32 format.
+3) Graphcore IPU [4]: The Graphcore Intelligence Pro-
+cessing Unit (IPU) is designed for efficient execution of fine-
+grained operations across a large number of parallel threads.
+By design, the IPU offers true Multiple Instruction, Multi-
+ple Data (MIMD) parallelism. This unique style of parallel-
+processor design adapts well to fine-grained computations that
+exhibit irregular data-access patterns. Each IPU contains 1,472
+tiles, containing 1 core and 624KiB of SRAM memory. A sin-
+gle core can only access the memory in its own tile. Intra-IPU
+communication relies on a powerful, low-latency interconnect
+called IPU exchange. For inter-IPU communications, each
+chip contains 10 so-called IPU links. The IPU compute
+units, called Accumulating Matrix Product (AMP)
+units, support FP32 arithmetic and are designed to accelerate
+matrix multiplications and convolutions. With respect to the
+programming model, the IPU adopts the Bulk Synchronous
+Parallel (BSP) model [33] through which it organizes its
+compute and data-exchange operations. This abstraction for
+parallel computations consists of multiple sequential super-
+steps. A superstep consists of a local computation phase;
+every process (tile, in the IPU case) operates in isolation
+performing compute only on its local memory, followed by a
+communication phase where each process can exchange values
+needed by other tiles. These activities are concluded with a
+barrier synchronization phase; only when all processes have
+reached the barrier can the next superstep be started. Because
+of this, the IPU can be described as a true BSP machine.
+4) Google TPU [7]: The TPU (version 1) was designed
+as a systolic-array processor for inference, only supporting
+8/16-bit operations. By supporting only matrix-multiply and
+4
+
+TABLE I: Overview of all hardware used in experimental setups
+Device
+On-chip Memory
+Process node
+Transistor count (Bn)
+Base-boost freq. (MHz)
+TDP (W) Software
+AMD 3955WX CPU *
+128 GB DDR4
+7 nm
+19.94
+3900 - 4300
+280
+TF 2.11.0
+GroqChip TSP
+230 MB on-chip
+14 nm
+26.8
+900
+-
+Groq SDK 0.9.1 ***
+Nvidia A100 GPU
+80 GB HBM2e
+7 nm
+54.2
+1275 - 1410
+400
+TF 2.11.0
+Graphcore IPU (GC200) **
+900 MB on-chip
+7 nm
+59.4
+1330
+185
+TF IPU 2.6.3+gc3.0.0
+Google TPUv3
+32 GiB HBM
+16 nm
+(est.) 11
+940
+450
+TF 2.11.0
+*AMD Ryzen Threadripper PRO 3955WX (16-Core) | **Single M2000 in IPU-POD16 (with 4 GC200 chips) | ***TF2ONNX 1.13.0 and ONNX opset 16
+basic nonlinear activation functions, it was unfit for training
+neural networks. Consequentially, an HPC application – for
+example, the one demonstrated in this paper – would also
+not be a suitable fit for this processor. However, with the
+TPUv2, Google shifted their focus towards supporting training
+on their TPU chips. Google added a vector-processing unit
+(VPU) and changed the matrix-multiply units to support the
+FP16 format (FP32, with only a 7-bit mantissa). The VPU
+most likely supports higher precision, as can be deducted from
+results in this work but no confirmation of this is found in the
+public domain. These two major (micro)architectural changes
+made it possible to run a wider range of applications including
+training neural models on the TPU. All are supported through
+the Google XLA compiler taking TensorFlow as input. The
+TPUv3, assessed in this work, is an upgrade in terms of
+functional-unit count, higher memory speed, and optimized
+chip layout, but did not include any fundamental changes.
+A. Performance Predictions
+The IO application has two components that map differently
+onto different types of hardware: i) a part with embarrassingly
+parallel computations for updating local neuron states; and
+ii) a part with SM computations for exchanging membrane
+voltages over gap junctions. Before we proceed to the actual
+experiments, we attempt performance predictions, driven by
+the idiosyncrasies of the different AI-chip architectures.
+Embarrassing parallelism: These are calculations for up-
+dating the state of every single neuron. This boils down to
+elementwise vector operations. The GPU architecture featuring
+one Warp execution per Streaming Multiprocessor or multiple
+Tensor Cores is very well-suited for this type of parallelization.
+The TPU and the GroqChip are both based upon systolic-array
+architectures, both natively supporting Matrix-Multiplication
+but also Vector-Operation operations that can be utilized
+for these calculations. In fact, since neuron updates require
+only
+1-D
+data, the Matrix-Multiplication units (which is
+the focus of these chips) are effectively underutilized in these
+architectures. The IPU, with a large amount of very small
+general-purpose cores, should also do well on parallelizing
+neuron-state calculations, however, its architecture is geared
+towards irregular data-access patterns, which is not essential
+to the particular task. The extra overhead of such advanced
+features, therefore, will not help performance in terms of
+computing this embarrassingly parallel part of the simulation.
+Communication:
+As described previously, gap-junction
+communication employs the gather-scatter operations (essen-
+tially, SM operations) from TensorFlow. For either the GPU or
+the IPU, such operations are handled better due to the different
+execution paths that can be handled within the architecture by
+design. In contrast, the GroqChip and TPU need to handle
+these differently: a naive approach would be to enforce dense-
+matrix operations via one-hot encoding of operands and,
+then, utilizing the matrix-multiplication hardware. In case the
+GroqChip or the TPU happen to use such a strategy, we expect
+that performance will deteriorate very rapidly or memory will
+be depleted with increasing IO-network sizes.
+1) CPU + TensorFlow: For this platform, JIT compila-
+tion through the XLA compiler [34] will be used; it will
+automatically utilize the many threads nowadays available in
+CPUs. We expect decent performance and very accurate results
+because of full FP32 support. Since it is the hardware on which
+brain models are traditionally executed and gives accurate
+results, the CPU will form our baseline. Accelerators should
+outperform this implementation in terms of runtime, especially
+for larger network sizes.
+2) GPU + TensorFlow: The XLA compiler is used, which
+optimizes the graph resulting in a single kernel launch. Among
+others, it does this by “fusing” the calculations. Moreover, this
+fusion keeps intermediate values stored in GPU registers [35].
+The TensorFlow backend for CUDA use Tensor Cores, at
+a loss of FP32 accuracy. However, this only happens when
+explicit matrix-multiplications are requested and not as an
+optimization. So in our case, the compiler will only use float32
+CUDA operations.
+3) IPU + TensorFlow: The IPU architecture is not a perfect
+fit for the embarrassingly parallel part of the computation.
+For the interneuron-communication part, the BSP model is a
+better fit and thus is expected to perform better. However, as
+the topology is given as an unknown parameter to the model,
+the IPU compiler can not be expected to allocate neighboring
+cells on adjacent tiles, resulting in sub-par communication
+performance. Available memory should easily be able to
+handle large problem sizes.
+4) TPUv3 + TensorFlow: The TPU supports FP32 and
+is expected to handle our workload, especially for the un-
+connected case, very well. As Google put much effort into
+TensorFlow support, gather-scatter operations are expected to
+be optimized, to the best of the hardware capabilities. Because
+of FP32 support in the v3 model, we expect correct numerics
+in the output, as well.
+5) CPU/GPU + ONNX: Expectations are the same as for
+CPU/GPU + TensorFlow. We expect the XLA compiler to
+outperform the ONNX runtime slightly for the CPU case
+simply because it can perform whole-program optimization.
+For the GPU, this effect is expected to be much larger and
+the TensorFlow is expected to dominate ONNX as the ability
+5
+
+to fuse kernels will be a big advantage for TensorFlow over
+single-kernel invocations in ONNX. Especially the invocation
+overhead for small GPU kernels will hurt the performance
+of the ONNX-GPU-runtime. TensorRT is also a supported
+backend in ONNX that is expected to outperform the CUDA
+runtime in performance; it will, however, drop precision as the
+backend switches to TF32 numerics.
+6) GroqChip + ONNX: The GroqChip is a new, upcoming
+modified systolic-array processor. Its compiler takes in the
+ONNX graph but is not limited to executing this on an
+operation-per-operation basis as it recompiles the full ONNX
+graph at once. Therefore, it can potentially perform the same
+optimizations as the XLA compiler for the TPU. As the first
+version of the architecture, current compiler development is
+still exploring ways to map non-standard ML-operations to
+the hardware. Besides, the GroqChip VXM is not capable of
+doing all operations in IEEE FP32 arithmetic. Because of this,
+it can be expected to perform slightly better than the TPU at
+the cost of reduced accuracy.
+V. EXPERIMENTAL SETUP
+A. Benchmarking Parameters
+Each platform is benchmarked for performance on a set
+problem (i.e., network) size as well as for its performance
+scalability by simulating the IO network for small population
+sizes in the range [43, 53, . . . , 203] and, again, for larger sizes
+in the range [303, 403, . . . , 1003], where the third power is
+an artifact of the cubic network-topology generation method.
+These experiments are focusing on four different aspects of
+each AI platform, discussed next.
+1) Unconnected Network: By removing the communica-
+tion step (gap junctions) from the model, we obtain a (bio-
+logically unrealistic) compute-heavy, embarrassingly parallel
+workload. First, we measure the setup time for each AI
+platform, including on-chip buffer allocation, Ahead-Of-Time
+(AOT) compilation or definition of Just-In-Time (JIT)-enabled
+functions. Next, we simulate an IO network for 100ms of
+biological time and take the minimum wall-clock time from
+5 runs (including data-transfer times). For JIT targets, the
+first runtime (if outside the other runtimes’ standard deviation)
+minus follow-up runtimes is taken as the JIT compilation time,
+such that we can compare setup times between AOT and JIT
+targets.
+2) Connected Network: By restoring gap junctions into the
+IO network, we assess communication overhead. Runtimes are
+obtained in an identical way as before, yet the expectation here
+is that they are markedly longer than the unconnected case.
+3) Numerical Validation: Measuring performance is our
+main focus, yet this must not come at the cost of functional
+correctness. Here, we simulate connected networks up to 729
+neurons for 10 seconds of biological time and numerically
+compare the various results to the reference CPU output.
+4) Numerical Stress-test: Here, we simulate the IO in a
+more biologically realistic way that is of interest to neurosci-
+entists: We add more variance to the neural parameters and,
+most importantly, a lot of external current inputs (simulating
+other brain regions) that will evoke action potentials (spikes)
+in the IO dynamics. These fast transients will stress-test the
+numerical performance of the AI hardware, especially non-
+IEEE754 targets (Tensor Cores and GroqChip). We perform
+this experiment on the smallest 64-neuron network and then
+compare for numerical accuracy against the CPU.
+Benchmarking
+is
+implemented
+in
+a
+publicly
+avail-
+able
+and
+modular,
+extensible
+framework,
+downloadable
+from GitLab https://gitlab.com/neurocomputing-lab/Inferior
+OliveEMC/ioperf.
+The
+main
+benchmarking
+script
+auto-
+discovers available hardware, runs the appropriate benchmarks
+and records results. Used software versions are also shown in
+Tab. I.
+B. Hardware-specific Optimizations
+While our original goal was not to write platform-specific
+code, we found that by default some of the AI platforms did
+not perform very well. For example, most platforms defaulted
+to copying over the entire parameters arrays for each kernel
+invocation, which was not needed for this mostly constant data.
+For a fair comparison between hardware platforms, we allowed
+optimizations to be applied to hardware-specific code that
+either led to operation fusion across different execution kernels
+or prevented unnecessary device-host data transfers. The exact
+optimizations have been applied in close collaboration with
+Graphcore and Groq for the respective chips, and are as
+follows:
+1) TensorFlow XLA: The TensorFlow graph executor typi-
+cally performs each operation separately when a graph is run
+with a corresponding kernel invocation. A different way to
+run TensorFlow models is made available by XLA, which
+turns a TensorFlow graph into a series of kernels created for
+a particular application. These kernels can take advantage of
+application-specific information for performing optimizations,
+e.g., operation fusion. The CPU, GPU, and TPU are the
+three available backends for the XLA compiler. For the IO
+application, a TensorFlow wrapper function was implemented
+that fuses up to 40 timesteps together for each call in order to
+fully exploit the XLA compiler.
+2) ONNX: Except for the GroqChip, all ONNX imple-
+mentations build on top of onnxruntime or onnxruntime-
+gpu. We enable all backend-supported graph-optimizations.
+Explicit IOBindings are used to prevent unneeded host-
+device data copies. Parameters are copied once to the device
+at simulation start. Then, state is allocated twice, with each
+timestep toggling between two buffers, one as the input state
+and the other as the output (next) state. For TensorRT, we
+leave the default behavior of using TF32 enabled, otherwise,
+it will not utilize its Tensor Cores.
+3) Groq: After the compilation of an ONNX graph with
+the Groq Compiler, the binary can be executed directly on the
+GroqChip. A naive approach here would be to invoke this
+binary 40 times for 40 timesteps and move the data back
+and forth continuously since the GroqChip only has SRAM
+which is fully managed at compile time. However, the Groq
+Compiler is able to tie input and output tensors together into
+6
+
+10
+1
+100
+101
+102
+103
+CPU (ONNX)
+CPU (TF)
+10
+1
+100
+101
+102
+103
+GPU (TF)
+IPU (TF)
+102
+103
+104
+105
+106
+10
+1
+100
+101
+102
+103
+TPU (TF)
+102
+103
+104
+105
+106
+GroqChip (ONNX)
+Network size (#neurons)
+Time required to simulate one biological second (s)
+Fig. 2: Runtime performance (lower is better), comparison between CPU baseline, GPU and AI chips. For scale, the mouse (•) and human (▲) Inferior Olive
+are shown as running in realtime in all figures. The CPU is included twice to explain the observed switching behavior of the IPU. On the CPU, while the XLA
+optimizer builds a single-core, connected-network simulation, it builds a multicore, unconnected-network simulation (as observed by load-testing), leading to
+an unexpectedly slow simulation for the latter case. The same behavior can be observed for the IPU, which uses the XLA compiler as well.
+a persistent memory buffer in the on-chip SRAM. Utilizing
+this results still in 40 invocations of the binary but skips the
+continuous I/O between host and accelerator. A more radical
+way to improve the performance is to compile the 40 timesteps
+into a single ONNX graph that can then be converted with the
+Groq Compiler; this method will reduce 40 invocations to a
+single invocation. We implemented all optimizations as long
+as the compiler was able to compile them. The 40 timesteps at
+once quickly ran into compiler errors with growing networks.
+4) Graphcore:
+The IPU has architectural support for
+streaming memory. This means that we can run a single
+program on-chip for the entire simulation that will stream
+out samples every 40 timesteps. The inner, unsampled 1-
+msec 40-timestep loop, is run using ipu.loops.repeat,
+after
+which
+the
+recorded
+voltages
+are
+pushed
+to
+an
+IPUOutfeedQueue with a 200-sample size. This is, then,
+looped once more using ipu.loops.repeat for the re-
+quired amount of milliseconds to simulate and wrapped in
+a TensorFlow JIT function. Furthermore, the fast-math op-
+timization is enabled, 128 IPU tiles are reserved for I/O
+with place_ops_on_io_tiles = True and program
+execution is limited to a single IPU.
+VI. EXPERIMENTAL RESULTS
+With the exception of the reference CPU, for brevity we
+report here either TensorFlow or ONNX results, depending
+on which of the two leads to better performance. Overall
+performance plots are shown in Fig. 2 and will be detailed
+in the next sections. In general, it is found that, for the
+IO application, the ONNX ports are outperformed by their
+TensorFlow counterparts. This is due to the fact that the
+onnx-runtime library currently does not perform as extensive
+optimizations as the XLA compiler. For example, the CUDA
+target translates each compute step into a single predefined
+kernel call. The TensorRT backend performs operator fusion,
+resulting in multiple kernels that chain arithmetic operations.
+Still, the CUDA XLA-backend vastly outperforms both ONNX
+CUDA targets, and as such we removed the corresponding
+findings from the main analysis. Note that the Groq platform
+only supports AOT compilation of ONNX models.
+A. Compilation Time
+Both software stack and hardware influence program setup
+time, as illustrated in Fig. 3 for the largest network (729 cells)
+that could fit in all AI chips. The CPU compiles the fastest
+across the board as we have a direct translation of ONNX
+operations to their CPU-optimized callbacks. The TensorFlow
+(XLA) version, not included in the figure, was much slower
+7
+
+100
+101
+102
+103
+Compile time (s)
+(A). Network type at compile time (n=729)
+Unconnected
+Connected
+CPU
+IPU
+GPU
+TPU
+GroqChip
+10
+2
+10
+1
+100
+101
+102
+Run time (s)
+(B).Connected network size at run time
+n=64
+n=729
+n=125000
+1.0x
+2.3x
+8.7x
+7.7x
+32.9x
+1.0x
+4.3x
+17.4x
+16.4x
+2.6x
+1.0x
+28.6x
+269.5x
+1208.5x
+Fig. 3: (A) Setup (AOT+JIT compile + memory allocation) times for a network
+of 729 neurons in both Unconnected- and Connected-network configurations.
+JIT compile times are extracted from the first run of 5 performance runs
+and added to the initial setup time (if outside one standard deviation). (B)
+Performance and speedup of different AI chips vs. the CPU reference on
+the Connected benchmark, for different network sizes. Sizes were chosen to
+be the smallest (64) and largest connected networks that could fit on the
+GroqChip (729) and the TPUv3 (125,000). The rightmost GroqChip bar is
+absent, corresponding to the model that could not be compiled.
+due to the increased compiler complexity. Both the IPU and
+GPU exhibit similar JIT compilation speeds. The GroqChip’s
+AOT compiler takes significantly longer for this workload due
+to the explicitly concatenated 40 timesteps. The GroqChip
+version with a single timestep per program compiles much
+faster than the Graphcore or A100 versions, but at a small
+performance loss.
+B. Runtime Performance
+1) Unconnected Network (embarrassingly parallel): For
+unconnected cells, neural dynamics are expressed only using
+vectorized operations. As predicted, this fits the compute
+paradigm of the GPU very well. Performance scales linearly
+with problem size (horizontal line), showing that the GPU
+cores are underutilized for all simulated network sizes.
+The TPU and GroqChip, as systolic-array-based processors,
+were expected to be a poorer architectural fit because large
+parts of the chip would be left unused. Still, the focus on
+efficient vector operations could result in speedups. We can
+indeed observe this in Fig. 2, although in different ways. The
+TPU, similar to the GPU, flatlines across all problem sizes,
+although being 4.1x slower. Consequently, memory capacity
+is not a problem for the TPU but performance capping in raw
+single-cell computations due to architectural design choices.
+In contrast, the GroqChip starts out 2.0x faster than the GPU,
+quickly loses this edge and, between 103 and 104 cells, starts
+to hit its memory-capacity limits, degrading performance with
+higher problem sizes. Networks of more than 640, 000 cells
+simply do not fit on the chip. The GroqView analyzer confirms
+that the problem is core-to-core-memory communication and
+that most dedicated cores are not used most of the time.
+The IPU was expected to perform well given its large core
+count but the very homogeneous compute load proved a poor
+fit for its MIMD design, leading to large under-utilization
+of the chip. With respect to real-time performance, only the
+GPU followed by the GroqChip (ignoring memory issues) and
+marginally the TPU makes the 1-sec cut.
+2) Connected Network (high communication overhead): As
+predicted, communication patterns induced by a small number
+of gap junctions lead to a large performance reduction of 4.6x
+for small networks on the GPU. For higher problem sizes,
+performance drops at a growing rate, with a 141x degradation
+for networks of 106 cells. The AI chips fare much better here,
+most of which initially shows a less than 20% reduction in
+performance against their unconnected counterparts.
+As an exception, the GroqChip’s connected-network sim-
+ulation runs 2.5x slower than the unconnected version; even
+so, it outperforms the GPU on very small, connected neural
+networks by a 3.7x speedup. However, the GroqChip (as
+expected) converts the SM communication into a dense-
+matrix multiplication, making the best out of its deterministic-
+execution hardware. This quickly leads to prohibitively large
+matrix multiplications and, beyond 729 cells, the scheduler
+is unable to allocate the necessary instructions. In effect, the
+GroqChip loses its edge over the GPU for larger networks.
+In contrast, the TPU shows nearly identical behavior to
+the unconnected case and its performance does still not scale
+with problem size. This changes around networks larger than
+105, where the JIT compiler seems to run into performance
+problems. Here, we observed large random fluctuations in
+performance that either led to approx. 1-sec or very long more
+than 400-sec run-times over the 5 repeated runs. We expect
+that these originate from memory limits of the TPU and had
+to stop benchmarking due to impractically large run times.
+However, we could not determine the true source of variation.
+The IPU, severely underutilized for the unconnected case,
+sees in fact a performance improvement when we increase
+the communication overhead in small networks. While coun-
+terintuitive, this is actually the same effect we see on the
+XLA-based CPU backend. Here, we see that gap junctions
+force the simulation to become single core, which actually
+becomes faster than the parallel, multi-core, unconnected case,
+due to the lower synchronization overhead. Around 104 cells,
+this behavior changes, gap-junction communication becomes a
+fixed overhead on top of normal simulation. At a certain point,
+this growth becomes exponential and the largest simulated
+network does not fit on a single IPU anymore.
+C. Numerical Validation
+While all AI chips outperform the CPU baseline, it is wise
+to explore also any potential decrease in numerical accuracy
+of the different runs with respect to that of the same CPU.
+Here, we compare 1-msec sampled cell somatic voltages of an
+extended, 10-sec simulation for a 729-cell, connected network
+8
+
+IPU
+GPU
+TPU
+GroqChip
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Abs. difference to CPU reference (mV)
+Left
+boxp.
+Right
+boxp.
+Vgroqchip
+Vcpu
+Fig. 4: Numerical-accuracy validation (lower is better). Box plots show
+deviations from CPU baseline, as recorded over two 1-sec timespans, one
+at the start (left) and one at the end (right) of the 10-sec numerical-validation
+simulation. The GroqChip result, showing the largest deviation, is plotted in
+the upper left corner together with the two recording spans .
+(the largest population supported by all platforms); results are
+shown in the box plots of Fig. 4.
+As expected, platforms supporting IEEE754 floating-point
+numerics (IPU, GPU, TPU) show accurate reproduction of
+voltage traces. The IPU, even with fast-math enabled, is the
+most faithful to the CPU baseline. The GPU and TPU exhibit
+increasingly large deviations but still fall within limits explain-
+able by floating-point instruction reordering. The GroqChip,
+while supporting FP32 number storage, implements certain
+operations at lower precision including exponent calculation.
+This is visible by a quite large mV -order deviation from the
+CPU baseline, for a process that happens at the 10 − 100mV -
+scales. This voltage difference mostly stems from a slowly
+accrued phase difference for the oscillating cells. TensorRT
+(not shown in this plot) is by default using Nvidia’s TF32, for
+which accuracy was found similar to that of the GroqChip.
+D. Numerical Stress-test
+The numerical stress test increases neuronal variation and
+adds external inputs that lead the neurons to spike. These
+fast transients can not be simulated using FP16 precision, but
+reduced-accuracy FP32 operations as used in Tensor Cores
+or GroqChip are still untested. Once more, we compare the
+deviation of the somatic-voltage traces of the various AI chips
+against the CPU baseline.
+Again, the platforms with native FP32 support show the
+lowest deviation: For the IPU this is 0.087mV , for the
+GPU this is 0.135mV and for the TPU this is a 0.672mV
+maximum absolute difference from the CPU baseline. These
+moderate, mV -order differences can be explained by small
+spike-time differences which due to the large neuronal-spike
+sizes quickly lead to large voltage discrepancies. Importantly,
+all simulations run stably; i.e., do not cause this chaotic IO-
+model simulator to crash. The GroqChip simulation initially
+starts out the same as in the numerical-validation test, but as
+soon as input perturbations are applied, it becomes unstable
+and settles on voltage deviation at a measured maximum
+of 8.51 × 1036mV , unacceptable for scientific applications.
+Notably, the error stabilizes at this point and does not explode
+to infinity or NaN values, as observed with FP16 simulations.
+To regain numerical stability, we tried lowering the time-
+stepping constant ∆t 10-fold and 100-fold for the GroqChip
+simulation, but this did not lead to results more closely in
+range with the CPU ones.
+VII. DISCUSSION
+As this work has shown, utilizing AI platforms for executing
+highly biologically plausible SNN workloads is made exceed-
+ingly user-friendly when using a ML-library like TensorFlow.
+Arguably, even better performances could be obtained by
+coding via the various hardware SDKs (Software Development
+Kits), but it is unrealistic to expect computational scientists
+to learn the low-level details of all hardware options made
+available to them these days.
+As shown, the added benefits from JIT compilation make
+a hand-coded CUDA implementation perform on par with the
+XLA-compiled TensorFlow version while, at the same time,
+allowing one to move easily to a new piece of hardware
+when this is released. We expect that, in the future, more
+classical HPC workloads will see ML-library, that is, tensor-
+based implementations.
+For promising upcoming accelerators like those by Graph-
+core and Groq, we believe that future speedups will chiefly
+come from software and compiler upgrades, as current SDKs
+are mostly optimized for ML workloads. For instance, gather-
+scatter operations on the GroqChip do not have to be im-
+plemented as dense-matrix operations, memory can be better
+utilized, and better support for iterative programs must also
+be introduced. The TPU which is architecturally similar to the
+GroqChip, clearly performs gather-scatter operations in a more
+efficient way than encoding indexing as one-hot vectors.
+Speed-ups could be gained by effective use of mixed
+precision on the IPU or reduced accuracy FP32 operations
+using Tensor Cores or GroqChip. For the IPU, this would
+constitute a separate numerical sensitivity analysis to find out
+which parts of the compute graph can be lowered to (stochastic
+rounded) FP16. As shown, the accuracy loss on Tensor Cores
+and GroqChip does in its current form not allow for brain
+simulation, but possible these could be put to use by switching
+the integration scheme or other numerical optimizations.
+Finally, this work has steered clear off multi-chip topologies.
+All discussed architectures do support specifically developed,
+low-latency, chip-to-chip hardware and assorted communica-
+tion protocols. In many ways, such coherent communication
+is a bigger and more timely challenge than acceleration speed
+itself, which would deliver massive benefits for large-scale
+SNN simulation (or training). However, tapping into those
+platform-specific interfaces requires SDK-specific coding of
+the IO application; relying on TensorFlow or ONNX frame-
+works will, generally, not work. Careful and platform-specific
+coding is necessary, which we leave as future work.
+VIII. CONCLUSION
+In this work, we built the first ML-library-based, effi-
+cient implementation of a large-scale, conductance-level brain
+9
+
+model, the Inferior Olive (IO). Subsequently, we benchmarked
+the performance of simulating this model on a 16-core AMD
+Ryzen Threadripper PRO 3955WX CPU, an Nvidia A100
+GPU, and different AI chips (Graphcore IPU M2000, Gro-
+qChip and Google TPU v3). We found that all accelerators
+provide significant speedups over the CPU implementation.
+For this specific problem, the GPU and TPU seem most
+fit for simulation, with the TPU setting a new record for
+real-time IO simulation. For small networks, the GroqChip
+outperforms the other accelerators, but large networks could
+not fit in the on-chip instruction memory. More generally, we
+hypothesize that modern ML-libraries possess the semantic
+power to model classical problems in scientific computing.
+These, then, map extremely well to ML-driven, novel AI-chip
+architectures, which apart from large performance benefits,
+also benefit from reduced development times. For example, the
+version of our IO application running on the TPU outperforms
+the handwritten and hand-optimized CUDA implementation by
+a large factor, at a fraction of the development cost. The exact
+hardware trade-off will vary on an application-by-application
+basis, and hardware selection also benefits significantly from
+the hardware-agnostic model description.
+ACKNOWLEDGEMENT
+This research would not have been possible without access
+to dedicated hardware: The RTX6000 was gifted from the
+NVIDIA Hardware Grant Program, Google provided free
+cloud credits for TPU access and Graphcore provided access
+to POD16 machines through Paperspace and Gcore cloud
+via its Academia program. Furthermore, we’d like to thank
+Graphcore employees for helping with optimizing the IPU
+code and Dr. Mario Negrello for neuroscientific insights.
+REFERENCES
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf,len=890
+page_content='Tricking AI chips into Simulating the Human Brain:  A Detailed Performance Analysis  Lennart P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Landsmeer∗  Quantum & Computer  Engineering Department  Delft University of Technology  Delft, The Netherlands  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' of Neuroscience  Erasmus Medical Center  Rotterdam, The Netherlands  ORCID 0000-0003-0010-7249  Max C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Engelen∗  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' of Neuroscience  Erasmus Medical Center  Rotterdam, The Netherlands  &  Maxeler IoT Labs  Delft, Netherlands  ORCID 0000-0002-5762-1276  Rene Miedema  Quantum & Computer  Engineering Department  Delft University of Technology  Delft, The Netherlands  &  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' of Neuroscience  Erasmus Medical Center  Rotterdam, The Netherlands  ORCID 0000-0002-0447-1083  Christos Strydis  Quantum & Computer  Engineering Department  Delft University of Technology  Delft, The Netherlands  &  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' of Neuroscience  Erasmus Medical Center  Rotterdam, The Netherlands  ORCID 0000-0002-0935-9322  Abstract—Challenging the Nvidia monopoly, dedicated AI-  accelerator chips have begun emerging for tackling the compu-  tational challenge that the inference and, especially, the training  of modern deep neural networks (DNNs) poses to modern  computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The field has been ridden with studies assessing  the performance of these contestants across various DNN model  types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' However, AI-experts are aware of the limitations of current  DNNs and have been working towards the fourth AI wave  which will, arguably, rely on more biologically inspired models,  predominantly on spiking neural networks (SNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' At the same  time, GPUs have been heavily used for simulating such models  in the field of computational neuroscience, yet AI-chips have not  been tested on such workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The current paper aims at filling  this important gap by evaluating multiple, cutting-edge AI-chips  (Graphcore IPU, GroqChip, Nvidia GPU with Tensor Cores and  Google TPU) on simulating a highly biologically detailed model of  a brain region, the inferior olive (IO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This IO application stress-  tests the different AI-platforms for highlighting architectural  tradeoffs by varying its compute density, memory requirements  and floating-point numerical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Our performance analysis  reveals that the simulation problem maps extremely well onto the  GPU and TPU architectures, which for networks of 125,000 cells  leads to a 28x respectively 1,208x speedup over CPU runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  At this speed, the TPU sets a new record for largest real-time IO  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GroqChip outperforms both platforms for small  networks but, due to implementing some floating-point operations  at reduced accuracy, is found not yet usable for brain simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Index Terms—AI accelerator, GPU, Brain simulation, com-  puter architecture  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' INTRODUCTION  To date, GPUs have achieved spectacularly better perfor-  mance in deep learning (DL) than CPUs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Recently, novel,  specialized AI hardware platforms have begun to emerge,  holding the promise of accelerating training and inference  even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The workloads targeted mainly are artificial, and  specifically, deep neural networks (DNNs), which have shown  great potential in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' On the other hand, highly  biologically plausible models such as conductance-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=',  Hodgkin-Huxley) neurons have not attracted similar atten-  tion from AI-chip manufacturers and analysts alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This is  strange, given that biological brains – the inspiration behind  these DNNs – are modeled using equations built on similar  elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' What is more, high-detail models are  touted as the next AI wave, which is intended to be more  biologically inspired than its predecessors [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Therefore,  it makes sense both for neuroscientists and for AI researchers  to reach for these AI accelerators and deploy them for brain  simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' yet no performance studies exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  In this work, we evaluate multiple, cutting-edge AI chips  (Graphcore IPU [4], GroqChip [5], TensorRT-capable GPU [6]  and Google TPU v3 [7]) on simulating a highly biologically  detailed model of a brain region, the Inferior-Olivary nucleus  (IO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Biologically detailed brain models, such as the IO,  chiefly involve addition, multiplication, division and expo-  nential operations, arranged as sparse computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' There  is, thus, a large operation overlap with artificial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Therefore, new AI chips seem a good fit for these types  of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This IO application represents timely, relevant  research and is constructed as an extended-Hodgkin-Huxley  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' It is very suitable for stress-testing the different AI  platforms and highlighting architectural tradeoffs by adjusting  the compute density, memory requirements and numerical  accuracy of the IO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Evaluation is performed using the  application encoded as a TensorFlow 2 [8] kernel, which in the  case of the GroqChip, is necessarily compiled to its ONNX [9]  equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' ONNX is an intermediary tool used to convert  models between different machine-learning (ML) frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  In the analysis of the different accelerators, the exact same  TensorFlow model is used, ensuring a fair comparison across  the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This is either used directly or ported to ONNX  via the Python package tf2onnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' TensorFlow is a high-level  API, requiring little to moderate intervention from the user  and is therefore suitable for a wide user base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A schematic  overview of this setup is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  While all accelerators in this study support chip-to-chip  communication, this work constrains the application to single-  chip performance comparisons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' multi-chip is left as future  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='13637v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='LG]  31 Jan 2023  IO-model  TensorFlow  CPU  GPU  IPU  ONNX  tf2onnx  CPU  GPU  Groq  TPU  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 1: Overview of the performance-analysis strategy followed in this work  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The contributions of this work are:  We take a deep dive into four cutting-edge AI architec-  tures with a focus on biologically plausible spiking neural  networks (SNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  We build the first ML-library-based, efficient implemen-  tation of a detailed brain model, the Inferior Olive (IO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  We deploy the IO model onto the four AI platforms and  benchmark their performance and numerical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  We demonstrate that modern ML libraries are seman-  tically able to model classical problems in scientific  computing, offering large performance gains and reduced  development times while remaining hardware-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Lastly, this work is the first to ever simulate a realistic  mouse-sized IO model with real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  The paper is organized as follows: Section II presents  related works in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Section III introduces the IO  model used as our benchmarking application, while Section IV  briefly presents the four AI architectures under evaluation and  attempts some performance predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Section V ensures  experiment reproducibility by detailing the experiment param-  eters and platform configurations used to acquire our results  presented in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A general discussion of our findings  is included in Section VII and conclusions in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' RELATED WORKS  Models of biological neurons come in various levels of  detail, ranging from population-level dynamics, from simpli-  fied models of single neurons to highly detailed biophysically  realistic neurons [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Coarse models of single neurons, no-  tably leaky-integrate-and-fire (LIF) type models have seen a  renewed interest in the DL-community (often referred here  to as SNNs) as an alternative to artificial neural networks  (ANNs) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In contrast, computational neuroscience is usu-  ally interested in biophysically accurate models that model the  underlying biological processes in a way that makes it possible  to gain insights about these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' These conductance  based models can be made more realistic by modeling of  their 3-dimensional structure (the morphology) using multiple  discretized compartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Multi-compartmental conductance-  based neurons are then simulated by explicit calculation of  electrical currents flowing within, between and into discretized  compartments [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Due to the computional resources needed for large-scale  conductance level brain simulations, computational neuro-  science was an early adopter of general-purpose GPU (GP-  GPU) in the HPC environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Notable GPU-based ex-  amples of large-scale, biologically detailed brain simulators  include CoreNeuron [13], which enabled porting of exist-  ing conductance-level NEURON [14] models to the GPU,  and more recently, Arbor [15], a library-based approach to  performance-portable, large-scale brain simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Their suc-  cess shows that the computational problems of neuroscience  map well to GP-GPU platforms and result in significant  speedups for large-scale brain models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Still, even with hand-  optimized CUDA code [16], the IO application (to be detailed  in the next section) at biological sizes runs order-of-magnitude  slower than the biological brain, hampering research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  With respect to TensorFlow-based implementations of  conductance-level models, there is PymoNNto [17], an attempt  to bring the Brian [18] API of neural models to TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  While faster than the Brian simulator on a GTX1080 GPU,  performance was not a primary goal and the architecture pro-  hibits optimizations using TensorFlow’s JIT compiler backend,  by scattering the computational definitions across the code-  base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Although this shows that TensorFlow does express the  right API surface for neural models, no efficient ML-library  based conductance-level GP-GPU brain simulators exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Simplified SNN models have readily available GP-GPU  implementations of LIF and similar models as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' High-  level ML-libraries like TensorFlow and PyTorch allowed for  the hardware-agnostic implementation of their neural dy-  namics, considerably lowering development efforts to build  SNN simulators for GP-GPU simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For example, Nengo  DL [19] allows for the GPU-based simulation of existing SNN  models defined in the Nengo framework using TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Beyond just simulating neural networks on the GPU, novel  developments in surrogate gradients for event-based SNNs  and automatic gradient calculation provided by ML-libraries  allowed for the nearly simultaneous appearance of similar  SNN deep-learning libraries Norse [20], snnTorch [21] and  SpikingJelly [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' BindsNET [23] is another, efficient SNN  implementation in PyTorch with a focus on reinforcement  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Again, these project show that not only ML-libraries  have the expressive power and performance needed to run  large-scale SNN models, also that this arguably can be devel-  oped faster than hardware-specific low-level code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' As these  libraries had DL-oriented goals in mind, none of these imple-  ments multi-compartmental, conductance-level neural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Simplified SNN models also led to the development of  specialized neuromorphic hardware to simulate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Numer-  ous publications show the benefits of using these chips for  simplified-SNN simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For a short review of the various  chips, we point the reader to [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' While exciting with respect  to low-power inference of SNN-based deep-learning models,  these chips, due to their hardwired dynamics, lack the ability  to simulate conductance level neural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  On AI chips that have the semantic power to capture  more general HPC workloads, little has been published about  both simplified and conductance-level SNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' With respect  to simplified SNN simulation, we find just one preprint tar-  geting an AI chip, introducing an IPU-optimized version of  2  snnTorch [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Training throughput of a dense 3-layer LIF  network on an image classification task is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='4x higher on the  IPU than on the A100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The reported performance benefits  decrease if the network size is increased, with the A100  apparently underutilized throughout the entire application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  This shows the potential of using the IPU for simple SNN  workloads, but the performance characteristics of other AI  chips or more complex SNNs are not yet obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  No works have been published targeting AI chips with  conductance-level models or other biologically realistic brain  simulation scenarios, neither using high-level ML libraries or  hardware-specific SDKs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' To the authors’ knowledge, this is the  first work to implement an efficient, conductance-level, multi-  compartmental neuron in an ML library and also the first to  benchmark multiple AI chips on this workload class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' THE INFERIOR-OLIVE APPLICATION  The IO is a intrinsically oscillating brain region located  in the brainstem, and is key to motor control and learn-  ing [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The estimated neuron population for the mouse  brain is approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 104 neurons [27] and for humans between  106 − 107 neurons [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' These numbers will be referred to  during hardware-performance evaluation (Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In this  work, we will capture in TensorFlow 2 the IO nucleus as  an extended Hodgkin-Huxley (eHH) model, conductance-level  brain model, first published in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The model is a good  example of the computational load of realistic brain models  and, also, a good fit for our benchmarking purposes, since  it captures complex neuron dynamics and fast interneural  communication (in the form of gap junctions), as will be  shown next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  We restate the IO-neuron main equations in this section, but  refer the reader to [29] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In addition, we model  connectivity based on the network described in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  1) The cable model:  Cm  dV (i)  dt  = −  �  k∈Channels  I(i)  k  −  �  i∈Compartments  Ik,j  −  �  i∈Gap junctions  Igj,k,j − I(i)  app  (1)  The eHH model describes the membrane that envelops the  neurons as a capacitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The cell internal voltage can thus be  calculated by integrating currents flowing into and out of the  cell (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Here, Iapp is an optional term describing externally  applied currents by the experimenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  2) Channel currents: Channels (CaL, h, KCa, Na, Kdr, K,  CaH, Na, K) allow currents to flow through the cell membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  They produce this current as function of internal state variables  changing over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In general, this current (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 2) results from  the potential difference to an channel specific reversal potential  E multiplied by the product of one or more internal gating  variables, each optionally raised to an integer power (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  The gating variables follow an Ordinary Differential Equation  (ODE), that brings them to a certain cell-voltage dependent  steady state at a given speed (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' These latter equations  Listing 1 Axonal sodium-channel current  m inf = 1/(1+ t f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' exp ( −(V axon+30) / 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 5 ) )  h  inf = 1/(1+ t f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' exp ( ( V axon+60) / 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 8 ) )  tau h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='5* t f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' exp ( −(V axon+40) /33)  dh dt = ( h inf −h ) / tau h  I na  = g Na * ( V axon−V Na) * m inf **3* h  Listing 2 Sparse gap-junction current  V d i f f  =  t f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' gather ( V dend ,  gj  src ) \\  −  t f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' gather ( V dend ,  g j  t g t )  I  per  gj = V d i f f g gj (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='2 + \\  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='8 t f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' exp ( −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='01* V d i f f * V d i f f ) )  I gapp  =  t f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' tensor scatter nd add (  t f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' zeros  like (V) ,  t f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' reshape ( gj  tgt ,  ( −1 ,1) ) ,  I  per  gj )  are usually gaussian or sigmoidal functions of the voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  For certain fast operating channels we set n(t) = n∞(V ) as  a numerical stability optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Ii = ¯gi  ��  k  ni,k(t)mk  �  (V − Ei)  (2)  τn (V ) dn  dt = n∞ (V ) − n (t)  (3)  3) Compartmental currents: A single IO cell consist of  three separate compartments, the axon, soma and dendrite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Currents flowing between different compartments are modeled  resistively as: Ii,j = gi,j (Vj − Vi)  4) Gap-junction currents: Gap junctions are direct electri-  cal connections between different IO cells and allow current  to flow between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' They follow experimentally determined  Connexin-36 protein dynamics:  Igj = ggj∆V  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='8 exp  �  −∆V 2/100  ��  (4)  with ∆V the potential-difference between two connected cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  5) Topology: The real IO looks like a large, folded sheet  with mostly local connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' As approximating this structure  is not a focus of this paper, our model neurons are assumed  to exist on a discrete 3-D grid with wrap-around connectivity  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=', a hypertorus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This should exhibit the same non-local  memory-access patterns as a more realistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Connec-  tions are sampled as a function of inter-neuron distance r on  a radially symmetric distribution: p(r) ∝ u(rmax − r)(e−r2 −  e−r2  max)n(r), where n(r) is the density of neurons in the  volume shell around r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This distribution is sampled until we  have 10 connections per neuron on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  6) TensorFlow Translation: The previous equations sum up  to a total of 14 ODEs per neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This system of ODEs is  translated to a series of TensorFlow operators in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' By  defining the model in TensorFlow instead of using platform-  specific APIs, we make sure all platforms have equal op-  timization opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Furthermore, TensorFlow naturally  translates to ONNX models, which is the only high-level  API available for GroqChip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Straightforward translation to  TensorFlow is achieved by storing all state in a large 2d-array  and direct substitution of mathematical expressions by their  3  TensorFlow counterparts (see Listing 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' When certain model  parameters need to be user-specified (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=', gi or Iapp), these  are passed to the TensorFlow kernel, which then needs to be  recompiled before running again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Translating gap junctions to both TensorFlow and ONNX  in a performant way requires expressing them as vector  operations, as opposed to more traditional for-loop-based  approaches [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' With just 10 connections per IO neuron on  average, cell-to-cell communication is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The effective  operation from a TensorFlow perspective is two sparse-matrix  (SM) multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' As a novel contribution in computa-  tional neuroscience, we model those as tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='gather and  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='tensor_scatter_nd_add operations (see Listing 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Apart from being more specific and memory-efficient in  describing SM multiplications, these functions have a direct  mapping to ONNX operators as Gather and ScatterND since  ONNX specification opset 11, contrary to SM multiplica-  tions which currently are not possible in ONNX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  At each timestep, ODEs are integrated using Forward-Euler  to produce the next state array, resulting in a hardware-  agnostic timestepping function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For TensorFlow backends, a  JIT-compilable TensorFlow function is constructed that exe-  cutes 40 timesteps at a ∆t of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='025ms, resulting in a 1ms  sampling accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For ONNX backends, the timestep function  is converted to an ONNX model and either the public onnx-  runtime library or Groq compiler is used to compile this  into executable code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This does not lead to the best possible  performance by default, thus hardware-specific optimizations  are discussed in Section V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' TARGET PLATFORMS  Hardware platforms were selected from the top-performing  AI accelerators in the MLCommons MLPerf training bench-  mark v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0 [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' From this, the Intel Habana Gaudi was not  available to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GroqChip was included as it was already  available through academic channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' An overview of all AI  chips is given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' I and will be presented next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A modern,  server-grade CPU is also included as a baseline for our  subsequent performance and numerical-accuracy comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  1) Nvidia GPU [6]: These are well-established accelerators  in the HPC world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' With the introduction of the Tensor Cores  in Nvidia GPUs, they also became well-known for their AI  capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Tensor Cores are capable of matrix multiplications  in a very efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The current generation of tensor  cores can support up to TensorFloat-32 (TF32) precision TF32  is a floating point with float 32 dynamic range but float 16  precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' There are multiple ways of interacting with them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=', via cuBLAS and TensorRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  2) GroqChip [5]: This is a deterministic Tensor Streaming  Processor (TSP), resembling a modified systolic array archi-  tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The chip layout is a conventional 2D mesh of cores,  each with its own dedicated functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A column of these  cores – all of the same type – is called a functional slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Data travels horizontally, executing 320 SIMD-style lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A  single instruction can control 16 lanes, effectively creating 20  superlanes that can all be operated independently from each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The functional slices consist of one vector processor  (VXM), two matrix execution modules (MXM), switch ex-  ecution modules (SXM) and memory modules (MEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Each  functional unit (core) accepts a set of instructions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' for example,  the MEM unit could receive the instruction to put a vector onto  one of the data streams or store the results from the data stream  in its available SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' As soon as data is loaded onto a data  stream, it automatically ‘flows’ in the direction of the stream,  which can be either EAST-bound or WEST-bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' When an  addition needs to be performed, both inputs need to arrive  at the same time as the add instruction at the corresponding  VXM core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This design choice puts the burden of optimization  on the software generating the instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This is either done  by the Groq compiler automatically from an ONNX-graph  input or manually controlled by a user through the exposed  Groq-API, which has different levels of abstraction on top of  the Groq-ISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' To support the creation of large-scale systems,  the GroqChip has dedicated Chip-to-Chip modules that are  capable of performing off-chip communication without losing  their determinism [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For this work, we will mainly utilize  the VXM and MEM units, The memory modules add up  to a total of 220 MiB of on-chip SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Each superlane  implements a 4x4 mesh of vector ALUs capable of doing  x16-SIMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Each ALU has a 32-bit input operand but with  the exception of additions and multiplications, instructions are  done in a reduced-precision FP32 format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  3) Graphcore IPU [4]: The Graphcore Intelligence Pro-  cessing Unit (IPU) is designed for efficient execution of fine-  grained operations across a large number of parallel threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  By design, the IPU offers true Multiple Instruction, Multi-  ple Data (MIMD) parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This unique style of parallel-  processor design adapts well to fine-grained computations that  exhibit irregular data-access patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Each IPU contains 1,472  tiles, containing 1 core and 624KiB of SRAM memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A sin-  gle core can only access the memory in its own tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Intra-IPU  communication relies on a powerful, low-latency interconnect  called IPU exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For inter-IPU communications, each  chip contains 10 so-called IPU links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The IPU compute  units, called Accumulating Matrix Product (AMP)  units, support FP32 arithmetic and are designed to accelerate  matrix multiplications and convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' With respect to the  programming model, the IPU adopts the Bulk Synchronous  Parallel (BSP) model [33] through which it organizes its  compute and data-exchange operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This abstraction for  parallel computations consists of multiple sequential super-  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A superstep consists of a local computation phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  every process (tile, in the IPU case) operates in isolation  performing compute only on its local memory, followed by a  communication phase where each process can exchange values  needed by other tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' These activities are concluded with a  barrier synchronization phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' only when all processes have  reached the barrier can the next superstep be started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Because  of this, the IPU can be described as a true BSP machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  4) Google TPU [7]: The TPU (version 1) was designed  as a systolic-array processor for inference, only supporting  8/16-bit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' By supporting only matrix-multiply and  4  TABLE I: Overview of all hardware used in experimental setups  Device  On-chip Memory  Process node  Transistor count (Bn)  Base-boost freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' (MHz)  TDP (W) Software  AMD 3955WX CPU *  128 GB DDR4  7 nm  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='94  3900 - 4300  280  TF 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0  GroqChip TSP  230 MB on-chip  14 nm  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='8  900 Groq SDK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='1 ***  Nvidia A100 GPU  80 GB HBM2e  7 nm  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='2  1275 - 1410  400  TF 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0  Graphcore IPU (GC200) **  900 MB on-chip  7 nm  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='4  1330  185  TF IPU 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='3+gc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0  Google TPUv3  32 GiB HBM  16 nm  (est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=') 11  940  450  TF 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0  AMD Ryzen Threadripper PRO 3955WX (16-Core) | **Single M2000 in IPU-POD16 (with 4 GC200 chips) | ***TF2ONNX 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0 and ONNX opset 16  basic nonlinear activation functions, it was unfit for training  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Consequentially, an HPC application – for  example, the one demonstrated in this paper – would also  not be a suitable fit for this processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' However, with the  TPUv2, Google shifted their focus towards supporting training  on their TPU chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Google added a vector-processing unit  (VPU) and changed the matrix-multiply units to support the  FP16 format (FP32, with only a 7-bit mantissa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The VPU  most likely supports higher precision, as can be deducted from  results in this work but no confirmation of this is found in the  public domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' These two major (micro)architectural changes  made it possible to run a wider range of applications including  training neural models on the TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' All are supported through  the Google XLA compiler taking TensorFlow as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The  TPUv3, assessed in this work, is an upgrade in terms of  functional-unit count, higher memory speed, and optimized  chip layout, but did not include any fundamental changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Performance Predictions  The IO application has two components that map differently  onto different types of hardware: i) a part with embarrassingly  parallel computations for updating local neuron states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' and  ii) a part with SM computations for exchanging membrane  voltages over gap junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Before we proceed to the actual  experiments, we attempt performance predictions, driven by  the idiosyncrasies of the different AI-chip architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Embarrassing parallelism: These are calculations for up-  dating the state of every single neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This boils down to  elementwise vector operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GPU architecture featuring  one Warp execution per Streaming Multiprocessor or multiple  Tensor Cores is very well-suited for this type of parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  The TPU and the GroqChip are both based upon systolic-array  architectures, both natively supporting Matrix-Multiplication  but also Vector-Operation operations that can be utilized  for these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In fact, since neuron updates require  only  1-D  data, the Matrix-Multiplication units (which is  the focus of these chips) are effectively underutilized in these  architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The IPU, with a large amount of very small  general-purpose cores, should also do well on parallelizing  neuron-state calculations, however, its architecture is geared  towards irregular data-access patterns, which is not essential  to the particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The extra overhead of such advanced  features, therefore, will not help performance in terms of  computing this embarrassingly parallel part of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Communication:  As described previously, gap-junction  communication employs the gather-scatter operations (essen-  tially, SM operations) from TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For either the GPU or  the IPU, such operations are handled better due to the different  execution paths that can be handled within the architecture by  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In contrast, the GroqChip and TPU need to handle  these differently: a naive approach would be to enforce dense-  matrix operations via one-hot encoding of operands and,  then, utilizing the matrix-multiplication hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In case the  GroqChip or the TPU happen to use such a strategy, we expect  that performance will deteriorate very rapidly or memory will  be depleted with increasing IO-network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  1) CPU + TensorFlow: For this platform, JIT compila-  tion through the XLA compiler [34] will be used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' it will  automatically utilize the many threads nowadays available in  CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' We expect decent performance and very accurate results  because of full FP32 support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Since it is the hardware on which  brain models are traditionally executed and gives accurate  results, the CPU will form our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Accelerators should  outperform this implementation in terms of runtime, especially  for larger network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  2) GPU + TensorFlow: The XLA compiler is used, which  optimizes the graph resulting in a single kernel launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Among  others, it does this by “fusing” the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Moreover, this  fusion keeps intermediate values stored in GPU registers [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  The TensorFlow backend for CUDA use Tensor Cores, at  a loss of FP32 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' However, this only happens when  explicit matrix-multiplications are requested and not as an  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' So in our case, the compiler will only use float32  CUDA operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  3) IPU + TensorFlow: The IPU architecture is not a perfect  fit for the embarrassingly parallel part of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  For the interneuron-communication part, the BSP model is a  better fit and thus is expected to perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' However, as  the topology is given as an unknown parameter to the model,  the IPU compiler can not be expected to allocate neighboring  cells on adjacent tiles, resulting in sub-par communication  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Available memory should easily be able to  handle large problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  4) TPUv3 + TensorFlow: The TPU supports FP32 and  is expected to handle our workload, especially for the un-  connected case, very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' As Google put much effort into  TensorFlow support, gather-scatter operations are expected to  be optimized, to the best of the hardware capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Because  of FP32 support in the v3 model, we expect correct numerics  in the output, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  5) CPU/GPU + ONNX: Expectations are the same as for  CPU/GPU + TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' We expect the XLA compiler to  outperform the ONNX runtime slightly for the CPU case  simply because it can perform whole-program optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  For the GPU, this effect is expected to be much larger and  the TensorFlow is expected to dominate ONNX as the ability  5  to fuse kernels will be a big advantage for TensorFlow over  single-kernel invocations in ONNX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Especially the invocation  overhead for small GPU kernels will hurt the performance  of the ONNX-GPU-runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' TensorRT is also a supported  backend in ONNX that is expected to outperform the CUDA  runtime in performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' it will, however, drop precision as the  backend switches to TF32 numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  6) GroqChip + ONNX: The GroqChip is a new, upcoming  modified systolic-array processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Its compiler takes in the  ONNX graph but is not limited to executing this on an  operation-per-operation basis as it recompiles the full ONNX  graph at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Therefore, it can potentially perform the same  optimizations as the XLA compiler for the TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' As the first  version of the architecture, current compiler development is  still exploring ways to map non-standard ML-operations to  the hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Besides, the GroqChip VXM is not capable of  doing all operations in IEEE FP32 arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Because of this,  it can be expected to perform slightly better than the TPU at  the cost of reduced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' EXPERIMENTAL SETUP  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Benchmarking Parameters  Each platform is benchmarked for performance on a set  problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=', network) size as well as for its performance  scalability by simulating the IO network for small population  sizes in the range [43, 53, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' , 203] and, again, for larger sizes  in the range [303, 403, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' , 1003], where the third power is  an artifact of the cubic network-topology generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  These experiments are focusing on four different aspects of  each AI platform, discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  1) Unconnected Network: By removing the communica-  tion step (gap junctions) from the model, we obtain a (bio-  logically unrealistic) compute-heavy, embarrassingly parallel  workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' First, we measure the setup time for each AI  platform, including on-chip buffer allocation, Ahead-Of-Time  (AOT) compilation or definition of Just-In-Time (JIT)-enabled  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Next, we simulate an IO network for 100ms of  biological time and take the minimum wall-clock time from  5 runs (including data-transfer times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For JIT targets, the  first runtime (if outside the other runtimes’ standard deviation)  minus follow-up runtimes is taken as the JIT compilation time,  such that we can compare setup times between AOT and JIT  targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  2) Connected Network: By restoring gap junctions into the  IO network, we assess communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Runtimes are  obtained in an identical way as before, yet the expectation here  is that they are markedly longer than the unconnected case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  3) Numerical Validation: Measuring performance is our  main focus, yet this must not come at the cost of functional  correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Here, we simulate connected networks up to 729  neurons for 10 seconds of biological time and numerically  compare the various results to the reference CPU output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  4) Numerical Stress-test: Here, we simulate the IO in a  more biologically realistic way that is of interest to neurosci-  entists: We add more variance to the neural parameters and,  most importantly, a lot of external current inputs (simulating  other brain regions) that will evoke action potentials (spikes)  in the IO dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' These fast transients will stress-test the  numerical performance of the AI hardware, especially non-  IEEE754 targets (Tensor Cores and GroqChip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' We perform  this experiment on the smallest 64-neuron network and then  compare for numerical accuracy against the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Benchmarking  is  implemented  in  a  publicly  avail-  able  and  modular,  extensible  framework,  downloadable  from GitLab https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='com/neurocomputing-lab/Inferior  OliveEMC/ioperf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  The  main  benchmarking  script  auto-  discovers available hardware, runs the appropriate benchmarks  and records results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Used software versions are also shown in  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Hardware-specific Optimizations  While our original goal was not to write platform-specific  code, we found that by default some of the AI platforms did  not perform very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For example, most platforms defaulted  to copying over the entire parameters arrays for each kernel  invocation, which was not needed for this mostly constant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  For a fair comparison between hardware platforms, we allowed  optimizations to be applied to hardware-specific code that  either led to operation fusion across different execution kernels  or prevented unnecessary device-host data transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The exact  optimizations have been applied in close collaboration with  Graphcore and Groq for the respective chips, and are as  follows:  1) TensorFlow XLA: The TensorFlow graph executor typi-  cally performs each operation separately when a graph is run  with a corresponding kernel invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A different way to  run TensorFlow models is made available by XLA, which  turns a TensorFlow graph into a series of kernels created for  a particular application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' These kernels can take advantage of  application-specific information for performing optimizations,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=', operation fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The CPU, GPU, and TPU are the  three available backends for the XLA compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For the IO  application, a TensorFlow wrapper function was implemented  that fuses up to 40 timesteps together for each call in order to  fully exploit the XLA compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  2) ONNX: Except for the GroqChip, all ONNX imple-  mentations build on top of onnxruntime or onnxruntime-  gpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' We enable all backend-supported graph-optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Explicit IOBindings are used to prevent unneeded host-  device data copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Parameters are copied once to the device  at simulation start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Then, state is allocated twice, with each  timestep toggling between two buffers, one as the input state  and the other as the output (next) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For TensorRT, we  leave the default behavior of using TF32 enabled, otherwise,  it will not utilize its Tensor Cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  3) Groq: After the compilation of an ONNX graph with  the Groq Compiler, the binary can be executed directly on the  GroqChip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A naive approach here would be to invoke this  binary 40 times for 40 timesteps and move the data back  and forth continuously since the GroqChip only has SRAM  which is fully managed at compile time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' However, the Groq  Compiler is able to tie input and output tensors together into  6  10  1  100  101  102  103  CPU (ONNX)  CPU (TF)  10  1  100  101  102  103  GPU (TF)  IPU (TF)  102  103  104  105  106  10  1  100  101  102  103  TPU (TF)  102  103  104  105  106  GroqChip (ONNX)  Network size (#neurons)  Time required to simulate one biological second (s)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 2: Runtime performance (lower is better), comparison between CPU baseline, GPU and AI chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For scale, the mouse (•) and human (▲) Inferior Olive  are shown as running in realtime in all figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The CPU is included twice to explain the observed switching behavior of the IPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' On the CPU, while the XLA  optimizer builds a single-core, connected-network simulation, it builds a multicore, unconnected-network simulation (as observed by load-testing), leading to  an unexpectedly slow simulation for the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The same behavior can be observed for the IPU, which uses the XLA compiler as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  a persistent memory buffer in the on-chip SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Utilizing  this results still in 40 invocations of the binary but skips the  continuous I/O between host and accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' A more radical  way to improve the performance is to compile the 40 timesteps  into a single ONNX graph that can then be converted with the  Groq Compiler;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' this method will reduce 40 invocations to a  single invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' We implemented all optimizations as long  as the compiler was able to compile them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The 40 timesteps at  once quickly ran into compiler errors with growing networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  4) Graphcore:  The IPU has architectural support for  streaming memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This means that we can run a single  program on-chip for the entire simulation that will stream  out samples every 40 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The inner, unsampled 1-  msec 40-timestep loop, is run using ipu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='repeat,  after  which  the  recorded  voltages  are  pushed  to  an  IPUOutfeedQueue with a 200-sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This is, then,  looped once more using ipu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='repeat for the re-  quired amount of milliseconds to simulate and wrapped in  a TensorFlow JIT function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Furthermore, the fast-math op-  timization is enabled, 128 IPU tiles are reserved for I/O  with place_ops_on_io_tiles = True and program  execution is limited to a single IPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' EXPERIMENTAL RESULTS  With the exception of the reference CPU, for brevity we  report here either TensorFlow or ONNX results, depending  on which of the two leads to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Overall  performance plots are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 2 and will be detailed  in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In general, it is found that, for the  IO application, the ONNX ports are outperformed by their  TensorFlow counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This is due to the fact that the  onnx-runtime library currently does not perform as extensive  optimizations as the XLA compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For example, the CUDA  target translates each compute step into a single predefined  kernel call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The TensorRT backend performs operator fusion,  resulting in multiple kernels that chain arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Still, the CUDA XLA-backend vastly outperforms both ONNX  CUDA targets, and as such we removed the corresponding  findings from the main analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Note that the Groq platform  only supports AOT compilation of ONNX models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Compilation Time  Both software stack and hardware influence program setup  time, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 3 for the largest network (729 cells)  that could fit in all AI chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The CPU compiles the fastest  across the board as we have a direct translation of ONNX  operations to their CPU-optimized callbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The TensorFlow  (XLA) version, not included in the figure, was much slower  7  100  101  102  103  Compile time (s)  (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Network type at compile time (n=729)  Unconnected  Connected  CPU  IPU  GPU  TPU  GroqChip  10  2  10  1  100  101  102  Run time (s)  (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='Connected network size at run time  n=64  n=729  n=125000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0x  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='3x  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='7x  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='7x  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='9x  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0x  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='3x  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='4x  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='4x  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='6x  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0x  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='6x  269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='5x  1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='5x  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 3: (A) Setup (AOT+JIT compile + memory allocation) times for a network  of 729 neurons in both Unconnected- and Connected-network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  JIT compile times are extracted from the first run of 5 performance runs  and added to the initial setup time (if outside one standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' (B)  Performance and speedup of different AI chips vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' the CPU reference on  the Connected benchmark, for different network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Sizes were chosen to  be the smallest (64) and largest connected networks that could fit on the  GroqChip (729) and the TPUv3 (125,000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The rightmost GroqChip bar is  absent, corresponding to the model that could not be compiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  due to the increased compiler complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Both the IPU and  GPU exhibit similar JIT compilation speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GroqChip’s  AOT compiler takes significantly longer for this workload due  to the explicitly concatenated 40 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GroqChip  version with a single timestep per program compiles much  faster than the Graphcore or A100 versions, but at a small  performance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Runtime Performance  1) Unconnected Network (embarrassingly parallel): For  unconnected cells, neural dynamics are expressed only using  vectorized operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' As predicted, this fits the compute  paradigm of the GPU very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Performance scales linearly  with problem size (horizontal line), showing that the GPU  cores are underutilized for all simulated network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  The TPU and GroqChip, as systolic-array-based processors,  were expected to be a poorer architectural fit because large  parts of the chip would be left unused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Still, the focus on  efficient vector operations could result in speedups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' We can  indeed observe this in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 2, although in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The  TPU, similar to the GPU, flatlines across all problem sizes,  although being 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='1x slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Consequently, memory capacity  is not a problem for the TPU but performance capping in raw  single-cell computations due to architectural design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  In contrast, the GroqChip starts out 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='0x faster than the GPU,  quickly loses this edge and, between 103 and 104 cells, starts  to hit its memory-capacity limits, degrading performance with  higher problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Networks of more than 640, 000 cells  simply do not fit on the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GroqView analyzer confirms  that the problem is core-to-core-memory communication and  that most dedicated cores are not used most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  The IPU was expected to perform well given its large core  count but the very homogeneous compute load proved a poor  fit for its MIMD design, leading to large under-utilization  of the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' With respect to real-time performance, only the  GPU followed by the GroqChip (ignoring memory issues) and  marginally the TPU makes the 1-sec cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  2) Connected Network (high communication overhead): As  predicted, communication patterns induced by a small number  of gap junctions lead to a large performance reduction of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='6x  for small networks on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For higher problem sizes,  performance drops at a growing rate, with a 141x degradation  for networks of 106 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The AI chips fare much better here,  most of which initially shows a less than 20% reduction in  performance against their unconnected counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  As an exception, the GroqChip’s connected-network sim-  ulation runs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='5x slower than the unconnected version;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' even  so, it outperforms the GPU on very small, connected neural  networks by a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='7x speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' However, the GroqChip (as  expected) converts the SM communication into a dense-  matrix multiplication, making the best out of its deterministic-  execution hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This quickly leads to prohibitively large  matrix multiplications and, beyond 729 cells, the scheduler  is unable to allocate the necessary instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In effect, the  GroqChip loses its edge over the GPU for larger networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  In contrast, the TPU shows nearly identical behavior to  the unconnected case and its performance does still not scale  with problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This changes around networks larger than  105, where the JIT compiler seems to run into performance  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Here, we observed large random fluctuations in  performance that either led to approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 1-sec or very long more  than 400-sec run-times over the 5 repeated runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' We expect  that these originate from memory limits of the TPU and had  to stop benchmarking due to impractically large run times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  However, we could not determine the true source of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  The IPU, severely underutilized for the unconnected case,  sees in fact a performance improvement when we increase  the communication overhead in small networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' While coun-  terintuitive, this is actually the same effect we see on the  XLA-based CPU backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Here, we see that gap junctions  force the simulation to become single core, which actually  becomes faster than the parallel, multi-core, unconnected case,  due to the lower synchronization overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Around 104 cells,  this behavior changes, gap-junction communication becomes a  fixed overhead on top of normal simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' At a certain point,  this growth becomes exponential and the largest simulated  network does not fit on a single IPU anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Numerical Validation  While all AI chips outperform the CPU baseline, it is wise  to explore also any potential decrease in numerical accuracy  of the different runs with respect to that of the same CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Here, we compare 1-msec sampled cell somatic voltages of an  extended, 10-sec simulation for a 729-cell, connected network  8  IPU  GPU  TPU  GroqChip  10  5  10  4  10  3  10  2  10  1  100  101  Abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' difference to CPU reference (mV)  Left  boxp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Right  boxp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Vgroqchip  Vcpu  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 4: Numerical-accuracy validation (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Box plots show  deviations from CPU baseline, as recorded over two 1-sec timespans, one  at the start (left) and one at the end (right) of the 10-sec numerical-validation  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GroqChip result, showing the largest deviation, is plotted in  the upper left corner together with the two recording spans .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  (the largest population supported by all platforms);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' results are  shown in the box plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  As expected, platforms supporting IEEE754 floating-point  numerics (IPU, GPU, TPU) show accurate reproduction of  voltage traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The IPU, even with fast-math enabled, is the  most faithful to the CPU baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GPU and TPU exhibit  increasingly large deviations but still fall within limits explain-  able by floating-point instruction reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GroqChip,  while supporting FP32 number storage, implements certain  operations at lower precision including exponent calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  This is visible by a quite large mV -order deviation from the  CPU baseline, for a process that happens at the 10 − 100mV -  scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' This voltage difference mostly stems from a slowly  accrued phase difference for the oscillating cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' TensorRT  (not shown in this plot) is by default using Nvidia’s TF32, for  which accuracy was found similar to that of the GroqChip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Numerical Stress-test  The numerical stress test increases neuronal variation and  adds external inputs that lead the neurons to spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' These  fast transients can not be simulated using FP16 precision, but  reduced-accuracy FP32 operations as used in Tensor Cores  or GroqChip are still untested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Once more, we compare the  deviation of the somatic-voltage traces of the various AI chips  against the CPU baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Again, the platforms with native FP32 support show the  lowest deviation: For the IPU this is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='087mV , for the  GPU this is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='135mV and for the TPU this is a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='672mV  maximum absolute difference from the CPU baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' These  moderate, mV -order differences can be explained by small  spike-time differences which due to the large neuronal-spike  sizes quickly lead to large voltage discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Importantly,  all simulations run stably;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=', do not cause this chaotic IO-  model simulator to crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The GroqChip simulation initially  starts out the same as in the numerical-validation test, but as  soon as input perturbations are applied, it becomes unstable  and settles on voltage deviation at a measured maximum  of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='51 × 1036mV , unacceptable for scientific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Notably, the error stabilizes at this point and does not explode  to infinity or NaN values, as observed with FP16 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  To regain numerical stability, we tried lowering the time-  stepping constant ∆t 10-fold and 100-fold for the GroqChip  simulation, but this did not lead to results more closely in  range with the CPU ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' DISCUSSION  As this work has shown, utilizing AI platforms for executing  highly biologically plausible SNN workloads is made exceed-  ingly user-friendly when using a ML-library like TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Arguably, even better performances could be obtained by  coding via the various hardware SDKs (Software Development  Kits), but it is unrealistic to expect computational scientists  to learn the low-level details of all hardware options made  available to them these days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  As shown, the added benefits from JIT compilation make  a hand-coded CUDA implementation perform on par with the  XLA-compiled TensorFlow version while, at the same time,  allowing one to move easily to a new piece of hardware  when this is released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' We expect that, in the future, more  classical HPC workloads will see ML-library, that is, tensor-  based implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  For promising upcoming accelerators like those by Graph-  core and Groq, we believe that future speedups will chiefly  come from software and compiler upgrades, as current SDKs  are mostly optimized for ML workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For instance, gather-  scatter operations on the GroqChip do not have to be im-  plemented as dense-matrix operations, memory can be better  utilized, and better support for iterative programs must also  be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The TPU which is architecturally similar to the  GroqChip, clearly performs gather-scatter operations in a more  efficient way than encoding indexing as one-hot vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Speed-ups could be gained by effective use of mixed  precision on the IPU or reduced accuracy FP32 operations  using Tensor Cores or GroqChip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For the IPU, this would  constitute a separate numerical sensitivity analysis to find out  which parts of the compute graph can be lowered to (stochastic  rounded) FP16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' As shown, the accuracy loss on Tensor Cores  and GroqChip does in its current form not allow for brain  simulation, but possible these could be put to use by switching  the integration scheme or other numerical optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  Finally, this work has steered clear off multi-chip topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  All discussed architectures do support specifically developed,  low-latency, chip-to-chip hardware and assorted communica-  tion protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' In many ways, such coherent communication  is a bigger and more timely challenge than acceleration speed  itself, which would deliver massive benefits for large-scale  SNN simulation (or training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' However, tapping into those  platform-specific interfaces requires SDK-specific coding of  the IO application;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' relying on TensorFlow or ONNX frame-  works will, generally, not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Careful and platform-specific  coding is necessary, which we leave as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' CONCLUSION  In this work, we built the first ML-library-based, effi-  cient implementation of a large-scale, conductance-level brain  9  model, the Inferior Olive (IO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Subsequently, we benchmarked  the performance of simulating this model on a 16-core AMD  Ryzen Threadripper PRO 3955WX CPU, an Nvidia A100  GPU, and different AI chips (Graphcore IPU M2000, Gro-  qChip and Google TPU v3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' We found that all accelerators  provide significant speedups over the CPU implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  For this specific problem, the GPU and TPU seem most  fit for simulation, with the TPU setting a new record for  real-time IO simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For small networks, the GroqChip  outperforms the other accelerators, but large networks could  not fit in the on-chip instruction memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' More generally, we  hypothesize that modern ML-libraries possess the semantic  power to model classical problems in scientific computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  These, then, map extremely well to ML-driven, novel AI-chip  architectures, which apart from large performance benefits,  also benefit from reduced development times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' For example, the  version of our IO application running on the TPU outperforms  the handwritten and hand-optimized CUDA implementation by  a large factor, at a fraction of the development cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' The exact  hardware trade-off will vary on an application-by-application  basis, and hardware selection also benefits significantly from  the hardware-agnostic model description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  This research would not have been possible without access  to dedicated hardware: The RTX6000 was gifted from the  NVIDIA Hardware Grant Program, Google provided free  cloud credits for TPU access and Graphcore provided access  to POD16 machines through Paperspace and Gcore cloud  via its Academia program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Furthermore, we’d like to thank  Graphcore employees for helping with optimizing the IPU  code and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Mario Negrello for neuroscientific insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
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+page_content=' Wong-VanHaren, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Baker,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Hawkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Bell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Thompson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Kahsai, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Kimmell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Hwang,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Leslie-Hurd, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Bye, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
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+page_content=' Boyd, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Venigalla,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Laforge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Purdy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
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+page_content=' Ahmad, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Gagarin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Czekalski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
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+page_content=' Werner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Sproch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Macias, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
+page_content=' Kurtz, “Think fast: A tensor  streaming processor (tsp) for accelerating deep learning workloads,” in  Proceedings of the ACM/IEEE 47th Annual International Symposium on  Computer Architecture, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
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+page_content='  IEEE Press, 2020, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFRT4oBgHgl3EQfwDjC/content/2301.13637v1.pdf'}
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diff --git a/DdAyT4oBgHgl3EQfSPd1/content/tmp_files/2301.00082v1.pdf.txt b/DdAyT4oBgHgl3EQfSPd1/content/tmp_files/2301.00082v1.pdf.txt
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+SURFACES OF MINIMUM CURVATURE VARIATION
+LUIS A. CAFFARELLI, PABLO RA´UL STINGA, AND HERN´AN VIVAS
+Abstract. We establish the analytical theory of surfaces of minimum curvature variation.
+We construct classical, G2 continuous surfaces, as well as weak solutions in the context of
+geometric measure theory.
+1. Introduction
+Computer-aided design (CAD) and computer-aided manufacturing (CAM) are widely pop-
+ular techniques whose basic feature is the use of computer software to create or modify shapes
+in such a way that some aspects of the design process, such as quality of the object or produc-
+tivity of the process, are optimized, see, for example, [9]. Their origins can be traced back
+to the 1950s and 60s and their development have been continuous since then. Nowadays,
+CAD/CAM are used in contexts as varied as engineering, particularly in automotive, ship-
+building and aerospace industries; architectural design; and computer animation for creation
+of special effects in movies, among many other applications.
+Within this realm, of particular interest are geometric problems in computer-aided geomet-
+ric design (CAGD). The goal of CAGD is the creation of complex smoothly shaped models
+and surfaces with specified geometric constraints. The resulting surfaces have to accurately
+reflect these specifications and be free of unwanted wrinkles, bulges and ripples. In many
+instances, the aim is to create fair surfaces that are aesthetically pleasing to the eye. As it
+turns out, many of these problems can be approached via a variational principle, that is, by
+looking for a surface that minimizes an appropriate functional or fairness energy subject to
+adequate geometric boundary conditions, see [10].
+The most commonly used fairness energy functionals can be split into two groups: physical-
+based or geometric-based. The first group roughly corresponds to interpreting the surface as
+an ideal elastic membrane or plate and minimize energies such as
+´
+|∇u|2 dx or
+´
+|∆u|2 dx.
+The second group aims at minimizing energies that relate to geometric invariants of the
+surface such as the area or curvature, see [11] and the references therein. In 1992, Moreton
+and S´equin proposed in [8] a numerical algorithm for the creation of 2-dimensional fair
+surfaces M as minimizers of the energy functional
+ˆ
+M
+��dκ1
+de1
+�2
++
+�dκ2
+de2
+�2�
+dA.
+Here e1 and e2 are the principal directions corresponding to the principal curvatures κ1 and
+κ2 of M and dA is the differential of surface area.
+It is a key aspect in CAGD to be able to construct fair surfaces that preserve several degrees
+of geometric continuity. This is particularly important at the boundary of the domains where
+2010 Mathematics Subject Classification. Primary: 35B65, 49Q10, 53A10.
+Secondary: 49Q20, 65D17,
+68U07.
+Key words and phrases. Curvature variation, computer-aided design, prescribed mean curvature, regularity.
+Research partially supported by NSF grant 1500871 (USA), Simons Foundation grant 580911 (USA), and
+Agencia Nacional de Promoci´on Cient´ıfica y Tecnol´ogica under grant PICT 2019-3530 (Argentina).
+1
+arXiv:2301.00082v1  [math.DG]  31 Dec 2022
+
+2
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+the surfaces meet. The notions of geometric continuity are referred to as G0 continuity, where
+two surfaces meet in a continuous fashion, without jumps; G1 continuity, where the tangent
+planes of the surfaces meet with continuity; and G2 continuity, where the curvatures meet
+with continuity. These are not the same as the classical notions of C0, C1 and C2 continuities,
+as those require some specific combination of the derivatives of the solutions to be continuous
+up to the boundary. In particular, G2 continuity turns out to be crucial in applications such
+as the streamlined surfaces of aircrafts, ships and cars, and this was the main motivation
+for the numerical study in [8].
+In [11], a numerical finite difference method is proposed
+to construct surfaces that would enjoy G2 continuity as steady states of a sixth order flow
+derived from the Euler–Lagrange equation of the energy functional
+ˆ
+M
+|∇H|2 dA
+where H is the mean curvature of M. Numerical evidence of G2 continuity is observed in [11],
+while G1 continuity is expected according to [8]. To the best of our knowledge, the analytical
+theory of surfaces of minimum curvature variation in general is missing. Furthermore, no
+proof of G2 continuity is available thus far.
+The aim of this paper is to fill these gaps and to develop the analytical foundation from
+the PDE perspective of the theory of surfaces of minimum curvature variation. We give
+two constructions of surfaces: classical solutions that are G2-continuous, and weak solutions
+through geometric measure theory methods.
+Therefore, we consider the minimization problem
+(1.1)
+min
+M
+1
+2
+ˆ
+M
+|∇MH|2 dA
+where M ranges over all n−dimensional manifolds in Rn+1, n ≥ 1, with prescribed bound-
+ary, H is the mean curvature of M and dA is the differential of surface area. Notice that
+(1.1) minimizes the (quadratic) variation of the mean curvature of M so that surfaces with
+constant mean curvature such as planes, circles and cylinders are minimizers.
+If M is the graph of a function defined on a bounded domain Ω ⊂ Rn, that is,
+M = {(x, u(x)) : x ∈ Ω}
+for some u : Ω → R, then the values of u at ∂Ω prescribe the boundary ∂M of M. For a
+point x0 ∈ Ω, the tangent plane to M at (x0, u(x0)) and its upward pointing unit normal are
+P(x) = u(x0) + ∇u(x0) · (x − x0)
+and
+ν(x0) =
+(−∇u(x0), 1)
+(1 + |∇u(x0)|2)1/2 ,
+respectively. The mean curvature H of M at a point is defined as the average of the n
+principal curvatures of M at that point. In the coordinates given by u, it takes the form
+H ≡ H(u) = 1
+n div
+�
+∇u
+(1 + |∇u|2)1/2
+�
+.
+If we set
+D(u) := (1 + |∇u|2)1/2
+we then have that dA = D(u) dx.
+Let f be a function in C1(Ω × R). The tangential gradient of f on M is obtained by
+projecting the gradient of f in Rn+1 onto the plane orthogonal to ν:
+∇Mf = ∇Rn+1f − (ν · ∇Rn+1f)ν
+on M.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+3
+Clearly, ν · ∇Mf = 0 and
+(1.2)
+|∇Mf|2 = |∇Rn+1f|2 − |ν · ∇Rn+1f|2.
+Furthermore, ∇Mf depends only on the values of f on M, see, for instance, [6, Section 16.1].
+To compute ∇MH we extend H as a function of (x, xn+1) ∈ Ω × R by making it constant
+in xn+1: H(x, xn+1) ≡ H(x). This is enough to compute ∇MH because the resulting value
+is independent of the extension. Since ∇Rn+1H = (∇H, Hxn+1) = (∇H, 0), by (1.2) we get
+|∇MH|2 = |(∇H, 0)|2 −
+����(−∇u · ∇H)(−∇u, 1)
+D(u)2
+����
+2
+= |∇H|2 −
+����
+∇u · ∇H
+D(u)
+����
+2
+.
+With this formula the energy in (1.1) becomes
+(1.3)
+E[M] = 1
+2
+ˆ
+Ω
+�
+|∇H|2 −
+����
+∇u · ∇H
+D(u)
+����
+2�
+D(u) dx.
+We will call this the geometric energy. It follows from the Cauchy–Schwartz inequality that
+|∇H|2
+D(u)2 ≤ |∇MH|2 ≤ |∇H|2.
+Therefore, we will also study the (larger) simplified energy functional
+(1.4)
+E[H, u] := 1
+2
+ˆ
+Ω
+|∇H|2D(u) dx.
+In Section 2 we consider (1.4) and show how to construct smooth solutions that satisfy the
+prescribed mean curvature equation for a curvature of minimum variation. Section 3 shows
+how to modify the argument to construct solutions of the geometric energy functional (1.3).
+Finally, in Section 4 we provide a weak formulation of the problem and prove existence of
+minimizers in the context of geometric measure theory.
+2. Existence of G2 surfaces for the simplified energy
+In this section we work with the simplified energy functional (1.4). Let Ω ⊂ Rn be a
+bounded domain such that ∂Ω ∈ C3,α for some 0 < α < 1 fixed. We assume that we are
+given prescribed boundary values g ∈ C3,α(Ω) for u and h ∈ C1,α(Ω) for H on ∂Ω.
+We address the following problem: given Ω and the boundary datum g, find a surface
+given by the graph of a function u such that its mean curvature H is a minimizer of (1.4)
+among all functions with prescribed boundary values h ∈ C1,α(Ω).
+We will use Schauder’s fixed point theorem:
+Theorem 2.1 (see [6, Corollary 11.2]). Let G be a closed convex set in a Banach space B
+and let T be a continuous mapping of G into itself such that the image T(G) is precompact.
+Then T has a fixed point.
+Consider the Banach space B = C1,α(Ω) and its subset
+G :=
+�
+v ∈ C1,α(Ω) : v = g on ∂Ω
+�
+.
+Observe that G is nonempty because g ∈ C3,α(Ω). By classical global Schauder estimates,
+we see that another example of function in G is the harmonic extension v of g inside of Ω:
+�
+∆v = 0
+in Ω
+v = g
+on ∂Ω.
+
+4
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+It is clear that G is convex and closed.
+For any v ∈ G, we define the functional
+(2.1)
+E[H, v] := 1
+2
+ˆ
+Ω
+|∇H|2D(v) dx.
+The map T : G → G is constructed in a 2-step process.
+Step 1. Given any v ∈ G, we find the unique minimizer H ∈ W 1,2(Ω) to (2.1) such that
+H − h ∈ W 1,2
+0 (Ω). This can be done because the coefficient D(v) satisfies
+1 ≤ D(v) ≤ (1 + ∥∇v∥2
+L∞(Ω))1/2 < ∞,
+so that (2.1) is a coercive functional. Then H is the unique weak solution to
+�
+div(D(v)∇H) = 0
+in Ω
+H = h
+on ∂Ω.
+Since v ∈ C1,α(Ω), the coefficient D(v) ∈ C0,α(Ω). Thus, by global Schauder estimates (see
+[6, Section 8.11]),
+(2.2)
+∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥D(v)∥C0,α(Ω)∥h∥C1,α(∂Ω)
+where Cn > 0 is a constant that depends only on dimension n.
+Step 2. Given H ∈ C1,α(Ω) from Step 1, we find the solution u to the prescribed mean
+curvature equation
+(2.3)
+�
+�
+�
+div
+� ∇u
+D(u)
+�
+= nH
+in Ω
+u = g
+on ∂Ω.
+For this, we use the following result (where we use nH instead of H in [7]).
+Theorem 2.2 ([7, Theorem 3.4.1] and its proof). Let 0 < α < 1 and Ω ⊂ Rn be a bounded
+domain with C3,α boundary. Suppose that H ∈ C1,α(Ω) satisfies
+(2.4)
+∥H∥Ln(Ω) <
+�ˆ
+Rn(1 + |p|2)− n+2
+2
+dp
+�1/n
+and, for any y ∈ ∂Ω,
+(2.5)
+|H(y)| ≤ H∂Ω(y),
+where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω.
+Then for any g ∈ C3,α(Ω) there exists a unique solution u ∈ C3,α(Ω) to (2.3). In particular,
+there exists a constant C∗ > 0, depending only on n, α, ∥H∥Ln(Ω), ∥H∥C1(Ω), ∥g∥C2,α(Ω) and
+Ω, such that
+(2.6)
+∥u∥C2,α(Ω) ≤ C∗.
+The constant in the right-hand side of (2.4) can be simplified. Recall the definition of the
+Beta function and its relation with the Gamma function: for x, y > 0,
+B(x, y) :=
+ˆ ∞
+0
+tx−1
+(1 + t)x+y dt = Γ(x)Γ(y)
+Γ(x + y) .
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+5
+We have that Γ(1) = 1 and xΓ(x) = Γ(x + 1), for all x > 0. By passing to polar coordinates
+p = rθ, for r > 0 and θ ∈ Sn−1, performing the change of variables t = r2 which makes
+2dr/r = dt/t, and using that |Sn−1| = n|B1|, we get
+ˆ
+Rn(1 + |p|2)− n+2
+2 dp = |Sn−1|
+ˆ ∞
+0
+rn
+(1 + r2)
+n+2
+2
+dr
+r
+= n|B1|
+2
+ˆ ∞
+0
+tn/2
+(1 + t)
+n+2
+2
+dt
+t
+= n|B1|
+2
+B(n/2, 1) = n|B1|
+2
+Γ(n/2)
+Γ(n/2 + 1) = |B1|.
+Therefore, (2.4) reads
+(2.7)
+∥H∥Ln(Ω) < |B1|1/n.
+Now (2.5) and (2.7) impose further restrictions on the boundary values h of H. Condition
+(2.5) is natural to assume and cannot be avoided (see, for example, the nonexistence results [6,
+Theorem 16.11] and [7, Theorem 3.4.5]). Therefore, we assume that h ∈ C1,α(Ω) additionally
+satisfies
+(2.8)
+|h(y)| ≤ H∂Ω(y)
+for all y ∈ ∂Ω.
+On the other hand, by the maximum principle (see [6, Section 8.1]), we can estimate
+(2.9)
+ˆ
+Ω
+|H|n dx ≤ |Ω|
+�
+max
+∂Ω |h|
+�n
+.
+or
+∥H∥Ln(Ω) ≤ |Ω|1/n max
+∂Ω |h|.
+Thus, in order to ensure (2.7), we further assume that
+(2.10)
+max
+∂Ω |h| <
+�|B1|
+|Ω|
+�1/n
+.
+From the computer science point of view, this means that for large boundary curvatures h
+the domain Ω for the reconstruction should be sufficiently small.
+Therefore, under the additional assumptions (2.8) and (2.10), we can apply Theorem 2.2
+and find the unique solution u ∈ C3,α(Ω) to (2.3). This completes Step 2.
+Using these two steps, we define T : G → G by T(v) = u. In order to apply Theorem 2.1,
+we need to verify that
+(1) T is continuous, and
+(2) T(G) is precompact.
+Let us begin with (1). Fix v1 ∈ G. We need to show that given any ε > 0 there exists
+δ = δ(ε, v1) > 0 such that for any v2 ∈ G satisfying ∥v1 − v2∥C1,α(Ω) < δ we have ∥u1 −
+u2∥C1,α(Ω) < ε, where uj = Tvj, for j = 1, 2.
+Let Hj denote the minimizer of E[·, vj],
+j = 1, 2, as constructed in Step 1. Then the difference H = H1 − H2 is the unique weak
+solution to
+�
+div(D(v1)∇H) = div
+�
+(D(v2) − D(v1))∇H2
+�
+in Ω
+H = 0
+on ∂Ω.
+
+6
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+By global Schauder estimates (see [6, Section 8.11]),
+(2.11)
+∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥D(v1)∥C0,α(Ω)∥(D(v2) − D(v1))∇H2∥C0,α(Ω)
+≤ C(n, α, Ω, v1, ∇H2)∥v1 − v2∥C1,α(Ω)
+=: C1∥v1 − v2∥C1,α(Ω).
+Let us now estimate the difference u = u1 − u2 ∈ C3,α(Ω). Since
+�
+�
+�
+div
+� ∇ui
+D(ui)
+�
+= nHi
+in Ω, for i = 1, 2
+u1 = u2 = g
+on ∂Ω
+we find that
+�
+�
+�
+div
+� ∇u1
+D(u1) − ∇u2
+D(u2)
+�
+= nH
+in Ω
+u = 0
+on ∂Ω.
+In order to apply global Schauder estimates one more time we need to find an equation for
+u. Set
+F(p) :=
+p
+�
+1 + |p|2
+p ∈ Rn.
+Then F is a smooth, bounded vector field with entries Fi(p) =
+pi
+√
+1+|p|2 , for i = 1, . . . , n. Note
+that, for j = 1, . . . , n,
+∂jFi(p) =
+�
+�
+�
+1
+√
+1+|p|2 −
+p2
+i
+(1+|p|2)3/2
+if i = j
+−
+pipj
+(1+|p|2)3/2
+if i ̸= j
+=
+δij
+D(p) − pipj
+D(p)3 .
+In particular,
+(2.12)
+∂jFi(p) = ∂iFj(p)
+so that ∇F is a symmetric matrix. It is clear that ∇F is bounded. To see that it is locally
+strictly elliptic, observe that, for any ξ ∈ Rn, by the Cauchy–Schwartz inequality,
+n
+�
+i,j=1
+∂jFi(p)ξiξj =
+n
+�
+i,j=1
+�δijξiξj
+D(p) − pipjξiξj
+D(p)3
+�
+≥ |ξ|2
+�
+1
+D(p) −
+|p|2
+D(p)3
+�
+=
+|ξ|2
+D(p)3 ≥ θ(R)|ξ|2
+for all |p| < R, where θ(R) → 0 as R → ∞. Furthermore, we can write
+Fi(∇u1) − Fi(∇u2) =
+ˆ 1
+0
+d
+dtFi(t∇u1 + (1 − t)∇u2) dt
+=
+ˆ 1
+0
+∇Fi(t∇u1 + (1 − t)∇u2) · ∇(u1 − u2) dt
+so that
+F(∇u1) − F(∇u2) = A(x)∇u
+with
+Aij(x) =
+ˆ 1
+0
+∂jFi(t∇u1 + (1 − t)∇u2) dt.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+7
+The matrix A is symmetric thanks to (2.12), as well as bounded. Recall that ∇F is locally
+strictly elliptic. Now, u1 ∈ C3,α(Ω) is fixed. By (2.6), the C2,α(Ω) norm of u2 is uniformly
+controlled by the C1(Ω) norm of H2, which in turn is uniformly close to the C1(Ω) norm of
+the initially fixed H1. These facts imply that that A(x) is strictly elliptic. Moreover, we have
+the following technical lemma.
+Lemma 2.3. Let U, V : Ω → Rn, U, V ∈ C0,α(Ω) and let ψ : Rn → R be a smooth function
+such that
+∥ψ∥L∞(Rn) + ∥∇ψ∥L∞(Rn) < ∞.
+Define
+φ(x) :=
+ˆ 1
+0
+ψ(tU(x) + (1 − t)V (x)) dt
+for every x ∈ Ω.
+Then φ ∈ C0,α(Ω), with
+∥φ∥C0,α(Ω) ≤ ∥ψ∥L∞(Rn) + ∥∇ψ∥L∞(Rn)
+�
+[U]Cα(Ω) + [V ]Cα(Ω)
+�
+.
+Proof. The boundedness of ψ implies that φ is bounded with ∥φ∥L∞(Ω) ≤ ∥ψ∥L∞(Rn). To
+bound the H¨older seminorm of φ, let x, y ∈ Ω. Then
+|φ(x) − φ(y)| =
+����
+ˆ 1
+0
+�
+ψ(tU(x) + (1 − t)V (x)) − ψ(tU(y) + (1 − t)V (y))
+�
+dt
+����
+≤ ∥∇ψ∥L∞(Rn)
+ˆ 1
+0
+|t(U(x) − U(y)) + (1 − t)(V (x) − V (y))| dt
+≤ ∥∇ψ∥L∞(Rn) (|U(x) − U(y)| + |V (x) − V (y)|)
+≤ ∥∇ψ∥L∞(Rn)
+�
+[U]Cα(Ω) + [V ]Cα(Ω)
+�
+|x − y|α.
+□
+Lemma 2.3 gives the H¨older continuity of A(x). Indeed, let ψ be any of the entries of the
+gradient matrix of F:
+(∇F(p))ij =
+δij
+D(p) − pipj
+D(p)3
+for i, j = 1, . . . , n,
+which are smooth and bounded, so ∥ψ∥L∞(Rn) ≤ M1, where M1 is independent of i and j.
+For any k = 1, . . . , n, we have
+∂k(∇F(p))ij = −δijpk + δikpj + δjkpi
+D(p)3
++ pipjpk
+D(p)5
+and these are all bounded. Therefore, ∥∇ψ∥L∞(Rn) ≤ M2, where M2 > 0 is independent of i
+and j. By setting U = ∇u1 and V = ∇u2 in Lemma 2.3, we get
+∥A∥C0,α(Ω) ≤ M1 + M2
+�
+[∇u1]Cα(Ω) + ([∇u2]Cα(Ω)
+�
+≤ M3
+with M3 > 0 a constant depending only on n, α, ∥H1∥Ln(Ω), ∥H1∥C1(Ω), ∥g∥C2,α(Ω), and
+Ω, see (2.6). Observe that all these quantities are independent of u2 if v2 is close to v1 in
+C1,α(Ω). In summary, we have found that u is a solution to
+�
+div(A(x)∇u) = nH
+in Ω
+u = 0
+on ∂Ω
+and so, by Schauder estimates,
+(2.13)
+∥u∥C1,α(Ω) ≤ Cn[∂Ω]C1,αM3∥H∥C1,α(Ω) =: C2∥H∥C1,α(Ω).
+
+8
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+Therefore, by collecting estimates (2.11) and (2.13), and recalling that u = u1 − u2 = Tv1 −
+Tv2, and H = H1 − H2, we obtain
+∥Tv1 − Tv2∥C1,α(Ω) ≤ C1C2∥v1 − v2∥C1,α(Ω).
+If we choose δ = ε/(C1C2) then we see that T is continuous, as desired.
+Let us now turn to (2), which will follow from a priori estimates for prescribed mean
+curvature equations. Let {vk}k≥1 be a sequence in G such that
+sup
+k≥1
+∥vk∥C1,α(Ω) ≤ N1 < ∞
+and consider the corresponding solutions Hk ∈ C1,α(Ω) found in Step 1. Set uk = Tvk. By
+(2.6),
+∥uk∥C2,α(Ω) ≤ Ck
+where Ck > 0 is a constant depending only on n, α, ∥Hk∥Ln(Ω), ∥Hk∥C1(Ω), ∥h∥C2,α(Ω), and
+Ω. Since all Hk have the same boundary values h, by (2.9), we get that
+sup
+k≥1
+∥Hk∥Ln(Ω) = N2 < ∞.
+Furthermore, from the C1,α estimate in (2.2),
+sup
+k≥1
+∥Hk∥C1(Ω) ≤ Cn[∂Ω]C1,α∥h∥C1,α(∂Ω) sup
+k≥1
+∥D(vk)∥C0,α(Ω) = N3 < ∞.
+Consequently,
+sup
+k≥1
+∥uk∥C2,α(Ω) ≤ sup
+k≥1
+Ck = N4 < ∞.
+By the Arzel`a–Ascoli compact embedding C2,α(Ω) ⊂⊂ C1,α(Ω), there exist a subsequence
+{ukj}j≥1 of {uk}k≥1 and u ∈ G such that ukj → u in C1,α(Ω). We conclude that T(G) is
+precompact and (2) is proved.
+Thus, by Theorem 2.1, there exists u ∈ G such that Tu = u. We have proved the following:
+Theorem 2.4 (Existence for the simplified energy). Let Ω ⊂ Rn be a bounded domain with
+C3,α boundary ∂Ω, for some 0 < α < 1. Fix g ∈ C3,α(Ω). Let h ∈ C1,α(Ω) such that
+(2.14)
+|h(y)| ≤ H∂Ω(y)
+for all y ∈ ∂Ω,
+where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω,
+and
+(2.15)
+max
+∂Ω |h| <
+�|B1|
+|Ω|
+�1/n
+.
+Then there exist u ∈ C3,α(Ω) and H ∈ C1,α(Ω) such that H minimizes the energy
+1
+2
+ˆ
+Ω
+|∇H|2D(u) dx
+among all H ∈ W 1,2(Ω) such that H − h ∈ W 1,2
+0 (Ω), or, equivalently, H is the unique weak
+solution to
+�
+div(D(u)∇H) = 0
+in Ω
+H = h
+on ∂Ω,
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+9
+and, in addition, H is the mean curvature of the graph of u with prescribed values on ∂Ω,
+that is,
+�
+�
+�
+1
+n div
+� ∇u
+D(u)
+�
+= H
+in Ω
+u = g
+on ∂Ω.
+Remark 2.5 (Nonexistence of solutions). The conditions imposed on the curvature at the
+boundary datum h in Theorem 2.4 come from restrictions already present when one seeks for
+solutions of the prescribed mean curvature equation. Indeed, the divergence form equation for
+H is uniformly elliptic when u is, say, Lipschitz continuous and therefore is always solvable.
+On the other hand, if condition (2.14) is not satisfied, that is,
+|h(y0)| > H∂Ω(y0)
+for some y0 ∈ ∂Ω
+and h ≥ 0 (or h ≤ 0) on ∂Ω then H ≥ 0 (or H ≤ 0) in Ω and we have that for any ε > 0
+there exists g ∈ C∞(Ω) with |g| < ε such that the prescribed mean curvature equation with
+curvature H and boundary values h is not solvable (see [7, Theorem 3.4.5] or [6, Corollary
+14.13]) and hence neither is the minimum curvature variation system.
+On the other hand, a necessary condition for existence of solutions of the prescribed mean
+curvature equation is
+(2.16)
+����
+ˆ
+Ω
+Hη dx
+���� ≤ 1 − ε0
+n
+ˆ
+Ω
+|∇η| dx
+for all η ∈ C1
+0(Ω) and with
+1 − ε0 = sup
+Ω
+|∇u|
+�
+1 + |∇u|2 ,
+see [6, eq. (16.60)]. This condition implies ∥H∥Ln(Ω) < |B1|1/n, which is the structural con-
+dition on H that motivates (2.15). The requirement in Theorem 2.4 could be thus weakened,
+but (2.16) is the least requirement under which existence for the prescribed mean curvature
+equation can be obtained and hence also for the system at hand.
+3. Existence of G2 surfaces for the geometric energy
+In this section we discuss how the technique we developed in the previous section can be
+applied to the geometric energy functional
+E[M] = 1
+2
+ˆ
+Ω
+�
+|∇H|2 −
+����
+∇u · ∇H
+D(u)
+����
+2�
+D(u) dx.
+Let Ω, α, h and g be as in Section 2. Fix v ∈ C1,α(Ω) such that v = g on ∂Ω. Consider the
+energy
+Ev[H] := 1
+2
+ˆ
+Ω
+�
+|∇H|2 −
+����
+∇v · ∇H
+D(v)
+����
+2�
+D(v) dx =
+ˆ
+Ω
+L(∇H) dx
+where the smooth Lagrangian L is given by
+L(p) = 1
+2
+�
+|p|2 −
+����
+∇v · p
+D(v)
+����
+2�
+D(v)
+for p ∈ Rn.
+Then L is coercive, as
+L(p) ≥ 1
+2
+�
+|p|2 − |∇v|2|p|2
+D(v)2
+�
+D(v) = 1
+2
+�
+D(v) − |∇v|2
+D(v)
+�
+|p|2 =
+1
+2D(v)|p|2.
+
+10
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+To prove that L is convex, first observe that, for i = 1, . . . , n,
+Lpi(p) =
+�
+pi − (∇v · p)
+D(v)2 vxi
+�
+D(v) =
+n
+�
+j=1
+�
+δijD(v) − vxivxj
+D(v)
+�
+pj
+and, for i, j = 1, . . . , n,
+Lpipj(p) = δijD(v) − vxivxj
+D(v) .
+Then, for any ξ ∈ Rn,
+Lpipj(p)ξiξj = D(v)|ξ|2 − (∇v · ξ)2
+D(v)
+≥
+�
+D(v) − |∇v|2
+D(v)
+�
+|ξ|2 =
+1
+D(v)|ξ|2.
+Thus, D2
+pL is a positive definite matrix, and L is uniformly convex. It follows that there
+exists a unique minimizer H ∈ W 1,2(Ω) of the energy Ev[H] such that H − h ∈ W 1,2
+0 (Ω). In
+particular, H is the unique weak solution to
+�
+�
+�
+�
+�
+n
+�
+i=1
+(Lpi(∇H))xi = 0
+in Ω
+H = h
+on ∂Ω.
+Since
+Lpi(∇H) =
+n
+�
+j=1
+�
+δijD(v) − vxivxj
+D(v)
+�
+Hxj
+we find that H is the unique weak solution to the linear problem
+�
+div(a(x)∇H) = 0
+in Ω
+H = h
+on ∂Ω
+where
+aij(x) = δijD(v) − vxivxj
+D(v) = Lpipj.
+Observe that
+|aij(x)| ≤ C
+�
+D(v) + |∇v|2
+D(v)
+�
+≤ C(D(v) + |∇v|) ≤ C(n, ∥∇v∥L∞(Ω)).
+We have already seen that aij(x) is uniformly elliptic. Moreover, if v ∈ C1,α(Ω) then aij(x) ∈
+C0,α(Ω). Hence, H ∈ C1,α(Ω), with
+∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥v∥C1,α(Ω)∥h∥C1,α(∂Ω).
+If h satisfies (2.8) and (2.10) then we can apply Theorem 2.2 and find the unique solution
+u ∈ C3,α(Ω) to (2.3). From here on we can continue with the fixed point arguments we did
+in Section 2 to conclude the following result.
+Theorem 3.1 (Existence for the geometric functional). Let Ω ⊂ Rn be a bounded domain
+with C3,α boundary ∂Ω, for some 0 < α < 1. Fix g ∈ C3,α(Ω). Let h ∈ C1,α(Ω) such that
+|h(y)| ≤ H∂Ω(y)
+for all y ∈ ∂Ω,
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+11
+where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω,
+and
+max
+∂Ω |h| <
+�|B1|
+|Ω|
+�1/n
+.
+Then there exist u ∈ C3,α(Ω) and H ∈ C1,α(Ω) such that H minimizes the energy
+1
+2
+ˆ
+Ω
+�
+|∇H|2 −
+����
+∇u · ∇H
+D(u)
+����
+2�
+D(u) dx
+among all H ∈ W 1,2(Ω) such that H − h ∈ W 1,2
+0 (Ω), or, equivalently, H is the unique weak
+solution to
+�
+div(a(x)∇H) = 0
+in Ω
+H = h
+on ∂Ω,
+where
+aij(x) = δijD(u) − uxiuxj
+D(u)
+and, in addition, H is the mean curvature of the graph of u with prescribed values on ∂Ω,
+that is,
+�
+�
+�
+1
+n div
+� ∇u
+D(u)
+�
+= H
+in Ω
+u = g
+on ∂Ω.
+4. Weak solutions
+In this section we develop the weak formulation of the minimum curvature variation prob-
+lem in the context of geometric measure theory.
+Given a Lipschitz bounded domain Ω, we denote by BV(Ω) the space of functions of
+bounded variation in Ω. We start by recalling that u ∈ BV(Ω) is a generalized solution
+to the prescribed mean curvature equation with (weak) mean curvature H ∈ L1(Ω) and
+boundary value g ∈ L1(∂Ω) if
+(WPMC)
+J [u] =
+min
+v∈BV(Ω) J [v]
+where
+J [v] :=
+ˆ
+Ω
+D(v) +
+ˆ
+Ω
+nHv dx +
+ˆ
+∂Ω
+|v − g| dS
+and
+(4.1)
+ˆ
+Ω
+D(v) := sup
+� ˆ
+Ω
+�
+v
+n
+�
+i=1
+∂xiφi + φn+1
+�
+dx : φi ∈ C1
+c (Ω),
+n+1
+�
+i=1
+φ2
+i ≤ 1
+�
+.
+Note that
+´
+Ω
+�
+1 + |∇u|2 dx does not make usual sense a priori for a function of bounded
+variation and so (4.1) is indeed a definition. Furthermore, this definition is consistent in the
+sense that for v ∈ W 1,1(Ω) we have
+ˆ
+Ω
+D(v) =
+ˆ
+Ω
+�
+1 + |∇v|2 dx,
+see the proof of Lemma 4.1.
+
+12
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+In [3, Theorem 1.1], Giaquinta proved that if H is a measurable function then (WPMC)
+is solvable in BV(Ω) if and only if there exists ε0 > 0 such that, for every measurable subset
+A ⊂ Ω,
+(4.2)
+����
+ˆ
+A
+H dx
+���� ≤ (1 − ε0) 1
+nP(∂A)
+where P(∂A) denotes the perimeter of A. Clearly, (4.2) is significant only when A is a set of
+finite perimeter (or Caccioppoli set).
+We need a generalized measure of surface area. In that regard, we recall that the dis-
+tributional gradient of u ∈ BV(Ω) is a vector valued Radon measure whose total variation
+is identified with |∇u|. This is again consistent in the sense that if u ∈ W 1,1(Ω) then the
+total variation equals
+´
+Ω |∇u| dx (see [2, Chapter 5] for this and other properties of the space
+BV(Ω) used hereafter). In general, for an open set U ⊂⊂ Ω the variation measure of ∇u
+over U is given by
+|∇u|(U) = sup
+�ˆ
+U
+u div φ dx : φ ∈ C1
+c (U; Rn), |φ| ≤ 1
+�
+and, for an arbitrary set V ⊂ Ω,
+|∇u|(V ) = inf
+�
+|∇u|(U) : V ⊂ U and U is open
+�
+.
+Taking this into account, and in analogy with (4.1), we define for the area measure by
+(4.3)
+D(u)(U) = sup
+� ˆ
+Ω
+�
+u
+n
+�
+i=1
+∂xiφi + φn+1
+�
+dx : φi ∈ C1
+c (U),
+n+1
+�
+i=1
+φ2
+i ≤ 1
+�
+for any U ⊂⊂ Ω open and, for an arbitrary set V ⊂ Ω,
+D(u)(V ) = inf {D(u)(U) : V ⊂ U and U is open} .
+Although (4.3) could be defined, in principle, for functions in L1(Ω), it is easy to check that
+(4.3) is finite if and only if u ∈ BV(Ω).
+Similarly as for the variation measure, D(u) is a Radon measure, namely, a locally finite,
+Borel regular measure in Rn (to prove that it is locally finite, see the ideas in [5, eq. (14.2)]).
+The following observation will be useful.
+Lemma 4.1. Let U ⊂ Ω be a Borel set. Then
+(4.4)
+|U| ≤ D(u)(U).
+Proof. Due to the Borel regularity of both D(u) and the Lebesugue measure it suffices to
+prove (4.4) for open sets. Let U ⊂ Rn be open.
+First, we note that
+(4.5)
+D(u)(U) =
+ˆ
+U
+�
+1 + |∇u|2 dx
+for any u ∈ C1(Ω).
+Indeed, an integration by parts yields
+ˆ
+Ω
+�
+u
+n
+�
+i=1
+∂xiφi + φn+1
+�
+dx =
+ˆ
+U
+(−∇u, 1) · Φ dx
+where Φ = (φ1, . . . , φn, φn+1). Then the Cauchy-Schwartz inequality in Rn+1 and the condi-
+tion |Φ| ≤ 1 give
+D(u)(U) ≤
+ˆ
+U
+�
+1 + |∇u|2 dx.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+13
+On the other hand,
+�
+1 + |∇u|2 ∈ L1(U) and so there exists a sequence Φj = (φj
+1, . . . , φj
+n, φj
+n+1)
+with φj
+i ∈ C1
+c (U), j ≥ 1, that converges in L1(U) and almost everywhere to
+(−∇u,1)
+√
+1+|∇u|2 . Fur-
+thermore,
+(−∇u,1)
+√
+1+|∇u|2 is a unit vector so we may assume that �n+1
+i=1 (φj
+i)2 ≤ 1. Since
+��Φj · (−∇u, 1)
+�� ≤ |Φj|
+�
+1 + |∇u|2 ≤
+�
+1 + |∇u|2 ∈ L1(U)
+we can use the dominated convergence theorem to get
+lim
+j→∞
+ˆ
+Ω
+�
+u
+n
+�
+i=1
+∂xiφj
+i + φj
+n+1
+�
+dx = lim
+j→∞
+ˆ
+U
+(−∇u, 1) · Φj dx =
+ˆ
+U
+�
+1 + |∇u|2 dx
+and the supremum is achieved. Thus (4.5) holds.
+Second, we have that u ∈ BV(U) and there exists {uk}k≥1 ⊂ BV(U) ∩ C∞(U) such that
+uk → u in L1(Ω) and
+(4.6)
+lim
+k→∞ D(uk)(U) = D(u)(U),
+see [2, Theorem 5.3]. Since, by (4.5), the conclusion (4.4) is trivial for C1 functions, we have
+|U| ≤ lim
+k→∞ D(uk)(U) = D(u)(U)
+as desired.
+□
+From now on, we fix a bounded, C1,1 domain Ω. We consider the minimization problem
+min
+(u,H)∈A I[u, H]
+where
+(4.7)
+I[u, H] :=
+ˆ
+Ω
+|∇H|2 dD(u)
+and dD(u) stands for the area measure defined in (4.3). The admissible set A is defined as
+follows. Let h ∈ W 2,2(Ω) ∩ Lip(∂Ω) satisfying
+(4.8)
+|h(y)| ≤ n − 1
+n
+Λ(y), y ∈ ∂Ω,
+and
+max
+∂Ω |h| ≤ (1 − ε0)
+�|B1|
+|Ω|
+�1/n
+,
+where Λ(y) is the weak mean curvature of ∂Ω at y ∈ ∂Ω and
+(4.9)
+n − 1
+n
+< ε0 < 1.
+Define
+(4.10)
+A :=
+� (u, H) ∈ BV(Ω) × (Ln(Ω) ∩ W 2,2(Ω)) : u solves (WPMC)
+and ∥H∥Ln(Ω) + ∥H∥W 2,2(Ω) ≤ C0, H = h on ∂Ω
+�
+with C0 > 0 is to be appropriately chosen. The equality H = h is understood in the sense of
+traces.
+Remark 4.2. The condition H ∈ W 2,2(Ω) is certainly natural for applications to the design
+of fair G2-continuous surfaces in CAD/CAM/CAGD. Indeed, in dimensions n = 1, 2, 3, the
+Sobolev embedding gives that the curvature H is H¨older continuous.
+The main result of this section is the following:
+Theorem 4.3 (Existence of weak solutions). Let I be defined by (4.7), h ∈ W 2,2(Ω)∩Lip(∂Ω)
+satisfying (4.8) and ε0 ∈ (0, 1) satisfying (4.9). Then the set of admissible functions A in
+(4.10) is nonempty and there exists a minimizer of I within the class A.
+
+14
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+To prove Theorem 4.3 we recall the notion and properties of Γ−convergence in our context,
+referring the reader to [1] for an introduction to the topic.
+Let Jk, k ≥ 1, and J∞ be
+functionals defined on the common space BV(Ω) and taking values in [−∞, ∞]. The sequence
+{Jk}k≥1 is said to Γ−converge to J∞ if the following two conditions hold:
+(a) For every v ∈ BV(Ω) and every sequence {vk}k≥1 ⊂ BV(Ω) such that vk → v in BV(Ω)
+it holds
+lim inf
+k→∞ Jk(vk) ≥ J∞(v).
+(b) For every v ∈ BV(Ω) there exists a sequence {vk}k≥1 ⊂ BV(Ω) such that vk → v in
+BV(Ω) for which
+lim sup
+k→∞
+Jk(vk) ≤ J∞(v).
+We will use the following result.
+Theorem 4.4 (see [1, Theorem 1.21]). Let (X, d) be a metric space and let {fk}k≥1 be an
+equi-mildly coercive sequence of functions on X that Γ−converges to f∞. Then, there exits
+min
+X f∞ = lim
+k→∞ inf
+X fk.
+Moreover, if {xk}k≥1 ⊂ X is a precompact sequence such that
+lim
+k→∞ fk(xk) = lim
+k→∞ inf
+X fk
+then every limit of {xk}k≥1 is a minimum point for f∞.
+Here f is said to be mildly coercive if there exists a nonempty compact set K ⊂ X such
+that infK f = infX f, and equi-mild coercivity means that the set K is the same for the whole
+sequence {fk}k≥1.
+Proof of Theorem 4.3. The proof is divided into 4 steps.
+Step 1. A ̸= ∅. We can extend h to Ω by solving
+�
+∆H = 0
+in Ω
+H = h
+on ∂Ω.
+By classical elliptic regularity, H ∈ W 2,2(Ω) and
+∥H∥W 2,2(Ω) ≤ C0
+where C0 = C0(∂Ω, ∥h∥L∞(∂Ω)) > 0. Moreover, by the H¨older and isoperimetric inequalities,
+����
+ˆ
+A
+H dx
+���� ≤ ∥H∥Ln(Ω)|A|
+n−1
+n
+≤ ∥H∥Ln(Ω)
+P(∂A)
+n|B1|1/n .
+By the maximum principle and (4.8) we have
+∥H∥Ln(Ω) ≤ |Ω|1/n max
+∂Ω |h| ≤ (1 − ε0)|B1|1/n,
+where we make C0 larger if needed. Therefore,
+����
+ˆ
+A
+H dx
+���� ≤ (1 − ε0)
+n
+P(∂A)
+and (WPMC) is solvable for this H. Let u ∈ BV(Ω) be the corresponding minimizer of J .
+We have that A ̸= ∅. We further point out that
+´
+Ω D(u) < ∞ and H ∈ Lip(Ω) so that
+ˆ
+Ω
+|∇H|2 dD(u) ≤ ∥∇H∥2
+L∞(Ω)D(u)(Ω) < ∞.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+15
+In particular,
+0 ≤
+inf
+(u,H)∈A I[u, H] < ∞.
+Step 2. Construction of a minimizer. Let {(uk, Hk)}k≥1 ⊂ A be a minimizing sequence:
+m :=
+inf
+(u,H)∈A I[u, H] = lim
+k→∞ I[uk, Hk].
+To get a convergent subsequence of {uk}k≥1 we show its uniform boundedness in BV(Ω) and
+use that BV(Ω) embedds compactly in L1(Ω). Since every uk is a minimizer of the functional
+Jk defined by
+(4.11)
+Jk[v] :=
+ˆ
+Ω
+D(v) +
+ˆ
+Ω
+nHkv dx +
+ˆ
+∂Ω
+|v − g| dS
+we have that, for any u0 ∈ BV(Ω),
+ˆ
+Ω
+D(uk) +
+ˆ
+Ω
+nHkuk dx +
+ˆ
+∂Ω
+|uk − g| dS ≤
+ˆ
+Ω
+D(u0) +
+ˆ
+Ω
+nHku0 dx +
+ˆ
+∂Ω
+|u0 − g| dS
+from where
+(4.12)
+ˆ
+Ω
+D(uk) +
+ˆ
+Ω
+nHkuk dx ≤ C +
+ˆ
+Ω
+nHku0 dx
+for C > 0 independent of k. Reasoning as in [3, eq. (1.4)] we have that
+(4.13)
+ˆ
+Ω
+Hkuk dx ≥ −(1 − ε0)
+ˆ
+Ω
+|∇uk| − C.
+Furthermore, BV(Ω) ⊂ L
+n
+n−1 (Ω) so (4.13) and the H¨older inequality in (4.12) give
+ˆ
+Ω
+D(uk) ≤ −n
+ˆ
+Ω
+Hkuk dx + C +
+ˆ
+Ω
+nHku0 dx
+≤ n(1 − ε0)
+ˆ
+Ω
+|∇uk| + n∥Hk∥Ln(Ω)∥u0∥L
+n
+n−1 (Ω) + C
+for a new constant C > 0 that is independent of k. Moreover, the uniform bound on the
+Ln(Ω) norm of {Hk}k≥1 (they all belong to A) gives
+ˆ
+Ω
+|∇uk| ≤ n(1 − ε0)
+ˆ
+Ω
+|∇uk| + nC0∥u0∥L
+n
+n−1 (Ω) + C.
+Thus, after rearranging terms and recalling (4.9),
+ˆ
+Ω
+|∇uk| ≤
+1
+(1 − n(1 − ε0))
+�
+nC0∥u0∥L
+n
+n−1 (Ω) + C
+�
+.
+Hence, by compactness in BV(Ω), there exists a subsequence of {uk}k≥1, still denoted by the
+same indexes, and u∞ ∈ BV(Ω) such that
+uk → u∞ in L1(Ω) as k → ∞, and
+|∇u∞|(Ω) ≤ lim inf
+k→∞ |∇uk|.
+Note that we also have
+(4.14)
+D(u∞) ≤ lim inf
+k→∞ D(uk).
+By Poincar´e’s inequality and the Rellich–Kondrachov compactness theorem, there exist a
+subsequence of {Hk}k≥1, still denoted by the same indexes, and H∞ ∈ W 2,2(Ω) such that
+(4.15)
+∇Hk → ∇H∞ in L2(Ω), as k → ∞.
+
+16
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+Further, due to the uniform bound on ∥Hk∥Ln(Ω), we may assume that Hk converges weakly
+in Ln(Ω) to H∞. Finally, the weak convergence ensures that
+∥H∞∥Ln(Ω) + ∥H∞∥W 2,2(Ω) ≤ C0.
+Step 3. (u∞, H∞) ∈ A. For this step we use Γ−convergence. Recall the functionals Jk
+defined in (4.11) (for the subsequence Hk we found in Step 2) and define J∞ analogously.
+We want to show that u∞ is a solution of (WPMC), namely, that u∞ is a minimizer of J∞
+over BV(Ω). Let us show that {Jk}k≥1 Γ−converges to J∞. A first remark is that it is
+enough to prove the Γ−convergence of
+�
+Jk(v) :=
+ˆ
+Ω
+vHk dx
+to
+�
+J∞(v) :=
+ˆ
+Ω
+vH∞ dx
+since the other two terms do not depend on k and can be considered as continuous pertur-
+bations of Jk, see [1, Remark 1.7]. To prove the liminf inequality (a), let {vk}k≥1 ⊂ BV(Ω)
+and v ∈ BV(Ω) such that vk → v in BV(Ω). We write
+ˆ
+Ω
+vkHk dx −
+ˆ
+Ω
+vH∞ dx = Ik + IIk + IIIk
+with
+Ik =
+ˆ
+Ω
+(vk − v)H∞ dx
+IIk =
+ˆ
+Ω
+(vk − v)(Hk − H∞) dx
+IIIk =
+ˆ
+Ω
+v(Hk − H∞) dx.
+By lower semicontinuity [4, Proposition 2.1],
+lim inf
+k→∞ Ik ≥ 0
+Next, we bound
+|IIk| ≤ ∥vk − v∥L
+n
+n−1 (Ω)
+�
+∥Hk∥Ln(Ω) + ∥H∞∥Ln(Ω)
+�
+.
+Since vk converge to v in BV(Ω), by the isoperimetric embedding, the convergence also holds
+in L
+n
+n−1 (Ω). This and the uniform bound of Hk in Ln(Ω) give
+lim
+k→∞ IIk = 0.
+Finally, limk→∞ IIIk = 0 by the weak convergence of Hk to H∞ in Ln(Ω). As for the limsup
+inequality (b), given any v ∈ BV(Ω), consider the constant sequence vk = v for all k ≥ 1 and
+notice that, using the weak convergence of Hk to H∞ in Ln(Ω), we have that
+lim
+k→∞
+�
+Jk(vk) = �
+J∞(v).
+Hence, {Jk}k≥1 converges to J∞ in the Γ sense. Therefore, we can apply Theorem 4.4 with
+X = BV(Ω), fk = Jk, f∞ = J∞ and {xk}k≥1 and x∞ given by {uk}k≥1 and u∞, respectively
+(note that the sequence {Jk}k is equi-mildly coercive), to conclude that u∞ is a minimizer
+of J∞. We have thus shown that (u∞, H∞) ∈ A.
+
+SURFACES OF MINIMUM CURVATURE VARIATION
+17
+Step 4. (u∞, H∞) is a minimizer. Recall that L(p) = 1
+2|p|2, p ∈ Rn, is convex, that is,
+1
+2|p|2 ≥ 1
+2|p0|2 + p0 · (p − p0)
+for every p, p0 ∈ Rn. Then we can write
+1
+2
+ˆ
+Ω
+|∇Hk|2 dD(uk) ≥ 1
+2
+ˆ
+Ω
+|∇H∞|2 dD(uk)
++
+ˆ
+Ω
+∇H∞ · (∇Hk − ∇H∞) dD(uk).
+As k → ∞, the left hand side of this inequality converges to m. As for the right hand side,
+(4.6) implies that
+lim inf
+k→∞
+1
+2
+ˆ
+Ω
+|∇H∞|2 dD(uk) ≥ 1
+2
+ˆ
+Ω
+|∇H∞|2 dD(u∞).
+It remains to analyze the second term on the right hand side. For this, notice that Lemma
+4.1 implies that dD(uk) is absolutely continuous with respect to the Lebesgue measure, see
+(4.4). This and H¨older’s inequality give
+����
+ˆ
+Ω
+∇H∞ · (∇Hk − ∇H∞) dD(uk)
+���� ≤
+ˆ
+Ω
+|∇H∞||∇Hk − ∇H∞| dD(uk)
+≤ C
+ˆ
+Ω
+|∇H∞||∇Hk − ∇H∞| dx
+≤ C∥∇H∞∥L2(Ω)∥∇Hk − ∇H∞∥L2(Ω).
+In view of (4.15), this term goes to 0 as k → ∞. We have shown that
+m ≥
+ˆ
+Ω
+|∇H∞|2 dD(u∞).
+Since (u∞, H∞) ∈ A equality must be attained and (u∞, H∞) is a minimizer, as desired.
+□
+References
+[1] A. Braides, Γ-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications 22,
+Oxford University Press, Oxford, 2002.
+[2] L. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, Text-
+books in Mathematics, CRC Press, Boca Raton, FL, 2015.
+[3] M. Giaquinta, On the Dirichlet problem for surfaces of prescribed mean curvature, Manuscripta Math.
+12 (1974), 73–86.
+[4] E. Giusti, Boundary value problems for non-parametric surfaces of prescribed mean curvature, Ann.
+Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 501–548.
+[5] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics 80,
+Birkh¨auser Verlag, Basel, 1984.
+[6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in
+Mathematics, Springer-Verlag, Berlin, 2001.
+[7] Q. Han, Nonlinear Elliptic Equations of the Second Order, Graduate Studies in Mathematics 171, Amer-
+ican Mathematical Society, Providence, RI, 2016.
+[8] H. P. Moreton and C. H. S´equin Functional optimization for fair surface design, ACM SIGGRAPH
+Computer Graphics 2 (1992), 167-176.
+[9] K. L. Narayan, K. M. Rao and M. M. M. Sacar, Computer Aided Design and Manufacturing, PHI Learning
+Pvt. Ltd., 2008.
+[10] W. Welch and A. Witkin, Variational surface modeling, ACM SIGGRAPH computer graphics 2 (1992),
+157-166.
+
+18
+L. A. CAFFARELLI, P. R. STINGA, AND H. VIVAS
+[11] G. Xu and Q. Zhang, Minimal mean-curvature-variation surfaces and their applications in surface mod-
+eling, in: International Conference on Geometric Modeling and Processing, 357–370, Springer, Berlin,
+Heidelberg, 2006.
+Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, TX
+78712, United States of America
+Email address: caffarel@math.utexas.edu
+Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, United
+States of America
+Email address: stinga@iastate.edu
+Centro Marplatense de Investigaciones Matem´aticas/CONICET, Dean Funes 3350, 7600 Mar
+del Plata, Argentina
+Email address: havivas@mdp.edu.ar
+
diff --git a/DdAyT4oBgHgl3EQfSPd1/content/tmp_files/load_file.txt b/DdAyT4oBgHgl3EQfSPd1/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,558 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf,len=557
+page_content='SURFACES OF MINIMUM CURVATURE VARIATION  LUIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' CAFFARELLI, PABLO RA´UL STINGA, AND HERN´AN VIVAS  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We establish the analytical theory of surfaces of minimum curvature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  We construct classical, G2 continuous surfaces, as well as weak solutions in the context of  geometric measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Introduction  Computer-aided design (CAD) and computer-aided manufacturing (CAM) are widely pop-  ular techniques whose basic feature is the use of computer software to create or modify shapes  in such a way that some aspects of the design process, such as quality of the object or produc-  tivity of the process, are optimized, see, for example, [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Their origins can be traced back  to the 1950s and 60s and their development have been continuous since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Nowadays,  CAD/CAM are used in contexts as varied as engineering, particularly in automotive, ship-  building and aerospace industries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' architectural design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' and computer animation for creation  of special effects in movies, among many other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Within this realm, of particular interest are geometric problems in computer-aided geomet-  ric design (CAGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The goal of CAGD is the creation of complex smoothly shaped models  and surfaces with specified geometric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The resulting surfaces have to accurately  reflect these specifications and be free of unwanted wrinkles, bulges and ripples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In many  instances, the aim is to create fair surfaces that are aesthetically pleasing to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' As it  turns out, many of these problems can be approached via a variational principle, that is, by  looking for a surface that minimizes an appropriate functional or fairness energy subject to  adequate geometric boundary conditions, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  The most commonly used fairness energy functionals can be split into two groups: physical-  based or geometric-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The first group roughly corresponds to interpreting the surface as  an ideal elastic membrane or plate and minimize energies such as  ´  |∇u|2 dx or  ´  |∆u|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  The second group aims at minimizing energies that relate to geometric invariants of the  surface such as the area or curvature, see [11] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In 1992, Moreton  and S´equin proposed in [8] a numerical algorithm for the creation of 2-dimensional fair  surfaces M as minimizers of the energy functional  ˆ  M  ��dκ1  de1  �2  +  �dκ2  de2  �2�  dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Here e1 and e2 are the principal directions corresponding to the principal curvatures κ1 and  κ2 of M and dA is the differential of surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  It is a key aspect in CAGD to be able to construct fair surfaces that preserve several degrees  of geometric continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' This is particularly important at the boundary of the domains where  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Primary: 35B65, 49Q10, 53A10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Secondary: 49Q20, 65D17,  68U07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Curvature variation, computer-aided design, prescribed mean curvature, regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Research partially supported by NSF grant 1500871 (USA), Simons Foundation grant 580911 (USA), and  Agencia Nacional de Promoci´on Cient´ıfica y Tecnol´ogica under grant PICT 2019-3530 (Argentina).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='00082v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='DG]  31 Dec 2022  2  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' VIVAS  the surfaces meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The notions of geometric continuity are referred to as G0 continuity, where  two surfaces meet in a continuous fashion, without jumps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' G1 continuity, where the tangent  planes of the surfaces meet with continuity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' and G2 continuity, where the curvatures meet  with continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' These are not the same as the classical notions of C0, C1 and C2 continuities,  as those require some specific combination of the derivatives of the solutions to be continuous  up to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In particular, G2 continuity turns out to be crucial in applications such  as the streamlined surfaces of aircrafts, ships and cars, and this was the main motivation  for the numerical study in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  In [11], a numerical finite difference method is proposed  to construct surfaces that would enjoy G2 continuity as steady states of a sixth order flow  derived from the Euler–Lagrange equation of the energy functional  ˆ  M  |∇H|2 dA  where H is the mean curvature of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Numerical evidence of G2 continuity is observed in [11],  while G1 continuity is expected according to [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' To the best of our knowledge, the analytical  theory of surfaces of minimum curvature variation in general is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Furthermore, no  proof of G2 continuity is available thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  The aim of this paper is to fill these gaps and to develop the analytical foundation from  the PDE perspective of the theory of surfaces of minimum curvature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We give  two constructions of surfaces: classical solutions that are G2-continuous, and weak solutions  through geometric measure theory methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Therefore, we consider the minimization problem  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1)  min  M  1  2  ˆ  M  |∇MH|2 dA  where M ranges over all n−dimensional manifolds in Rn+1, n ≥ 1, with prescribed bound-  ary, H is the mean curvature of M and dA is the differential of surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Notice that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1) minimizes the (quadratic) variation of the mean curvature of M so that surfaces with  constant mean curvature such as planes, circles and cylinders are minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  If M is the graph of a function defined on a bounded domain Ω ⊂ Rn, that is,  M = {(x, u(x)) : x ∈ Ω}  for some u : Ω → R, then the values of u at ∂Ω prescribe the boundary ∂M of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' For a  point x0 ∈ Ω, the tangent plane to M at (x0, u(x0)) and its upward pointing unit normal are  P(x) = u(x0) + ∇u(x0) · (x − x0)  and  ν(x0) =  (−∇u(x0), 1)  (1 + |∇u(x0)|2)1/2 ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The mean curvature H of M at a point is defined as the average of the n  principal curvatures of M at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In the coordinates given by u, it takes the form  H ≡ H(u) = 1  n div  �  ∇u  (1 + |∇u|2)1/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  If we set  D(u) := (1 + |∇u|2)1/2  we then have that dA = D(u) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Let f be a function in C1(Ω × R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The tangential gradient of f on M is obtained by  projecting the gradient of f in Rn+1 onto the plane orthogonal to ν:  ∇Mf = ∇Rn+1f − (ν · ∇Rn+1f)ν  on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  SURFACES OF MINIMUM CURVATURE VARIATION  3  Clearly, ν · ∇Mf = 0 and  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2)  |∇Mf|2 = |∇Rn+1f|2 − |ν · ∇Rn+1f|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Furthermore, ∇Mf depends only on the values of f on M, see, for instance, [6, Section 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  To compute ∇MH we extend H as a function of (x, xn+1) ∈ Ω × R by making it constant  in xn+1: H(x, xn+1) ≡ H(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' This is enough to compute ∇MH because the resulting value  is independent of the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Since ∇Rn+1H = (∇H, Hxn+1) = (∇H, 0), by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2) we get  |∇MH|2 = |(∇H, 0)|2 −  ����(−∇u · ∇H)(−∇u, 1)  D(u)2  ����  2  = |∇H|2 −  ����  ∇u · ∇H  D(u)  ����  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  With this formula the energy in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1) becomes  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3)  E[M] = 1  2  ˆ  Ω  �  |∇H|2 −  ����  ∇u · ∇H  D(u)  ����  2�  D(u) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  We will call this the geometric energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' It follows from the Cauchy–Schwartz inequality that  |∇H|2  D(u)2 ≤ |∇MH|2 ≤ |∇H|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Therefore, we will also study the (larger) simplified energy functional  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4)  E[H, u] := 1  2  ˆ  Ω  |∇H|2D(u) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  In Section 2 we consider (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4) and show how to construct smooth solutions that satisfy the  prescribed mean curvature equation for a curvature of minimum variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Section 3 shows  how to modify the argument to construct solutions of the geometric energy functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Finally, in Section 4 we provide a weak formulation of the problem and prove existence of  minimizers in the context of geometric measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Existence of G2 surfaces for the simplified energy  In this section we work with the simplified energy functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let Ω ⊂ Rn be a  bounded domain such that ∂Ω ∈ C3,α for some 0 < α < 1 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We assume that we are  given prescribed boundary values g ∈ C3,α(Ω) for u and h ∈ C1,α(Ω) for H on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  We address the following problem: given Ω and the boundary datum g, find a surface  given by the graph of a function u such that its mean curvature H is a minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4)  among all functions with prescribed boundary values h ∈ C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  We will use Schauder’s fixed point theorem:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1 (see [6, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let G be a closed convex set in a Banach space B  and let T be a continuous mapping of G into itself such that the image T(G) is precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Then T has a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Consider the Banach space B = C1,α(Ω) and its subset  G :=  �  v ∈ C1,α(Ω) : v = g on ∂Ω  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Observe that G is nonempty because g ∈ C3,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' By classical global Schauder estimates,  we see that another example of function in G is the harmonic extension v of g inside of Ω:  �  ∆v = 0  in Ω  v = g  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  4  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' VIVAS  It is clear that G is convex and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  For any v ∈ G, we define the functional  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1)  E[H, v] := 1  2  ˆ  Ω  |∇H|2D(v) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  The map T : G → G is constructed in a 2-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Given any v ∈ G, we find the unique minimizer H ∈ W 1,2(Ω) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1) such that  H − h ∈ W 1,2  0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' This can be done because the coefficient D(v) satisfies  1 ≤ D(v) ≤ (1 + ∥∇v∥2  L∞(Ω))1/2 < ∞,  so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1) is a coercive functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Then H is the unique weak solution to  �  div(D(v)∇H) = 0  in Ω  H = h  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Since v ∈ C1,α(Ω), the coefficient D(v) ∈ C0,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Thus, by global Schauder estimates (see  [6, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='11]),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2)  ∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥D(v)∥C0,α(Ω)∥h∥C1,α(∂Ω)  where Cn > 0 is a constant that depends only on dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Given H ∈ C1,α(Ω) from Step 1, we find the solution u to the prescribed mean  curvature equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3)  �  �  �  div  � ∇u  D(u)  �  = nH  in Ω  u = g  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  For this, we use the following result (where we use nH instead of H in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2 ([7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1] and its proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let 0 < α < 1 and Ω ⊂ Rn be a bounded  domain with C3,α boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Suppose that H ∈ C1,α(Ω) satisfies  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4)  ∥H∥Ln(Ω) <  �ˆ  Rn(1 + |p|2)− n+2  2  dp  �1/n  and, for any y ∈ ∂Ω,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='5)  |H(y)| ≤ H∂Ω(y),  where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Then for any g ∈ C3,α(Ω) there exists a unique solution u ∈ C3,α(Ω) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In particular,  there exists a constant C∗ > 0, depending only on n, α, ∥H∥Ln(Ω), ∥H∥C1(Ω), ∥g∥C2,α(Ω) and  Ω, such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='6)  ∥u∥C2,α(Ω) ≤ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  The constant in the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4) can be simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Recall the definition of the  Beta function and its relation with the Gamma function: for x, y > 0,  B(x, y) :=  ˆ ∞  0  tx−1  (1 + t)x+y dt = Γ(x)Γ(y)  Γ(x + y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  SURFACES OF MINIMUM CURVATURE VARIATION  5  We have that Γ(1) = 1 and xΓ(x) = Γ(x + 1), for all x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' By passing to polar coordinates  p = rθ, for r > 0 and θ ∈ Sn−1, performing the change of variables t = r2 which makes  2dr/r = dt/t, and using that |Sn−1| = n|B1|, we get  ˆ  Rn(1 + |p|2)− n+2  2 dp = |Sn−1|  ˆ ∞  0  rn  (1 + r2)  n+2  2  dr  r  = n|B1|  2  ˆ ∞  0  tn/2  (1 + t)  n+2  2  dt  t  = n|B1|  2  B(n/2, 1) = n|B1|  2  Γ(n/2)  Γ(n/2 + 1) = |B1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Therefore, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4) reads  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='7)  ∥H∥Ln(Ω) < |B1|1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Now (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='7) impose further restrictions on the boundary values h of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Condition  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='5) is natural to assume and cannot be avoided (see, for example, the nonexistence results [6,  Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='11] and [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Therefore, we assume that h ∈ C1,α(Ω) additionally  satisfies  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='8)  |h(y)| ≤ H∂Ω(y)  for all y ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  On the other hand, by the maximum principle (see [6, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1]), we can estimate  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='9)  ˆ  Ω  |H|n dx ≤ |Ω|  �  max  ∂Ω |h|  �n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  or  ∥H∥Ln(Ω) ≤ |Ω|1/n max  ∂Ω |h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Thus, in order to ensure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='7), we further assume that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='10)  max  ∂Ω |h| <  �|B1|  |Ω|  �1/n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  From the computer science point of view, this means that for large boundary curvatures h  the domain Ω for the reconstruction should be sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Therefore, under the additional assumptions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='10), we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2  and find the unique solution u ∈ C3,α(Ω) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' This completes Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Using these two steps, we define T : G → G by T(v) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In order to apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1,  we need to verify that  (1) T is continuous, and  (2) T(G) is precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Let us begin with (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Fix v1 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We need to show that given any ε > 0 there exists  δ = δ(ε, v1) > 0 such that for any v2 ∈ G satisfying ∥v1 − v2∥C1,α(Ω) < δ we have ∥u1 −  u2∥C1,α(Ω) < ε, where uj = Tvj, for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Let Hj denote the minimizer of E[·, vj],  j = 1, 2, as constructed in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Then the difference H = H1 − H2 is the unique weak  solution to  �  div(D(v1)∇H) = div  �  (D(v2) − D(v1))∇H2  �  in Ω  H = 0  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  6  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' VIVAS  By global Schauder estimates (see [6, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='11]),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='11)  ∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥D(v1)∥C0,α(Ω)∥(D(v2) − D(v1))∇H2∥C0,α(Ω)  ≤ C(n, α, Ω, v1, ∇H2)∥v1 − v2∥C1,α(Ω)  =: C1∥v1 − v2∥C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Let us now estimate the difference u = u1 − u2 ∈ C3,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Since  �  �  �  div  � ∇ui  D(ui)  �  = nHi  in Ω, for i = 1, 2  u1 = u2 = g  on ∂Ω  we find that  �  �  �  div  � ∇u1  D(u1) − ∇u2  D(u2)  �  = nH  in Ω  u = 0  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  In order to apply global Schauder estimates one more time we need to find an equation for  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Set  F(p) :=  p  �  1 + |p|2  p ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Then F is a smooth, bounded vector field with entries Fi(p) =  pi  √  1+|p|2 , for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Note  that, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' , n,  ∂jFi(p) =  �  �  �  1  √  1+|p|2 −  p2  i  (1+|p|2)3/2  if i = j  −  pipj  (1+|p|2)3/2  if i ̸= j  =  δij  D(p) − pipj  D(p)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  In particular,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='12)  ∂jFi(p) = ∂iFj(p)  so that ∇F is a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' It is clear that ∇F is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' To see that it is locally  strictly elliptic, observe that, for any ξ ∈ Rn, by the Cauchy–Schwartz inequality,  n  �  i,j=1  ∂jFi(p)ξiξj =  n  �  i,j=1  �δijξiξj  D(p) − pipjξiξj  D(p)3  �  ≥ |ξ|2  �  1  D(p) −  |p|2  D(p)3  �  =  |ξ|2  D(p)3 ≥ θ(R)|ξ|2  for all |p| < R, where θ(R) → 0 as R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Furthermore, we can write  Fi(∇u1) − Fi(∇u2) =  ˆ 1  0  d  dtFi(t∇u1 + (1 − t)∇u2) dt  =  ˆ 1  0  ∇Fi(t∇u1 + (1 − t)∇u2) · ∇(u1 − u2) dt  so that  F(∇u1) − F(∇u2) = A(x)∇u  with  Aij(x) =  ˆ 1  0  ∂jFi(t∇u1 + (1 − t)∇u2) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  SURFACES OF MINIMUM CURVATURE VARIATION  7  The matrix A is symmetric thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='12), as well as bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Recall that ∇F is locally  strictly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Now, u1 ∈ C3,α(Ω) is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='6), the C2,α(Ω) norm of u2 is uniformly  controlled by the C1(Ω) norm of H2, which in turn is uniformly close to the C1(Ω) norm of  the initially fixed H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' These facts imply that that A(x) is strictly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Moreover, we have  the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let U, V : Ω → Rn, U, V ∈ C0,α(Ω) and let ψ : Rn → R be a smooth function  such that  ∥ψ∥L∞(Rn) + ∥∇ψ∥L∞(Rn) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Define  φ(x) :=  ˆ 1  0  ψ(tU(x) + (1 − t)V (x)) dt  for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Then φ ∈ C0,α(Ω), with  ∥φ∥C0,α(Ω) ≤ ∥ψ∥L∞(Rn) + ∥∇ψ∥L∞(Rn)  �  [U]Cα(Ω) + [V ]Cα(Ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The boundedness of ψ implies that φ is bounded with ∥φ∥L∞(Ω) ≤ ∥ψ∥L∞(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' To  bound the H¨older seminorm of φ, let x, y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Then  |φ(x) − φ(y)| =  ����  ˆ 1  0  �  ψ(tU(x) + (1 − t)V (x)) − ψ(tU(y) + (1 − t)V (y))  �  dt  ����  ≤ ∥∇ψ∥L∞(Rn)  ˆ 1  0  |t(U(x) − U(y)) + (1 − t)(V (x) − V (y))| dt  ≤ ∥∇ψ∥L∞(Rn) (|U(x) − U(y)| + |V (x) − V (y)|)  ≤ ∥∇ψ∥L∞(Rn)  �  [U]Cα(Ω) + [V ]Cα(Ω)  �  |x − y|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3 gives the H¨older continuity of A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Indeed, let ψ be any of the entries of the  gradient matrix of F:  (∇F(p))ij =  δij  D(p) − pipj  D(p)3  for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' , n,  which are smooth and bounded, so ∥ψ∥L∞(Rn) ≤ M1, where M1 is independent of i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  For any k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' , n, we have  ∂k(∇F(p))ij = −δijpk + δikpj + δjkpi  D(p)3  + pipjpk  D(p)5  and these are all bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Therefore, ∥∇ψ∥L∞(Rn) ≤ M2, where M2 > 0 is independent of i  and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' By setting U = ∇u1 and V = ∇u2 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3, we get  ∥A∥C0,α(Ω) ≤ M1 + M2  �  [∇u1]Cα(Ω) + ([∇u2]Cα(Ω)  �  ≤ M3  with M3 > 0 a constant depending only on n, α, ∥H1∥Ln(Ω), ∥H1∥C1(Ω), ∥g∥C2,α(Ω), and  Ω, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Observe that all these quantities are independent of u2 if v2 is close to v1 in  C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In summary, we have found that u is a solution to  �  div(A(x)∇u) = nH  in Ω  u = 0  on ∂Ω  and so, by Schauder estimates,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='13)  ∥u∥C1,α(Ω) ≤ Cn[∂Ω]C1,αM3∥H∥C1,α(Ω) =: C2∥H∥C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  8  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' VIVAS  Therefore, by collecting estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='13), and recalling that u = u1 − u2 = Tv1 −  Tv2, and H = H1 − H2, we obtain  ∥Tv1 − Tv2∥C1,α(Ω) ≤ C1C2∥v1 − v2∥C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  If we choose δ = ε/(C1C2) then we see that T is continuous, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Let us now turn to (2), which will follow from a priori estimates for prescribed mean  curvature equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let {vk}k≥1 be a sequence in G such that  sup  k≥1  ∥vk∥C1,α(Ω) ≤ N1 < ∞  and consider the corresponding solutions Hk ∈ C1,α(Ω) found in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Set uk = Tvk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' By  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='6),  ∥uk∥C2,α(Ω) ≤ Ck  where Ck > 0 is a constant depending only on n, α, ∥Hk∥Ln(Ω), ∥Hk∥C1(Ω), ∥h∥C2,α(Ω), and  Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Since all Hk have the same boundary values h, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='9), we get that  sup  k≥1  ∥Hk∥Ln(Ω) = N2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Furthermore, from the C1,α estimate in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2),  sup  k≥1  ∥Hk∥C1(Ω) ≤ Cn[∂Ω]C1,α∥h∥C1,α(∂Ω) sup  k≥1  ∥D(vk)∥C0,α(Ω) = N3 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Consequently,  sup  k≥1  ∥uk∥C2,α(Ω) ≤ sup  k≥1  Ck = N4 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  By the Arzel`a–Ascoli compact embedding C2,α(Ω) ⊂⊂ C1,α(Ω), there exist a subsequence  {ukj}j≥1 of {uk}k≥1 and u ∈ G such that ukj → u in C1,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We conclude that T(G) is  precompact and (2) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Thus, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1, there exists u ∈ G such that Tu = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We have proved the following:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4 (Existence for the simplified energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let Ω ⊂ Rn be a bounded domain with  C3,α boundary ∂Ω, for some 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Fix g ∈ C3,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let h ∈ C1,α(Ω) such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='14)  |h(y)| ≤ H∂Ω(y)  for all y ∈ ∂Ω,  where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω,  and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='15)  max  ∂Ω |h| <  �|B1|  |Ω|  �1/n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Then there exist u ∈ C3,α(Ω) and H ∈ C1,α(Ω) such that H minimizes the energy  1  2  ˆ  Ω  |∇H|2D(u) dx  among all H ∈ W 1,2(Ω) such that H − h ∈ W 1,2  0 (Ω), or, equivalently, H is the unique weak  solution to  �  div(D(u)∇H) = 0  in Ω  H = h  on ∂Ω,  SURFACES OF MINIMUM CURVATURE VARIATION  9  and, in addition, H is the mean curvature of the graph of u with prescribed values on ∂Ω,  that is,  �  �  �  1  n div  � ∇u  D(u)  �  = H  in Ω  u = g  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='5 (Nonexistence of solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The conditions imposed on the curvature at the  boundary datum h in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4 come from restrictions already present when one seeks for  solutions of the prescribed mean curvature equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Indeed, the divergence form equation for  H is uniformly elliptic when u is, say, Lipschitz continuous and therefore is always solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  On the other hand, if condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='14) is not satisfied, that is,  |h(y0)| > H∂Ω(y0)  for some y0 ∈ ∂Ω  and h ≥ 0 (or h ≤ 0) on ∂Ω then H ≥ 0 (or H ≤ 0) in Ω and we have that for any ε > 0  there exists g ∈ C∞(Ω) with |g| < ε such that the prescribed mean curvature equation with  curvature H and boundary values h is not solvable (see [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='5] or [6, Corollary  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='13]) and hence neither is the minimum curvature variation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  On the other hand, a necessary condition for existence of solutions of the prescribed mean  curvature equation is  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='16)  ����  ˆ  Ω  Hη dx  ���� ≤ 1 − ε0  n  ˆ  Ω  |∇η| dx  for all η ∈ C1  0(Ω) and with  1 − ε0 = sup  Ω  |∇u|  �  1 + |∇u|2 ,  see [6, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='60)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' This condition implies ∥H∥Ln(Ω) < |B1|1/n, which is the structural con-  dition on H that motivates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The requirement in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4 could be thus weakened,  but (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='16) is the least requirement under which existence for the prescribed mean curvature  equation can be obtained and hence also for the system at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Existence of G2 surfaces for the geometric energy  In this section we discuss how the technique we developed in the previous section can be  applied to the geometric energy functional  E[M] = 1  2  ˆ  Ω  �  |∇H|2 −  ����  ∇u · ∇H  D(u)  ����  2�  D(u) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Let Ω, α, h and g be as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Fix v ∈ C1,α(Ω) such that v = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Consider the  energy  Ev[H] := 1  2  ˆ  Ω  �  |∇H|2 −  ����  ∇v · ∇H  D(v)  ����  2�  D(v) dx =  ˆ  Ω  L(∇H) dx  where the smooth Lagrangian L is given by  L(p) = 1  2  �  |p|2 −  ����  ∇v · p  D(v)  ����  2�  D(v)  for p ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Then L is coercive, as  L(p) ≥ 1  2  �  |p|2 − |∇v|2|p|2  D(v)2  �  D(v) = 1  2  �  D(v) − |∇v|2  D(v)  �  |p|2 =  1  2D(v)|p|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  10  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' VIVAS  To prove that L is convex, first observe that, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' , n,  Lpi(p) =  �  pi − (∇v · p)  D(v)2 vxi  �  D(v) =  n  �  j=1  �  δijD(v) − vxivxj  D(v)  �  pj  and, for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' , n,  Lpipj(p) = δijD(v) − vxivxj  D(v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Then, for any ξ ∈ Rn,  Lpipj(p)ξiξj = D(v)|ξ|2 − (∇v · ξ)2  D(v)  ≥  �  D(v) − |∇v|2  D(v)  �  |ξ|2 =  1  D(v)|ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Thus, D2  pL is a positive definite matrix, and L is uniformly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' It follows that there  exists a unique minimizer H ∈ W 1,2(Ω) of the energy Ev[H] such that H − h ∈ W 1,2  0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In  particular, H is the unique weak solution to  �  �  �  �  �  n  �  i=1  (Lpi(∇H))xi = 0  in Ω  H = h  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Since  Lpi(∇H) =  n  �  j=1  �  δijD(v) − vxivxj  D(v)  �  Hxj  we find that H is the unique weak solution to the linear problem  �  div(a(x)∇H) = 0  in Ω  H = h  on ∂Ω  where  aij(x) = δijD(v) − vxivxj  D(v) = Lpipj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Observe that  |aij(x)| ≤ C  �  D(v) + |∇v|2  D(v)  �  ≤ C(D(v) + |∇v|) ≤ C(n, ∥∇v∥L∞(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  We have already seen that aij(x) is uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Moreover, if v ∈ C1,α(Ω) then aij(x) ∈  C0,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Hence, H ∈ C1,α(Ω), with  ∥H∥C1,α(Ω) ≤ Cn[∂Ω]C1,α∥v∥C1,α(Ω)∥h∥C1,α(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  If h satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='10) then we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2 and find the unique solution  u ∈ C3,α(Ω) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' From here on we can continue with the fixed point arguments we did  in Section 2 to conclude the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1 (Existence for the geometric functional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let Ω ⊂ Rn be a bounded domain  with C3,α boundary ∂Ω, for some 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Fix g ∈ C3,α(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let h ∈ C1,α(Ω) such that  |h(y)| ≤ H∂Ω(y)  for all y ∈ ∂Ω,  SURFACES OF MINIMUM CURVATURE VARIATION  11  where H∂Ω is the mean curvature of ∂Ω corresponding to the inner unit normal vector to ∂Ω,  and  max  ∂Ω |h| <  �|B1|  |Ω|  �1/n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Then there exist u ∈ C3,α(Ω) and H ∈ C1,α(Ω) such that H minimizes the energy  1  2  ˆ  Ω  �  |∇H|2 −  ����  ∇u · ∇H  D(u)  ����  2�  D(u) dx  among all H ∈ W 1,2(Ω) such that H − h ∈ W 1,2  0 (Ω), or, equivalently, H is the unique weak  solution to  �  div(a(x)∇H) = 0  in Ω  H = h  on ∂Ω,  where  aij(x) = δijD(u) − uxiuxj  D(u)  and, in addition, H is the mean curvature of the graph of u with prescribed values on ∂Ω,  that is,  �  �  �  1  n div  � ∇u  D(u)  �  = H  in Ω  u = g  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Weak solutions  In this section we develop the weak formulation of the minimum curvature variation prob-  lem in the context of geometric measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Given a Lipschitz bounded domain Ω, we denote by BV(Ω) the space of functions of  bounded variation in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We start by recalling that u ∈ BV(Ω) is a generalized solution  to the prescribed mean curvature equation with (weak) mean curvature H ∈ L1(Ω) and  boundary value g ∈ L1(∂Ω) if  (WPMC)  J [u] =  min  v∈BV(Ω) J [v]  where  J [v] :=  ˆ  Ω  D(v) +  ˆ  Ω  nHv dx +  ˆ  ∂Ω  |v − g| dS  and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1)  ˆ  Ω  D(v) := sup  � ˆ  Ω  �  v  n  �  i=1  ∂xiφi + φn+1  �  dx : φi ∈ C1  c (Ω),  n+1  �  i=1  φ2  i ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Note that  ´  Ω  �  1 + |∇u|2 dx does not make usual sense a priori for a function of bounded  variation and so (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1) is indeed a definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Furthermore, this definition is consistent in the  sense that for v ∈ W 1,1(Ω) we have  ˆ  Ω  D(v) =  ˆ  Ω  �  1 + |∇v|2 dx,  see the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  12  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' VIVAS  In [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1], Giaquinta proved that if H is a measurable function then (WPMC)  is solvable in BV(Ω) if and only if there exists ε0 > 0 such that, for every measurable subset  A ⊂ Ω,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2)  ����  ˆ  A  H dx  ���� ≤ (1 − ε0) 1  nP(∂A)  where P(∂A) denotes the perimeter of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Clearly, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2) is significant only when A is a set of  finite perimeter (or Caccioppoli set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  We need a generalized measure of surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In that regard, we recall that the dis-  tributional gradient of u ∈ BV(Ω) is a vector valued Radon measure whose total variation  is identified with |∇u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' This is again consistent in the sense that if u ∈ W 1,1(Ω) then the  total variation equals  ´  Ω |∇u| dx (see [2, Chapter 5] for this and other properties of the space  BV(Ω) used hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' In general, for an open set U ⊂⊂ Ω the variation measure of ∇u  over U is given by  |∇u|(U) = sup  �ˆ  U  u div φ dx : φ ∈ C1  c (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Rn), |φ| ≤ 1  �  and, for an arbitrary set V ⊂ Ω,  |∇u|(V ) = inf  �  |∇u|(U) : V ⊂ U and U is open  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Taking this into account, and in analogy with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1), we define for the area measure by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3)  D(u)(U) = sup  � ˆ  Ω  �  u  n  �  i=1  ∂xiφi + φn+1  �  dx : φi ∈ C1  c (U),  n+1  �  i=1  φ2  i ≤ 1  �  for any U ⊂⊂ Ω open and, for an arbitrary set V ⊂ Ω,  D(u)(V ) = inf {D(u)(U) : V ⊂ U and U is open} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Although (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3) could be defined, in principle, for functions in L1(Ω), it is easy to check that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3) is finite if and only if u ∈ BV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Similarly as for the variation measure, D(u) is a Radon measure, namely, a locally finite,  Borel regular measure in Rn (to prove that it is locally finite, see the ideas in [5, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  The following observation will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let U ⊂ Ω be a Borel set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Then  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4)  |U| ≤ D(u)(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Due to the Borel regularity of both D(u) and the Lebesugue measure it suffices to  prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4) for open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let U ⊂ Rn be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  First, we note that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='5)  D(u)(U) =  ˆ  U  �  1 + |∇u|2 dx  for any u ∈ C1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Indeed, an integration by parts yields  ˆ  Ω  �  u  n  �  i=1  ∂xiφi + φn+1  �  dx =  ˆ  U  (−∇u, 1) · Φ dx  where Φ = (φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' , φn, φn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Then the Cauchy-Schwartz inequality in Rn+1 and the condi-  tion |Φ| ≤ 1 give  D(u)(U) ≤  ˆ  U  �  1 + |∇u|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  SURFACES OF MINIMUM CURVATURE VARIATION  13  On the other hand,  �  1 + |∇u|2 ∈ L1(U) and so there exists a sequence Φj = (φj  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' , φj  n, φj  n+1)  with φj  i ∈ C1  c (U), j ≥ 1, that converges in L1(U) and almost everywhere to  (−∇u,1)  √  1+|∇u|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Fur-  thermore,  (−∇u,1)  √  1+|∇u|2 is a unit vector so we may assume that �n+1  i=1 (φj  i)2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Since  ��Φj · (−∇u, 1)  �� ≤ |Φj|  �  1 + |∇u|2 ≤  �  1 + |∇u|2 ∈ L1(U)  we can use the dominated convergence theorem to get  lim  j→∞  ˆ  Ω  �  u  n  �  i=1  ∂xiφj  i + φj  n+1  �  dx = lim  j→∞  ˆ  U  (−∇u, 1) · Φj dx =  ˆ  U  �  1 + |∇u|2 dx  and the supremum is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Second, we have that u ∈ BV(U) and there exists {uk}k≥1 ⊂ BV(U) ∩ C∞(U) such that  uk → u in L1(Ω) and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='6)  lim  k→∞ D(uk)(U) = D(u)(U),  see [2, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Since, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='5), the conclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4) is trivial for C1 functions, we have  |U| ≤ lim  k→∞ D(uk)(U) = D(u)(U)  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  □  From now on, we fix a bounded, C1,1 domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We consider the minimization problem  min  (u,H)∈A I[u, H]  where  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='7)  I[u, H] :=  ˆ  Ω  |∇H|2 dD(u)  and dD(u) stands for the area measure defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The admissible set A is defined as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let h ∈ W 2,2(Ω) ∩ Lip(∂Ω) satisfying  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='8)  |h(y)| ≤ n − 1  n  Λ(y), y ∈ ∂Ω,  and  max  ∂Ω |h| ≤ (1 − ε0)  �|B1|  |Ω|  �1/n  ,  where Λ(y) is the weak mean curvature of ∂Ω at y ∈ ∂Ω and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='9)  n − 1  n  < ε0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Define  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='10)  A :=  � (u, H) ∈ BV(Ω) × (Ln(Ω) ∩ W 2,2(Ω)) : u solves (WPMC)  and ∥H∥Ln(Ω) + ∥H∥W 2,2(Ω) ≤ C0, H = h on ∂Ω  �  with C0 > 0 is to be appropriately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The equality H = h is understood in the sense of  traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The condition H ∈ W 2,2(Ω) is certainly natural for applications to the design  of fair G2-continuous surfaces in CAD/CAM/CAGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Indeed, in dimensions n = 1, 2, 3, the  Sobolev embedding gives that the curvature H is H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  The main result of this section is the following:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3 (Existence of weak solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let I be defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='7), h ∈ W 2,2(Ω)∩Lip(∂Ω)  satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='8) and ε0 ∈ (0, 1) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Then the set of admissible functions A in  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='10) is nonempty and there exists a minimizer of I within the class A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  14  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' VIVAS  To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3 we recall the notion and properties of Γ−convergence in our context,  referring the reader to [1] for an introduction to the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Let Jk, k ≥ 1, and J∞ be  functionals defined on the common space BV(Ω) and taking values in [−∞, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The sequence  {Jk}k≥1 is said to Γ−converge to J∞ if the following two conditions hold:  (a) For every v ∈ BV(Ω) and every sequence {vk}k≥1 ⊂ BV(Ω) such that vk → v in BV(Ω)  it holds  lim inf  k→∞ Jk(vk) ≥ J∞(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  (b) For every v ∈ BV(Ω) there exists a sequence {vk}k≥1 ⊂ BV(Ω) such that vk → v in  BV(Ω) for which  lim sup  k→∞  Jk(vk) ≤ J∞(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  We will use the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4 (see [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let (X, d) be a metric space and let {fk}k≥1 be an  equi-mildly coercive sequence of functions on X that Γ−converges to f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Then, there exits  min  X f∞ = lim  k→∞ inf  X fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Moreover, if {xk}k≥1 ⊂ X is a precompact sequence such that  lim  k→∞ fk(xk) = lim  k→∞ inf  X fk  then every limit of {xk}k≥1 is a minimum point for f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Here f is said to be mildly coercive if there exists a nonempty compact set K ⊂ X such  that infK f = infX f, and equi-mild coercivity means that the set K is the same for the whole  sequence {fk}k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' The proof is divided into 4 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We can extend h to Ω by solving  �  ∆H = 0  in Ω  H = h  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  By classical elliptic regularity, H ∈ W 2,2(Ω) and  ∥H∥W 2,2(Ω) ≤ C0  where C0 = C0(∂Ω, ∥h∥L∞(∂Ω)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Moreover, by the H¨older and isoperimetric inequalities,  ����  ˆ  A  H dx  ���� ≤ ∥H∥Ln(Ω)|A|  n−1  n  ≤ ∥H∥Ln(Ω)  P(∂A)  n|B1|1/n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  By the maximum principle and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='8) we have  ∥H∥Ln(Ω) ≤ |Ω|1/n max  ∂Ω |h| ≤ (1 − ε0)|B1|1/n,  where we make C0 larger if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Therefore,  ����  ˆ  A  H dx  ���� ≤ (1 − ε0)  n  P(∂A)  and (WPMC) is solvable for this H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let u ∈ BV(Ω) be the corresponding minimizer of J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  We have that A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We further point out that  ´  Ω D(u) < ∞ and H ∈ Lip(Ω) so that  ˆ  Ω  |∇H|2 dD(u) ≤ ∥∇H∥2  L∞(Ω)D(u)(Ω) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  SURFACES OF MINIMUM CURVATURE VARIATION  15  In particular,  0 ≤  inf  (u,H)∈A I[u, H] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Construction of a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let {(uk, Hk)}k≥1 ⊂ A be a minimizing sequence:  m :=  inf  (u,H)∈A I[u, H] = lim  k→∞ I[uk, Hk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  To get a convergent subsequence of {uk}k≥1 we show its uniform boundedness in BV(Ω) and  use that BV(Ω) embedds compactly in L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Since every uk is a minimizer of the functional  Jk defined by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='11)  Jk[v] :=  ˆ  Ω  D(v) +  ˆ  Ω  nHkv dx +  ˆ  ∂Ω  |v − g| dS  we have that, for any u0 ∈ BV(Ω),  ˆ  Ω  D(uk) +  ˆ  Ω  nHkuk dx +  ˆ  ∂Ω  |uk − g| dS ≤  ˆ  Ω  D(u0) +  ˆ  Ω  nHku0 dx +  ˆ  ∂Ω  |u0 − g| dS  from where  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='12)  ˆ  Ω  D(uk) +  ˆ  Ω  nHkuk dx ≤ C +  ˆ  Ω  nHku0 dx  for C > 0 independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Reasoning as in [3, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4)] we have that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='13)  ˆ  Ω  Hkuk dx ≥ −(1 − ε0)  ˆ  Ω  |∇uk| − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Furthermore, BV(Ω) ⊂ L  n  n−1 (Ω) so (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='13) and the H¨older inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='12) give  ˆ  Ω  D(uk) ≤ −n  ˆ  Ω  Hkuk dx + C +  ˆ  Ω  nHku0 dx  ≤ n(1 − ε0)  ˆ  Ω  |∇uk| + n∥Hk∥Ln(Ω)∥u0∥L  n  n−1 (Ω) + C  for a new constant C > 0 that is independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Moreover, the uniform bound on the  Ln(Ω) norm of {Hk}k≥1 (they all belong to A) gives  ˆ  Ω  |∇uk| ≤ n(1 − ε0)  ˆ  Ω  |∇uk| + nC0∥u0∥L  n  n−1 (Ω) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Thus, after rearranging terms and recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='9),  ˆ  Ω  |∇uk| ≤  1  (1 − n(1 − ε0))  �  nC0∥u0∥L  n  n−1 (Ω) + C  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Hence, by compactness in BV(Ω), there exists a subsequence of {uk}k≥1, still denoted by the  same indexes, and u∞ ∈ BV(Ω) such that  uk → u∞ in L1(Ω) as k → ∞, and  |∇u∞|(Ω) ≤ lim inf  k→∞ |∇uk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Note that we also have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='14)  D(u∞) ≤ lim inf  k→∞ D(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  By Poincar´e’s inequality and the Rellich–Kondrachov compactness theorem, there exist a  subsequence of {Hk}k≥1, still denoted by the same indexes, and H∞ ∈ W 2,2(Ω) such that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='15)  ∇Hk → ∇H∞ in L2(Ω), as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  16  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' CAFFARELLI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' STINGA, AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' VIVAS  Further, due to the uniform bound on ∥Hk∥Ln(Ω), we may assume that Hk converges weakly  in Ln(Ω) to H∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Finally, the weak convergence ensures that  ∥H∞∥Ln(Ω) + ∥H∞∥W 2,2(Ω) ≤ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' (u∞, H∞) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' For this step we use Γ−convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Recall the functionals Jk  defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='11) (for the subsequence Hk we found in Step 2) and define J∞ analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  We want to show that u∞ is a solution of (WPMC), namely, that u∞ is a minimizer of J∞  over BV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Let us show that {Jk}k≥1 Γ−converges to J∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' A first remark is that it is  enough to prove the Γ−convergence of  �  Jk(v) :=  ˆ  Ω  vHk dx  to  �  J∞(v) :=  ˆ  Ω  vH∞ dx  since the other two terms do not depend on k and can be considered as continuous pertur-  bations of Jk, see [1, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' To prove the liminf inequality (a), let {vk}k≥1 ⊂ BV(Ω)  and v ∈ BV(Ω) such that vk → v in BV(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We write  ˆ  Ω  vkHk dx −  ˆ  Ω  vH∞ dx = Ik + IIk + IIIk  with  Ik =  ˆ  Ω  (vk − v)H∞ dx  IIk =  ˆ  Ω  (vk − v)(Hk − H∞) dx  IIIk =  ˆ  Ω  v(Hk − H∞) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  By lower semicontinuity [4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1],  lim inf  k→∞ Ik ≥ 0  Next, we bound  |IIk| ≤ ∥vk − v∥L  n  n−1 (Ω)  �  ∥Hk∥Ln(Ω) + ∥H∞∥Ln(Ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Since vk converge to v in BV(Ω), by the isoperimetric embedding, the convergence also holds  in L  n  n−1 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' This and the uniform bound of Hk in Ln(Ω) give  lim  k→∞ IIk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Finally, limk→∞ IIIk = 0 by the weak convergence of Hk to H∞ in Ln(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' As for the limsup  inequality (b), given any v ∈ BV(Ω), consider the constant sequence vk = v for all k ≥ 1 and  notice that, using the weak convergence of Hk to H∞ in Ln(Ω), we have that  lim  k→∞  �  Jk(vk) = �  J∞(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Hence, {Jk}k≥1 converges to J∞ in the Γ sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Therefore, we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4 with  X = BV(Ω), fk = Jk, f∞ = J∞ and {xk}k≥1 and x∞ given by {uk}k≥1 and u∞, respectively  (note that the sequence {Jk}k is equi-mildly coercive), to conclude that u∞ is a minimizer  of J∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We have thus shown that (u∞, H∞) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  SURFACES OF MINIMUM CURVATURE VARIATION  17  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' (u∞, H∞) is a minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Recall that L(p) = 1  2|p|2, p ∈ Rn, is convex, that is,  1  2|p|2 ≥ 1  2|p0|2 + p0 · (p − p0)  for every p, p0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Then we can write  1  2  ˆ  Ω  |∇Hk|2 dD(uk) ≥ 1  2  ˆ  Ω  |∇H∞|2 dD(uk)  +  ˆ  Ω  ∇H∞ · (∇Hk − ∇H∞) dD(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  As k → ∞, the left hand side of this inequality converges to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' As for the right hand side,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='6) implies that  lim inf  k→∞  1  2  ˆ  Ω  |∇H∞|2 dD(uk) ≥ 1  2  ˆ  Ω  |∇H∞|2 dD(u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  It remains to analyze the second term on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' For this, notice that Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='1 implies that dD(uk) is absolutely continuous with respect to the Lebesgue measure, see  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' This and H¨older’s inequality give  ����  ˆ  Ω  ∇H∞ · (∇Hk − ∇H∞) dD(uk)  ���� ≤  ˆ  Ω  |∇H∞||∇Hk − ∇H∞| dD(uk)  ≤ C  ˆ  Ω  |∇H∞||∇Hk − ∇H∞| dx  ≤ C∥∇H∞∥L2(Ω)∥∇Hk − ∇H∞∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='15), this term goes to 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' We have shown that  m ≥  ˆ  Ω  |∇H∞|2 dD(u∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Since (u∞, H∞) ∈ A equality must be attained and (u∞, H∞) is a minimizer, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  □  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Braides, Γ-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications 22,  Oxford University Press, Oxford, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Evans and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, Text-  books in Mathematics, CRC Press, Boca Raton, FL, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Giaquinta, On the Dirichlet problem for surfaces of prescribed mean curvature, Manuscripta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  12 (1974), 73–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content=' Giusti, Boundary value problems for non-parametric surfaces of prescribed mean curvature, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
+page_content='  Scuola Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
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+page_content=' Moreton and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
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+page_content='  Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, TX  78712, United States of America  Email address: caffarel@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
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+page_content='edu  Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, United  States of America  Email address: stinga@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfSPd1/content/2301.00082v1.pdf'}
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+page_content='MNRAS 000, 1–15 (2022)  Preprint 6 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='5 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  Rongjun Huang 黄钅容钧 ID 1,2★, Andrew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Battisti  ID 1,2†, Kathryn Grasha  ID 1,2,5‡, Elisabete da Cunha  ID 2,3,  Claudia del P Lagos ID 2,3, Sarah K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Leslie ID 2,4 and Emily Wisnioski ID 1,2  1Research School of Astronomy and Astrophysics, Australian National University, Cotter Road, Weston Creek, ACT 2611, Australia  2ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia  3International Centre for Radio Astronomy Research, University of Western Australia, 35 Stirling Hwy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Crawley, WA 6009, Australia  4Leiden Observatory, Leiden University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Box 9513, NL-2300 RA Leiden, The Netherlands  5Visiting Fellow, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA  Accepted 2023 January 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Received 2022 December 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' in original form 2022 October 14  ABSTRACT  Previous studies have shown that the normalization and scatter of the galaxy ‘main sequence’ (MS), the relation between star  formation rate (SFR) and stellar mass (𝑀∗), evolves over cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' However, such studies often rely on photometric redshifts  and/or only rest-frame UV to near-IR data, which may underestimate the SFR and 𝑀∗ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We use MAGPHYS+photo-z  to fit the UV to radio spectral energy distributions of 12,380 galaxies in the COSMOS field at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 and self-consistently  include photometric redshift uncertainties on the derived SFR and 𝑀∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We quantify the effect on the observed MS scatter from  (1) photometric redshift uncertainties (which are minor) and (2) fitting only rest-frame ultraviolet to near-infrared observations  (which are severe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' At fixed redshift and 𝑀∗, we find that the intrinsic MS scatter for our sample of galaxies is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='6 times  larger than the measurement uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The average intrinsic MS scatter has decreased by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='5 to ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' At  low-𝑧, the trend between the intrinsic MS scatter and 𝑀∗ follows a functional form similar to an inverse stellar mass-halo mass  relation (SMHM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 𝑀∗/𝑀halo vs 𝑀∗), with a minimum in intrinsic MS scatter at log(𝑀∗/𝑀⊙) ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 and larger scatter at both  lower and higher 𝑀∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' while this distribution becomes flatter for high-𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The SMHM is thought to be a consequence of feedback  effects and this similarity may suggest a link between galaxy feedback and the intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' These results favor a slight  evolution in the intrinsic MS scatter with both redshift and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Key words: methods: observational, galaxies: evolution, galaxies: general, galaxies: star formation  1 INTRODUCTION  The galaxy main sequence (MS) describes the empirical relation  between the star formation rate (SFR) of galaxies and their stellar  masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Noeske et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Renzini & Peng 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Barro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' These studies find that galaxies  have higher SFR with increasing redshifts at a fixed stellar mass  (𝑀∗) in the earlier universe, and more massive galaxies have higher  SFRs at a fixed redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Some of these studies show a flattening or  turnover in the relationship at high masses (log(𝑀∗/𝑀⊙) > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5)  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2021) and  suggest that this turnover is driven by the quenching of star formation  due to feedback processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  The galaxy MS is a powerful tool for understanding and constrain-  ing the distribution and evolution of galaxies (Katsianis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Curtis-Lake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Popesso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  According to theories of galaxy feedback, the existence of a rela-  tively tight MS is thought to be mainly driven by the dynamical  ★ E-mail: u6569836@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='au  † E-mail: andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='battisti@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='au  ‡ ARC DECRA Fellow  balance between inflows and outflows caused by self-regulated star-  formation and/or active galactic nuclei (AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Somerville & Davé  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Characterising this evolution is difficult because observations  only provide a single snapshot in time for each observed galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  However, the evolution in the scatter, slope, and normalisation in the  MS of large statistical samples of star-forming (SF) galaxies with  cosmic time provides an indirect way to study galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The  width (or scatter) of the MS at a single redshift is thought to reflect  the burstiness of the average star formation history (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Caplar & Tacchella  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Donnari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Katsianis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Matthee & Schaye  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Theories suggest that a small MS width (small scatter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=',  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 dex) is indicative of gradual, continuous star formation histo-  ries (SFHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In contrast, large MS widths (large scatter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4  dex) are indicative of more bursty, stochastic SFHs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Tacchella  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Sparre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  The question around whether the intrinsic MS scatter is constant  or evolving is actively debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Previous studies have found a time-  independent MS scatter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Noeske et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Ciesla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Pessa  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2021), while others suggest it evolves with redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Kur-  czynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Katsianis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Davies  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='01995v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='GA]  5 Jan 2023  2  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' As the width of MS is related to the SFH, the MS scatter  can provide useful constraints on the evolution of SF galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For  example, a larger burstiness or stochasticity in the SFH can lead to  an increase in MS scatter, and this may change over cosmic time  (Matthee & Schaye 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Improving our understanding of the galaxy MS and its scatter re-  quires using large samples of galaxies with accurate redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' How-  ever, it is observationally expensive to get spectroscopic redshifts for  every galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' A common solution is to instead use photometrically  derived redshifts (𝑧phot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Most previous studies of the galaxy MS have  relied on determining stellar masses and SFRs based on SED-fitting  at fixed photometric redshift 𝑧phot and ignore the uncertainty of the  𝑧phot (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Studies that do not account for zphot uncertainty will systematically  underestimate the uncertainties in all distance-dependent parameters  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', 𝑀∗ & SFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  In this study, we use MAGPHYS+photo-z (Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2019) to  study the intrinsic scatter of the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The improvement in using  MAGPHYS+photo-z is that it sets 𝑧phot as an unknown quantity and  finds its probability distribution (Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Hence, the un-  certainty in the 𝑧phot is incorporated into the overall uncertainty in the  derived physical properties of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This allows us to examine  how much of the scatter in the MS is driven by measurement uncer-  tainty as opposed to true intrinsic MS scatter or other measurement  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Simultaneously, MAGPHYS+photo-z also includes IR  information to resolve the effect of dust attenuation at UV-near-IR  wavelengths on the SED based on dust emission from mid-IR-radio,  which dramatically improves the accuracy of the derived properties,  particularly for SFRs (Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Therefore, the unique as-  pect of MAGPHYS+photo-z is that it uses broadband photometry to  predict the best-fitting properties in a self-consistent manner, which  helps to mitigate potential biases on the derived values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  This paper is organised as follows: Section 2 introduces the data  and methods used in this study, Section 3 summarises our results,  and Section 4 compares our results with some previous observa-  tional studies and simulations and Section 5 outlines our conclu-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Throughout this paper, the flat Lambda Cold Dark Matter  (ΛCDM) model is adopted by assuming the Hubble constant is 𝐻0 =  70km/s/Mpc and the mass density of the Universe is Ωm,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2 DATA AND METHODS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 COSMOS Sample  The multi-wavelength observations of galaxies used in this study  come from two catalogues: the COSMOS2020 catalogue (Weaver  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2022) and the COSMOS Super-deblended catalogue (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The COSMOS2020 catalogue contains photometric data for  ∼ 1 million sources in 13 filters from UV to near-IR (Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2022), and the COSMOS Super-deblended catalogue presents photo-  metric data for ∼ 200, 000 galaxies in 11 filters in the mid-IR, far-IR,  and radio (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We cross-match the galaxies’ ID and select  a subsample of galaxies with SEDs that are sampled well enough to  constrain their stellar mass and (dust-corrected) SFRs robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' To  achieve this, we use two criteria: (1) signal-to-noise ratio (𝑆/𝑁) of  𝑆/𝑁 > 3 in three or more UV–near-IR bands and (2) 𝑆/𝑁>3 in 2 or  more IR–radio bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' By virtue of criterion (2), all of the sources in  our sample have a match in the Super-deblended catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  It is important to note that criterion (2) roughly translates into  a cut in SFR such that only galaxies above a certain SFR will be  included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This SFR threshold increases as a function of increasing  redshift (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Additionally, AGNs are excluded based  on X-ray detections and IR & radio colour cuts (Seymour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Donley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This is done because  MAGPHYS+photo-z does not include AGN models, so the derived  properties are not accurate for these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Due to the limited  availability of data required for these AGN diagnostics, some AGNs  may not be identified and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Further details and references on  these cuts are in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3 of the Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' These selec-  tion criteria leave us with a photometric sample of 14,607 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  For later comparison (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1), only 3,873 of the whole 14,607  galaxies have spectroscopic redshifts (𝑧spec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2 Methods  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 MAGPHYS+photo-z  MAGPHYS fits the full SEDs of galaxies with known redshifts from  the ultraviolet to the radio (da Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2008, 2015) by com-  bining the emission from stellar populations with the attenua-  tion and re-emission of starlight by interstellar dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The recent  MAGPHYS+photo-z extension, described in Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2019), ex-  tends the code to fit the SEDs of galaxies with unknown redshifts, and  constrain the photometric redshift simultaneously with other galaxy  physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In practice, the code builds libraries of model  UV-to-radio SEDs at different redshifts and compares them with  the observed SEDs of galaxies, using a Bayesian method to obtain  the likelihood distributions of physical parameters such as redshifts,  stellar masses, and SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' There are two sets of libraries used in  MAGPHYS+photo-z: (1) an optical library that describes emissions  from stars, and (2) an infrared library that describes the emission  from dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The optical library uses the spectral population synthesis  models of (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Bruzual & Charlot 2003) and initial mass function  from (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Chabrier 2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' while the infrared library consists of  models for PAHs and hot dust emitting in the mid-IR, and warm and  cold dust components in thermal equilibrium that emit in the far-IR  to submillimeter (da Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' These two sets of model  libraries maintain the balance of the energy absorbed by dust (via  attenuation in UV to near-IR) and the energy re-emitted by dust (via  thermal emission in mid-IR to sub-mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Due to insufficient models  at 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 to compare to based on the redshift prior that is adopted in  the MAGPHYS+photo-z, galaxies with 𝑧phot < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 are not constrained  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Therefore, we exclude galaxies with 𝑧phot < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 after running  the code (Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' An example MAGPHYS+photo-z fit  is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We compare the distributions of photometric  redshifts of our sample with those of the full COSMOS2020 sample  in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  We fit the SEDs of 14,607 galaxies with MAGPHYS+photo-z to  determine the M*, SFR, 𝑧phot and respective errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We use the 𝜒2  value of the best-fit model from MAGPHYS+photo-z as an indicator  of the goodness of fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We fit the 𝜒2 distribution with a lognormal  function (see Figure 3) and convert the lognormal parameters 𝜇 and  𝜎 to the geometric parameters 𝜇geo and 𝜎geo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Finally, we perform  a 2𝜎 confidence cut (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', 𝜒2 < 𝜇geo + 2𝜎geo) to the histogram and  remove the high-𝜒2 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The remaining galaxies are reduced to  13,639, and their 𝜒2 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  However, some galaxies have problematic SEDs due to inconsis-  tencies in fluxes and/or upper limits between bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In these cases,  MAGPHYS+photo-z derives large uncertainties of 𝑧phot and distance-  dependent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In addition, some cases have multiple redshift  solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', degeneracies in Lyman vs Balmer break position),  which can lead to multi-peaked solutions for distance-dependent de-  rived properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We adopt the following selection criteria based on  MNRAS 000, 1–15 (2022)  Intrinsic MS Scatter at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Output of MAGPHYS+photo-z for one of the galaxies of our sample, COSMOS2020 ID 1587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The upper panel shows the best-fit SED (black curve),  the observed data (red square) and the predicted unattenuated SED (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The black open circle on the SED fitting curve is the corresponding model  photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The goodness of fit is presented by 𝜒2 in the upper right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The lower panel shows the likelihood distribution of 10 basic physical parameters:  𝑧phot, stellar mass (log[𝑀∗/𝑀⊙]), log[SFR/(𝑀⊙yr−1)], specific-SFR (log[sSFR/yr]), dust luminosity (log[𝐿dust/𝐿⊙]), dust mass (log[𝑀dust/𝑀⊙]), mass-  weighted stellar age (log[𝐴𝑔𝑒𝑚/yr]), V-band dust attenuation (𝐴𝑉 /mag), 2175 𝐴 bump strength (𝐸′  𝑏) and the effective dust temperature (𝑇𝑑𝑢𝑠𝑡/K) (Battisti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  key parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', 𝑧phot, 𝑀∗ and SFR):  ����  ����  𝜎(𝑧phot)  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25  𝜎(log(𝑀∗/𝑀⊙))  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3  𝜎(log(SFR/𝑀⊙yr−1))  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3,  where 𝜎(𝑧phot), 𝜎(log(𝑀∗/𝑀⊙)) and 𝜎(log(SFR/𝑀⊙yr−1)) are  measurement uncertainties for 𝑧phot, 𝑀∗ and SFR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The  measurement uncertainty is calculated by half of difference between  upper and lower 1𝜎 (68%) boundary of probability distribution func-  tion (PDF) for each parameters derived by MAGPHYS+photo-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We  restrict the measurement uncertainty on redshift based on the size of  our adopted redshift bins of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 dex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', 2 times larger than the un-  certainty boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The limits of uncertainties on 𝑀∗ and SFR are  set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3 dex (roughly a factor of 2) because we want the measure-  ment uncertainties to be lower than the typical intrinsic MS scatter,  which is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3 dex (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Ciesla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Speagle  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' These cuts remove 201 galaxies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5%) from our sample  and we are left with 13, 418 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2 Reference Main Sequence Relation  Numerous studies have examined the nature of the galaxy MS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=',  Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Tomczak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Pearson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Bisigello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Thorne  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Tomczak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2016) and Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2020) introduce  nonlinear fits to the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For our reference MS relation, we adopt  Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2020) which also used galaxies in the COSMOS field,  which has the form:  log(SFR(𝑀, 𝑡)/𝑀⊙yr−1) = 𝑆0 − 𝑎1𝑡 − log  �  1 + 10𝑀′  𝑡  10𝑀  �  𝑀′  𝑡 = 𝑀0 + 𝑎2𝑡,  (1)  MNRAS 000, 1–15 (2022)  t  ID:1587  Zfit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='81  12  x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='74  (7/77)60]  1f  Resid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  1  0  100  101  102  EOT  104  10  A/μm [obs-frame]  1DO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='86  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='56  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='12  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='44+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='50  kel  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='DO  0  2  4  12  2  12  i0  8  12  Zotat  lo-g[MMα ]  lo-g[SFR/(Mα yr-1]]  log[sSFR/yr-1]  og[LdrtfL  ]  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='07  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='50  ke  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0O  8  1  6  8  5  115  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content="5  152  40  8  log[Mdr/M a]  lo-g[Ageur]  Avfmag  Eb'  Taust/K4  R." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Normalized 𝑧phot histogram for the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The red histogram in-  dicates the distribution of photo-z’s from MAGPHYS+photo-z for the 14,607  galaxies in our sample, while the blue histogram represents the parent distri-  bution of the whole 964,506 galaxies from COSMOS2020 catalogue (Weaver  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2022), where the 𝑧phot are derived using LePhare (Arnouts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Ilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We show the corresponding look-back time 𝑡lb on the top  axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Distribution of MAGPHYS+photo-z fit 𝜒2 of our 14,607 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  The black curve represents the normalized lognormal distribution function  fitting to the 𝜒2 histogram, while the vertical black dashed line indicates the  normal 2𝜎 confidence cut within 𝜒2 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  where 𝑆0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='08  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='09, 𝑀0 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='15  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='16, 𝑎1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='22+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='01  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='01, 𝑎2 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='02, 𝑀 is log(𝑀∗/𝑀⊙) and 𝑡 is the age of the universe in  Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2020) separate their sample into two classes, ‘All’  and ‘SF’ (‘Star-forming’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We adopt the ‘SF’ relation, which should  coincide more closely with the sample used in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The Leslie  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2020) ‘SF’ sample applies a colour selection (NUV-r-J cut)  that will exclude ‘passive’ galaxies with low SFRs, which has a  similar role as our selection criterion described in next paragraph  (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The probed steller mass range in Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  (2020) is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 ≲ 𝑀∗ ≲ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 and redshift range is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3 < 𝑧 < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  They use radio data to derive SFRs, which provides a dust-unbiased  measurement of the SFR (Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Although the goal of this study is not to investigate the relation  between SFR and 𝑀∗, we note that the exact functional form of  the MS is still under debate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', Katsianis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Leja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Different methods of estimating SFRs are thought to be the  primary reason for differences between studies (Katsianis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Hence, despite using similar catalogues from the COSMOS field as  Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2020) that use radio continuum for robust SFRs (dust-  insensitive), there are some other systematic problems that can arise,  such as priors, metallicities, timescales, stellar masses, ages, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We  stress that the reference MS we show is intended only to guide the  eye and we do not use it for any selection cuts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', to define ‘on’ vs  ‘off’ the MS), which instead are based on sSFR (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Therefore, the choice of the reference MS has no impact on the results  of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3 sSFR Selection  We also adopt a specific-SFR (sSFR = SFR/𝑀∗) cut to eliminate  quenched galaxies (see the comparison to U-V-J selection in Ap-  pendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' These ‘passive’ galaxies form stars at a much lower rate  for a given stellar mass compared to SF galaxies (Renzini & Peng  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' By definition, quenched galaxies have low sSFR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The  purpose of the sSFR cut is to remove these red galaxies to avoid  overestimating the MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Since the sSFR of SF galaxies evolves  with cosmic time (Madau & Dickinson 2014), we adopt a redshift-  dependent cut1 in this selection criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In Figure 4, we use a linear  regression model fit to the median-3𝜎 values for sSFR bins (24 bins)  vs 𝑧phot and remove quenched galaxies, which we define as 3 − 𝜎  outliers lying below the equation:  log(sSFR/yr−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='57𝑧phot − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  (2)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 Influence of IR-selection on SFR- and mass-completeness  A galaxy’s SFR scales with the IR luminosity (𝐿IR) (Kennicutt &  Evans 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Due to this, the IR-selection criteria in our sample  only includes galaxies above a certain SFR (depending on redshift),  introducing an SFR bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 𝐿IR in this paper represents the integrated  dust emission from both dust components in MAGPHYS+photo-z  over all wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' As the luminosity distance (𝐷lum) increases,  the lowest SFR of the SF galaxies we can observe will increase  correspondingly2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Hence, the functional form for IR-selection in SFR  is similar to the relationship between luminosity and redshift:  log(SFRIR/𝑀⊙yr−1) ∝ log(𝐿IR)  = log(4𝜋𝐹IR𝐷2  lum)  = log(𝛼𝐷2  lum),  (3)  where 𝛼 is a constant factor determined by the data and 𝐷lum is  the luminosity distance in units of Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' By converting 𝐷lum to  𝑧phot and applying Equation 3 to log(SFR) 𝑣𝑠 𝑧phot, we obtain an  empirical estimate of our SFR limit with redshift based on the 1𝜎  lower boundary of our population, and the constant factor of the  function 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='50 × 10−7 corresponds to the lower boundary of the  68% (1𝜎) population enclosed curve (see Figure 5):  log(SFRIR/𝑀⊙yr−1) = log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='50 × 10−7(𝐷lum/Mpc)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  (4)  1 We explored adopting a constant cut at log(sSFR/yr−1) = −11, which  increases our sample by ∼100 galaxies, but this has a very small difference on  our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We suspect that this phenomenon could be driven by IR-selection,  which is described in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2 This excludes the negative-k correction effect  MNRAS 000, 1–15 (2022)  Look-back time tb (Gyr)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='71  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='24  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='95  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='31  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='55  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='71  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='83  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='8  Samples in this study  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7  Full COSMOS2020 catalog  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='6  Number density  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  2  5  1  3  4  6  0  7  8  Zphot400 μgeo=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='43, Ogeo=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='83  350  MAGPHYS x² histogram  300 -  250  nt  no  200  150  100  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='76  50  0  4  6  0  8  10  2Intrinsic MS Scatter at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  5  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' log(sSFR) vs z enclosed contour plot for the 13,071 selection  galaxies within 2𝜎 − 𝜒2 from redshifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The colours ranging  from blue to yellow indicate the increasing number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The number of  galaxies inside the enclosed blue and green curve is 68% (1𝜎) and 95%  (2𝜎) of the total population, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The magenta points indicate the  median value of log(sSFR/yr−1), while the red points are the median-3𝜎  values for 24 sSFR bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The red line is the sSFR cut adopted in this study,  and there are 64 galaxies identified as quenched galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Due to the mini-  mum timescale of star formation in MAGPHYS+photo-z, there is a maximum  value of log(sSFR/yr−1) ∼ −8, corresponding to the adopted SFR timescale  (100Myr), showing as a horizontal boundary in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' log(SFR) vs z enclosed contour plot for the 13,071 selection  galaxies within 𝜒2 selection from redshifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The equation 4 is  plotted as the red curve in this diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Due to the detection limits, we cannot trust our ability to detect  galaxies that are below Equation 4 in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This SFR incom-  pleteness translates to an incompleteness on stellar mass (via galaxy  MS relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We infer the corresponding mass-completeness thresh-  old at each redshift using the Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2020) MS relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For  subsequent analysis, we will refer to samples above and below this  threshold as our mass-complete and mass-incomplete samples, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  3 RESULTS AND ANALYSIS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 The Role of Redshift uncertainty and IR data on the  Measured Scatter of the MS  In this study, we use MAGPHYS+photo-z to constrain the stellar  masses and SFRs of our galaxies because it uses the full wave-  length range from UV to radio, and it constrains the photometric  redshifts jointly with the other physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Using the full  SED provides the tightest possible constraints on 𝑀∗ and SFR, thus  minimizing the main sequence scatter that is due to errors on these  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Obtaining the photometric redshift at the same time al-  lows us to fold in the redshift error into the errors on 𝑀∗ and SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  This improves our ability to quantify the ‘observational’ scatter on  the main sequence and, in turn, characterise its intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  In this section, we test the accuracy of MAGPHYS+photo-z and quan-  tify the influence that the redshift precision and inclusion of IR data  have on derived physical properties (for the COSMOS filter set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 Accuracy of 𝑧phot relative to 𝑧spec  To examine the 𝑧phot accuracy of MAGPHYS+photo-z, we use the  latest COSMOS master spectroscopic catalogue (curated by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Sal-  vato for internal use within the COSMOS collaboration), which is  the same dataset used to originally test the code (Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  There are spectroscopic redshifts, 𝑧spec, for 3,873 out of the 14,607  galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' After applying the 𝜒2 cut, we obtain 3,724  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Here we adopt some metrics defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 of  Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2019) to estimate the accuracy of 𝑧phot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We find  𝜎NMAD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='086, 𝜂 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2% and 𝑧bias = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='002, where 𝜎NMAD  (normalized median absolute deviation) is known as the precision or  scatter of the data, 𝜂 characterises the fraction of catastrophic failures  and 𝑧bias represents the accuracy of the redshift (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' systematic devi-  ation or bias).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The value of 𝑧bias is much smaller than 𝜎NMAD, and  hence we constrain the redshifts very well with the multiple UV to  radio bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Since we use a similar database as the one used in Bat-  tisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2019), the results of 𝜎NMAD, 𝜂 and 𝑧bias should be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  As a comparison, these values calculated in Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2019) are  𝜎NMAD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='032, 𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='037, 𝑧bias = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='004 for the COSMOS2015  samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  The upper panels of Figure 6 is a demonstration of 𝑧phot accuracy  of MAGPHYS+photo-z, which also shows a comparison between the  𝑀∗ and SFR derived from 𝑧phot and 𝑧spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The median values of  differences for 𝑀∗ and SFR are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='00 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='05 dex, respectively,  reflecting that MAGPHYS+photo-z does not affect the overall mea-  surement of 𝑀∗ and SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Therefore, we do not expect that relying  on 𝑧phot will introduce significant bias or dominate the uncertainty  of the other derived properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2 Uncertainties of 𝑧phot, 𝑀∗ and SFR  We characterise the measurement uncertainties of 𝑧phot, 𝑀∗ and SFR  for our sample of 13,418 galaxies in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In our analysis, the  data are separated into five bins of 𝑀∗ with a width of ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 dex at a  specific redshift epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The uncertainty in 𝑀∗ in each bin is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='06  dex, while the 𝑧phot uncertainty is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Both uncertainties are ∼ 10  times smaller than the bin size in this study (≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 dex for log(𝑀∗)  bin and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 for redshift epoch), therefore we do not anticipate  that these uncertainties will have a substantial impact on the derived  intrinsic scatter on the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The median value of SFR’s uncertainty  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='08 dex, which is comparable to the scatter of galaxies on the  MS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2 dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Thus, when measuring  MNRAS 000, 1–15 (2022)  log(sSFR/yr=1)  10  68%enclosedcurve  95%enclosedcurve  sSFR cut: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='57Zphot - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='60  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  Zphot3  2  log(SFR/Moyr-1)  0  IRlimit:log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='50e-07*D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  um  68%enclosedcurve  95%enclosedcurve  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  Zphot6  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  the intrinsic scatter of MS, we need to consider SFR’s uncertainty as  the component of the scatter in MS and remove it properly to obtain  the intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Our method for removing this component is  described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3 Contribution of including IR data to the Uncertainties of 𝑀∗  and SFR  IR wavelengths probe dust emission and provide information re-  garding the amount of dust-obscured star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' By excluding  IR observations from the SED fits, we can determine the impact  of these bands on the uncertainties of 𝑀∗ and SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We rerun the  MAGPHYS+photo-z without fitting the observational data for filters  at wavelengths longer than IRAC2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5um) for the same 14,607 galax-  ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' After rejecting the cases with bad fits (𝜒2 > 2𝜎), we compare  the uncertainties of 𝑧phot, 𝑀∗ and SFR derived from UV to near-IR  photo-z fitting to those results from fitting the full available SED in  the top panels of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' As expected (Battisti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2022), the  non-IR fits tend to come with larger measurement uncertainties be-  cause fewer observations are available to constrain the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For  𝑧phot and 𝑀∗, including the IR bands only leads to a relatively small  improvement (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', decrease) in the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In contrast, for SFR,  the median uncertainty when IR bands are not included is nearly  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 times larger than that with the IR bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This is because the IR  bands are important to distinguish the amount of dust-obscured star  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' It is harder to accurately measure the intrinsic scatter of  the MS with the larger measurement error in SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Therefore, by  restricting the sample to sources where SED fitting can be performed  that include IR filters, we significantly reduce the amount of scatter  of the MS arising from measurement uncertainty to accurately con-  strain the intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The lower panels of Figure 8 show the  difference in the values of 𝑧phot, 𝑀∗ and SFR with and without the  IR bands included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' It can be seen that the median of the difference  remains close to zero as a function of each property suggesting that  there is minimal bias occurring as a result of the MAGPHYS+photo-z  priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2 Measuring the intrinsic MS scatter  We  divide  our  sample  into  6  redshift  bins:  𝑧phot  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0, with widths of Δ𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 except for  the lowest bin, which spans 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75 due to the limitation on the red-  shift prior for MAGPHYS+photo-z (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' At each redshift,  we further divide the sample into 5 stellar mass bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We determine  the log(SFR) dispersion (standard deviation) of the galaxies within  each bin relative to the median log(SFR) measurement uncertain-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We set the following five bins for all selected galaxies according  to their mass: log(𝑀∗) < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 log(𝑀⊙), 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 log(𝑀⊙), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 log(𝑀⊙), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 log(𝑀⊙) and > 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 log(𝑀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The  values adopted for each bin is the median log(SFR) and 𝑀∗ of  each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We characterise the intrinsic MS scatter in the range of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 < 𝑧phot < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The upper boundary (𝑧phot = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25)  corresponds to where our sample size dramatically decreases such  that we do not have enough sources to properly characterize the  MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Then we take the further selections of 2𝜎-𝜒2 and sSFR  cut (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3) to reduce the effects of quiescent  galaxies on the intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  For each bin, we assume that any excess in the SFR disper-  sion relative to the median measurement uncertainty in SFR from  MAGPHYS+photo-z is due to the intrinsic scatter of the MS relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  We determine the intrinsic MS scatter by assuming the measured  scatter is a result of the measurement uncertainty and intrinsic MS  scatter being added in quadrature, which can be rearranged as:  log(𝜎int/𝑀⊙yr−1)  =  √︃  (log(𝜎tot/𝑀⊙yr−1))2 − (log(𝜎meas/𝑀⊙yr−1))2,  (5)  where 𝜎int, 𝜎tot, and 𝜎meas are intrinsic MS scatter of the galaxies,  the observed standard deviations, and the median MAGPHYS+photo-z  uncertainty, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Figure 9 displays the galaxy MS for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 ≲  𝑧phot ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0, showing that the dispersion in SFR of the galaxies are  significantly larger than the measurement uncertainties (representa-  tive error bars in lower-right of each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In each interval of 𝑀∗,  the size of the SFR intrinsic MS scatter is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='15−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='39 dex larger than  the measurement uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The horizontal dashed lines in Figure  9 indicate the limit on SFR defined in Equation 4 and Figure 5 at the  median 𝑧phot for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The vertical dashed lines correspond to the  stellar mass at this SFR from the reference MS relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We define the  right half in each panel as the mass-complete area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' There are 12,380  galaxies (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='71%) in the mass-complete regions and 691 galaxies  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='29%) in the mass-incomplete regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We show the values of the  scatter terms for our mass-complete bins in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  We observe the following three trends (see the left panel of Figure  10 and Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' First, the median SFR measurement uncertainties  are always smaller than the intrinsic MS scatter of the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The  minimum difference between intrinsic MS scatter and uncertainties  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='13 dex in the range of 1010 − 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5𝑀⊙ at 𝑧phot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5, while the  maximum occurring at 𝑧phot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 for galaxies with log(𝑀∗/𝑀⊙) >  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='39 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Second, excluding the mass-incomplete regions, our  galaxies roughly follow the same observed main sequence as shown  in Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2020) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Equation 4), but with slightly lower SFRs  than the reference MS for most redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Third, the intrinsic  dispersion in SFR at a given mass tends to decrease as the 𝑧phot  increases at a fixed 𝑀∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  The size of the 𝑧phot interval we selected may affect the behaviour  of the SFR intrinsic MS scatter evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The width of each 𝑧phot in  our criteria is Δ𝑧phot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 except 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='35 at 𝑧phot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' However, with  the increase of 𝑧phot, the cosmic time corresponding to the Δ𝑧phot is  decreasing because the look-back time 𝑡lb does not linearly increase  with 𝑧phot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' As a result, this reducing length of the binning interval in  cosmic time with increasing 𝑧phot may affect the SFR intrinsic MS  scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' To examine this issue, we adopt look-back time (𝑡lb) instead of  redshift as a more consistent way to measure the intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  We convert the 𝑧phot into 𝑡lb and rearrange our sample of 13,071  galaxies in 6 𝑡lb bins (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr) with the equal  length of time (Δ𝑡lb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We reproduce the log(SFR)-log(𝑀∗)  plane in the right panel of Figure 10 and present the results in Table  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We find that the 𝑡lb results share the same trends and features  as the previous 𝑧phot version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' However, because binning the data in  equal 𝑡lb width removes the unequal-length effect when measuring  the intrinsic MS scatter, we will adopt this for our main analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  The left panel of Figure 11 shows the relationship between intrinsic  MS scatter and 𝑡lb for both the redshift and look-back time binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  In this study, we adopt the weighted linear regression to the intrinsic  MS scatter versus 𝑡lb:  log(𝜎int/𝑀⊙yr−1) = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='002)𝑡lb + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='432 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='015),  (6)  where 𝜎int is the intrinsic scatter of the MS, and these parameters  are calculated with mass-complete sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We observe a trend of  MNRAS 000, 1–15 (2022)  Intrinsic MS Scatter at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  7  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Upper left panel: Comparison of measurement uncertainties between default MAGPHYS high-𝑧 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e, fixed to 𝑧spec) and MAGPHYS+photoz runs for  the subsample of 3,724 𝜒2-selection galaxies with spectroscopic redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Upper right panel: redshift accuracy ((𝑧phot − 𝑧spec)/(1 + 𝑧spec)) as a function of  𝑧spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The redshift scatter (𝜎NMAD), catastrophic failure rate (𝜂), and redshift bias (median((𝑧phot − 𝑧spec)/(1 + 𝑧spec))) values are shown at the upper right  corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Lower panels: Difference in 𝑀∗ and SFR derived by 𝑧phot and 𝑧spec as a function of the 𝑧spec-derived values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The 2D histogram/scatterplot colors range  from blue to yellow with increasing number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The black line in each sub-diagram is the one-to-one relation as reference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' red, green and blue curves  enclose 68%, 95% and 99% populations of sample galaxies within 2𝜎-𝜒2 cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  decreasing intrinsic MS scatter up to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7Gyr (𝑧 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7) as 𝑡lb increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  The error bars are derived by bootstrap resampling the data in each  bin 100 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Bins with smaller sample sizes have larger bootstrap  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Although the descending rate of intrinsic MS scatter over  look-back time is shallow, the Spearman and Pearson correlation  coefficients (𝑟𝑠 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='943 and 𝑟 𝑝 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='956, respectively) indicate  a strong monotonic decreasing correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Conversely, with 𝑟𝑠 =  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='486 and 𝑟 𝑝 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='837, there is a weaker and tentative correlation  when using redshift binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This is due to the redshift binning  having a potential upturning feature at 𝑧phot ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We suggest this  is a consequence of unequal-length binning for redshift, which will  be discussed in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Furthermore, there is a ‘upturn’  feature, and the intrinsic MS scatter tends to increase after 𝑡lb ∼  10Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Given the uncertainty in our intrinsic MS scatter and our  limited sample size at high-𝑧, it is difficult to assess the significance  of this upturn with our current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  The intrinsic MS scatter may vary with the adopted Δ𝑡lb of each  bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The right panel of Figure 11 presents the effect of binning the  MNRAS 000, 1–15 (2022)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  68% enclosed curve  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  95% enclosed curve  99% enclosed curve  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  Zphot(MAGPHYS)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  0  1  2  3  4  ZspecONMAD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='086,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4  n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2%,  Zbias = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4  Zphot = Zspec ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1(1 + Zspec)  0  1  2  3  4  Zspec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  9  10  11  12  log(M* /Mo)zspec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  +-log(SFR/Moyr-  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  log(SFR/Moyr-  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  1  0  1  2  3  log(SFR/Moyr  speo8  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Distribution of the values and uncertainties for the key parameters of our study for the 13,418 galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Red, green and blue curves  enclose 68%, 95% and 99% populations of sample galaxies within 2𝜎-𝜒2 cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The median values of 𝑧phot, log(𝑀∗/𝑀⊙) and log(SFR/𝑀⊙yr−1) are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='20,  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='64 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='42, while the median values of uncertainties are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='06 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='08 dex, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Upper panels: Comparison between the uncertainties on 𝑧phot, 𝑀∗ and SFR for MAGPHYS+photo-z runs including IR bands (labeled ‘phot,IR’) and  without IR or radio bands (‘phot,nonIR’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Differing from the cases for 𝑧phot and 𝑀∗, the SFR’s uncertainty increases dramatically for SED fits without the IR  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The median uncertainty values derived from MAGPHYS+photo-z including the IR bands (distributed along x-axis) are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='06 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='08 dex for 𝑧phot,  𝑀∗ and SFR, while the median uncertainty values derived from non-IR runs (distributed along y-axis) are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='08 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 dex, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The black line  in each sub-diagram is the one-to-one relation as reference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' red, green and blue curves enclose 68%, 95% and 99% populations of sample galaxies within 2𝜎-𝜒2  cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Lower panels: Difference in measurements between the SED fits with and without IR bands included as a function of the measurements derived from fitting  including IR bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For the COSMOS2020 data, we find that the dominant factor in accurately constraining the scatter in the main sequence is whether the IR  bands are included to constrain the dust-obscured SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  data in different Δ𝑡lb widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The data are binned into 3, 6, 12, and  24 groups (no less than 100 galaxies in each bin) in 4 different sets  with equal 𝑡lb widths, where 6 is our fiducial number of bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' It can  be seen that the data in the 12 and 24 bins have a similar distribution  statistically relative to our fiducial binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We find that as the num-  ber of bins increases, the normalisation (intrinsic MS scatter) slightly  decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' However, the coefficients of the corresponding equations  tend to converge somewhere closely below the current linear re-  gression equation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' the yellow line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Even though the decreasing  binning time scale causes more severe fluctuation along the linear  regression line, the similarity and high absolute values of 𝑟𝑠 and 𝑟 𝑝  still demonstrate a strong correlation between intrinsic MS scatter  MNRAS 000, 1–15 (2022)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='0  Zphot0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='4  (log(M* /Mo))  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='0  8  9  10  11  12  log(M* /Mo)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='0  1  0  3  log(SFR*/Moyr  phot,IRIntrinsic MS Scatter at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  9  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 𝑀∗-SFR relation of our galaxies in 6 redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' All bins are shown through 13,071 selection galaxies with colours from purple to red to green  coding the range of 𝑧phot at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The horizontal dot-dashed line indicates the IR-selection completeness cut in SFR, and the vertical  dashed line indicates the corresponding IR-selection completeness cut in 𝑀∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The open symbols indicate the mass-incomplete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We drop the data points  where the numbers of galaxies within a bin are less than 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The longer error bars indicate the standard deviation of the SFR distribution in each stellar mass  bin, while the shorter black ones represent the median MAGPHYS measurement uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The gray error bar in the bottom right of each panel denotes the  median uncertainties on 𝑀∗ (along the x-axis) and SFR (along the y-axis) from MAGPHYS+photo-z for the entire redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The curves are the MS relations  at each redshift epoch from Leslie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The maximum value of log(sSFR/yr−1) ∼ −8 due to the adopted SFR timescale shows as an upper boundary  of the slope in each SFR-𝑀∗ panel, while the bottom right slope represents the sSFR cut at each redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  and 𝑡lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' On the other hand, this phenomenon also partially explains  why redshift binning is not a good approach in this study, especially  at high redshifts: a narrower binning time scale may lead to larger  fluctuations in the intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Since there is no significant  discrepancy in intrinsic MS scatter for 𝑛bin ≥ 6, we expect that these  coefficients in linear regression lines approach some values slightly  smaller than the current binning one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Hence, we conclude that the  binning does not strongly affect the trend of intrinsic MS scatter  evolution and we adopt 𝑛bin ≥ 6 as the current 𝑡lb binning number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  Leslie+20  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='0  mass-incomplete  ★  mass-complete  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='5  Z=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='0  log(M* /Mo)10  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The MS evolution and scatter for our sample with the same colour scheme adopted in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The shaded regions shown in the diagrams indicate the  range of the reference MS between the upper and lower redshift boundaries for each bin and do not relate to the intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The thick and thin error bars  indicate the observed standard deviation in SFR and the median SFR uncertainty from MAGPHYS+photo-z for each bin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The left panel shows the  sample binned in equal redshift bins, with the mass-incomplete bins shown as open symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The right panel is similar to the left but binned in equal look-back  time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Adopting equal-width redshift bins (left-panel) instead of look-back time (right-panel) may impact the measured intrinsic MS scatter because of the  evolution in the MS relation over the redshift range contained within a single bin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', differing widths in shaded region in left-panel relative to right-panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Hence, we adopt look-back time bins for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The left panel is intrinsic MS scatter vs 𝑧phot and 𝑡lb binning with a linear regression fit to the look-back time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The yellow star-like scatter  points are the 𝑡lb binning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We take the average of the data points in each mass bin and obtain error bars of intrinsic MS scatter by bootstrap resampling the  distribution 100 times based on varying the individual values by their uncertainties and rebinning them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The solid yellow line is the linear fit of intrinsic MS  scatter versus 𝑡lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The right panel display the effect of binning number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' A total of 12,380 galaxies are regrouped in 3, 6, 12 and 24 bins with red circles, yellow  star, green squares and blue diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In each panel, we show the Spearman and Pearson correlation coefficients (𝑟𝑠 and 𝑟𝑝) as well as the corresponding  p-values (𝑝𝑠, 𝑝𝑝) for different binning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  𝑀∗  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 Log(M⊙)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 Log(M⊙)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 Log(M⊙)  > 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 Log(M⊙)  𝑧phot  N  𝜎tot  𝜎meas  𝜎int  N  𝜎tot  𝜎meas  𝜎int  N  𝜎tot  𝜎meas  𝜎int  N  𝜎tot  𝜎meas  𝜎int  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  428  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='34  897  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='34  922  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content=' "N" is the number of galaxies, "𝜎tot" is the galaxy SFR dispersion (standard deviation), "𝜎meas" is the MAGPHYS uncertainty in SFR and "𝜎int" is  the intrinsic MS scatter in each 𝑀∗ bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The mass-incomplete data are marked as "-", but the number of galaxies in the binning interval are still shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Since  all the 𝑀∗ < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 log(𝑀⊙) galaxies lie in the mass-incomplete regime, they are not included in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  log(SFR/Moyr-1)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  MS at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  MS at z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  z: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  MS at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  mass-incomplete  z: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75 ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25  MS at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  mass-complete  z: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  MS at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  z: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='40 ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75  z: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75 ~ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25  MS at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  z: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75 ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  log(M * /Mo)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='9Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='1Gyr(z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='27  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Look-back time version of Table 1 by collecting and reassigning the 12,380 mass-complete data to 6 𝑡lb bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We exclude bins in which the number of  galaxies is less than 50 (> 1011𝑀⊙ at 𝑡lb = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr), together with the mass-incomplete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In general, the intrinsic MS scatter is still significantly larger than  measurement uncertainty in new binning criteria, which avoids the effect of nonlinear 𝑡lb widths in previous 𝑧phot binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  4 DISCUSSION  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 Comparison to the halo mass-stellar mass relation  The stochastic events that occurred throughout an SF galaxy’s history,  including as galaxy mergers and supernova & AGN feedback, are  assumed to be responsible for the inherent MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The amount  of burstiness in SFH, which is probably related to galactic feedback  mechanisms, is reflected in the distribution and evolution of intrinsic  MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  In the left panel of Figure 12, we show the distribution of the  intrinsic MS scatter versus 𝑀∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We see a minimum intrinsic MS  scatter of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='35 dex at 𝑀∗ ∼ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We see a higher increase  of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 dex at higher mass (> 1011𝑀⊙) with decreasing look-  back time for low redshifts (𝑡lb: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 Gyr) and a relatively flat  relation at higher redshifts (𝑡lb ≳ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In the low mass end  (∼ 109𝑀⊙), where the galaxies are mass-incomplete, the intrinsic  scatter rises from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='35 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='6 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For some of the redshift bins,  this type of trend is qualitatively similar to the turnover that occurs  in the halo mass-stellar mass (HMSM) relation (see Figure 2 of  Wechsler & Tinker (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The shape of the HMSM relation is  thought to be a consequence of feedback, with SF feedback reducing  the SF efficiency in low-mass galaxies and AGN feedback reducing  the SF efficiency in high-mass galaxies, with a turnover at halo mass  𝑀h ∼ 1012𝑀⊙, which is coincidentally corresponding to the upturn  point of 𝑀∗-𝜎int panel at 𝑀∗ ∼ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25𝑀⊙ in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The intrinsic  scatter in the MS is also expected to be linked to feedback, and this  may account for similarities in the observed trends with the HMSM  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  In the right panel of Figure 12, we present a toy model where we  relate the HMSM relation with the intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We adopt  the best-fit functional form and the parameterised data of the stellar  mass halo mass (SMHM) relation from Equations 21 & 22 and Table  2 of Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2010) parameterize the  evolution of SMHM relation in terms of 𝑀1, 𝑀∗,0, 𝛽, 𝛿 and 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' All  these variables vary with the scale factor 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For our model, we convert  the standard SMHM relation into HMSM fraction vs stellar mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=',  log(𝑀h/𝑀∗) vs log(𝑀∗), instead of log(𝑀∗/𝑀h) vs log(𝑀h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This  change results in the HMSM relation having an upturn shape instead  of the usual downturn shape (because we invert the values in the y-  axis ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This modified HMSM relation presents a similar turnover  feature as the ‘U-shaped’ distribution shown in the left subplot of  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Considering time evolution, we renormalize the HMSM  relation to match our data by multiplying the HMSM fraction by an  arbitrary coefficient 𝑘𝑖 (𝑖 indicates different 𝑡lb), which is computed  by the non-linear regression:  log(𝜎int/𝑀⊙yr−1) = 𝑘𝑖 · 𝑓HMSM(𝑎),  (7)  𝑡lb  𝑘𝑖  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='48)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='20  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='66)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='19  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='90)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='17  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5Gyr (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='22)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='15  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7Gyr (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='70)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='12  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='51)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='10  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Best-fit values for the constant factor 𝑘𝑖 at different bins of look-back  time for our toy model given by Equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  where 𝜎int is the intrinsic MS scatter, 𝑘𝑖 is a constant factor (see  Table 3) and 𝑓HMSM(𝑎) = 𝑀h/𝑀∗ is the HMSM fraction that varies  with the scale factor, 𝑎 (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The Equation 7 does a  reasonable job of matching the trends for the first four time bins (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9-  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5Gyr), but the final two bins (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr) favor a flatter shape  than our toy model at high 𝑀∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' At the lower and higher mass ends,  the increasing intrinsic MS scatter may be driven by the feedback  of supernovas and AGNs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In contrast, in the mid-range  of stellar mass (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 < log(𝑀∗/𝑀⊙) < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5), the intrinsic MS  scatter relation reaches a minimum (maximum in HMSM relation),  reflecting the maximum conversion efficiency of gas to baryon and  is thought to be linked to a minimum in the influence of starburst  and galaxy feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We notice a large discrepancy between our data  with shifted HMSM fraction at higher redshifts, which may be due to  the low quality of observational data at high redshifts or differences  in feedback mechanisms in the earlier universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For example, high-𝑧  observational limitation can lead to larger uncertainties on SFR and  make it more difficult to constrain the intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' On the  other hand, weaker AGN feedback for high-𝑀∗ galaxies at high-𝑧  may also account for the almost constant intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2 Comparison to observational studies  We compare our measurements of the intrinsic MS scatter with some  previous studies in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Traditionally, a redshift-independent  width of MS at either 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 dex is founded in observations  (Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Noeske et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Ciesla  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Kurczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2016), which is  lower than our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In contrast, Kurczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Santini  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2022) report a redshift evolution of  intrinsic MS scatter, which spans a wider range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Our data show a similar trend to the ‘U-shaped’ distribution de-  scribed in Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' They used the DEVILS survey, con-  taining ∼60,000 galaxies with spectroscopic redshifts ranging from  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In the case of 𝑧phot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5, where our samples have red-  shift overlap, the intrinsic MS scatter trend along the 𝑀∗ is highly  consistent with the ‘U-shaped’ distribution, except we have signifi-  cantly smaller intrinsic MS scatter (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='24 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='40 dex in this study, while  ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='8 dex in Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' At low stellar mass, Davies  MNRAS 000, 1–15 (2022)  12  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Left panel: the intrinsic MS scatter 𝜎(log(SFR)) vs 𝑀∗ for our 6 look-back time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Solid lines connect the mass-complete data and dashed lines  indicate the mass-incomplete regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Right panel: we show a fit of our toy model (relation indicated in the panel), based on the halo mass to stellar mass fraction  vs stellar mass relation Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2010) normalized by a constant factor, normalized by a constant factor, relative to our intrinsic MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Similar to the  left, we indicate the mass-complete and mass-incomplete regions with solid and dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' A comparison of our intrinsic MS scatter to previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The first three panels display the distribution of intrinsic MS scatter over 𝑀∗ at  𝑧phot < 1, 1 < 𝑧phot < 2 and 𝑧phot > 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The star symbols are the results in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The pentagon symbols show the results from Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  (2022), while the curves in the top left panel are polynomial fitting curves to their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The filled pentagons are the fitting range of Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The  circles are the results from Kurczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2016), while the squares represent the data from Santini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The horizontal dashed line in the bottom  indicates a constant intrinsic MS scatter at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 dex (Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Ciesla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2014), while the solid line represents a non-evolving  scatter at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='20 dex (Noeske et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Pessa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Our data are quantitatively similar to Kurczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7  tib = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='48)  tib = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='66)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='6  tib = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='90)  tib = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5Gyr (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='22)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  tib = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7Gyr (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='70)  log(oint/Moyr-1  tib = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='51)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2  mass-incomplete  mass-complete  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  log(M* /Mo)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='6  log(oint/Moyr-1) = kj - fHMSM  log(oint/Moyr-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  log(M* /Mo)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='8  z<1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='6  log(oint/Moyr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2  7  8  9  10  11  12  log(M* /Mo)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='8  1<<2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7  log(int/Moyr-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2  8  9  10  12  11  log(M*/Mo)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='8  Z>2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7  (t-/w/ul0)60l  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2  8  9  10  12  7  11  log(M*/Mo)constant scatter = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='2 dex  Davies+22, z:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='85  constant scatter = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 dex  Kurczynski+16, z:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  ★★★★★★  tib = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='48)  Kurczynski+16, z:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  tib = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='66)  Kurczynski+16, z:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  tib = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3Gyr (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='90)  Kurczynski+16, z:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  tib = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5Gyr (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='22)  Kurczynski+16, z:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  tib = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7Gyr (z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='70)  Santini+17, z:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3-2  tib = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='51)  Santini+17, z:2-3  Davies+22, z:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25  Santini+17, z:3-4  Davies+22, z:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4  Santini+17, z:4-5  Davies+22, z:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='55  Santini+17, z:5-6  Davies+22, z:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='55-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='7Intrinsic MS Scatter at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  13  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2022) suggest that the intrinsic MS scatter is driven by stochas-  tic starbursts and stellar feedback events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' while the galaxies become  more massive and reach intermediate stellar mass (around 1010𝑀⊙),  the galaxies are too massive so that the effect of star formation and  galaxy feedback is less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In this study, intrinsic MS scatter  increases at high stellar mass (log(𝑀∗/𝑀⊙) ≳ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3), consistent with  the ‘U-shaped’ distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2022) conclude that AGN  feedback leads to a large scatter at the high mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We note that in  our selection, we removed sources with current AGN signatures (see  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1), but that the feedback from previous AGN will still affect  the MS scatter over longer timescales than the AGN duty cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' As the  redshift increases, more data in the low stellar mass end are identified  as mass-incomplete due to IR selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' However, we can still rec-  ognize that the intrinsic MS scatter tends to increase when galaxies  become more massive for 𝑀∗ > 1010𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' With increasing redshift,  we notice that the right half of the ‘U-shaped’ distribution becomes  flattered and even decreases at 𝑡lb = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='9Gyr (𝑧phot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This  suggests that the star-forming and feedback activity or efficiency for  high-mass galaxies in the early universe may differ relative to lower  redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Regarding the higher intrinsic MS scatter in Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2022)  relative to our results, we think this is due to the following reasons:  (1) They do not include photo-z uncertainties in the SED modeling,  which results in an underestimate of the measurement uncertainty  (on 𝑀∗ and SFR), and hence, an overestimation of the intrinsic  MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2) They derive galaxy properties for the D10 field of  DEVILS (Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2018) by using different SED fitting code,  PROSPECT (Robotham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Large differences in obtained  properties are produced by different derivation techniques applied  to various photometric data (see the comparison between PROSPECT  and MAGPHYS3 in Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (3) They adopt the U-V-J  selection rather than sSFR cut so that the samples in Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  (2022) contain low-sSFR galaxies (see the discussion of these two  selection criteria in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' However, in our selection criteria,  these galaxies are identified as quenched and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The addition  of quenched galaxies severely enlarges the MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' As will be  shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3 when comparing to simulations, the amplitude  of the intrinsic MS scatter is very sensitive to the choice of sSFR  cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For example, Leja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2022) demonstrate that fixed-sSFR  cuts may reduce the inferred MS scatter, particularly at the highest  stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (4) They do not strictly require detection in the IR  bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3, the SED fitting in the absence  of IR information results in considerable uncertainty of derived SFR,  which may increase the SFR standard deviation and inferred intrinsic  MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3 Comparison to theoretical studies  We first compare to results from the SHARK semi-analytic models (left  panel of Figure 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For SHARK, we select galaxies  with stellar masses between 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='25 −1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75𝑀⊙ within ±1 dex along  the MS for each redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We calculate the standard deviations of  the median SFRs for selected galaxies and present the results as  the dot-dashed lines in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' A notable difference from the  observational data is that the overall scatter in SHARK is larger than  observational data in the mass-complete range, particularly for the  highest redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We find a common trend that the SHARK results  3 We pick MAGPHYS+photo-z in this study because this code can treat red-  shift as a free parameter and derive the 𝑧phot and the corresponding measure-  ment uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  follow a similar ‘U-shaped’ distribution at each redshift, though the  minimum and maximum points occur at a stellar mass < 1010𝑀⊙ and  > 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75𝑀⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We suggest that the consistent ‘upturn’  feature in SHARK also indicates the effect of past AGNs for 𝑀∗ ≳  1010𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We also observe a flat or decreasing scatter in SHARK for  galaxies in the range of 𝑀∗ ≥ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This occurrence indicates  that galaxies in this mass range are shifted below the chosen sSFR  limit because AGNs have a more significant impact on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We also  investigate the effect of the sSFR cut on intrinsic MS scatter and find  that the stricter sSFR cut leads to a smaller amplitude of the intrinsic  MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For instance, the intrinsic MS scatter will be reduced by  ∼ 1 dex overall when we pick a selection with a narrower sSFR cut,  such as ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75 dex along the MS, rather than ±1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Next, we compare the results with EAGLE hydrodynamical sim-  ulations (right panel of Figure 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Matthee & Schaye 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We  redivide the data in redshift binning at 𝑧phot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  for a proper comparison with Figure 3 in Matthee & Schaye (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Matthee & Schaye (2019) adopt SF galaxies with evolved sSFR se-  lection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', log(sSFR/yr−1) = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 and increases to  −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4 at 𝑧 = 3) and measure the scatter from the residuals by us-  ing the non-parametric local polynomial regression method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Then  they obtain the intrinsic MS scatter by subtracting the observational  errors derived by median uncertainties of the observational sample  from Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The EAGLE results suggest lower intrinsic  MS scatter at higher redshift for 𝑀∗ < 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='8𝑀⊙ and similar intrinsic  MS scatter for 𝑀∗ > 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='8𝑀⊙, with a downward ‘U-shaped’ feature  appearing at z = 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This shape differs substantially from our  findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Unlike the decreasing trend with stellar mass from simu-  lation, the intrinsic MS scatter in this study decreases initially but  increases at the high mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Matthee & Schaye (2019) consider  that supermassive black hole growth accounts for the increasing scat-  ter at 𝑀∗ ≈ 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='8𝑀⊙ at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' However, our results suggest  the influence of the feedback mechanism from previous AGNs might  be more significant at higher stellar masses (𝑀∗ ≳ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='3𝑀⊙) at low  redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The intrinsic MS scatter at each redshift bin is also larger  than Matthee & Schaye (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We suspect that different physics  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', feedback, SF model) adopted in EAGLE give rise to the quanti-  tative difference to our results and from SHARK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  5 CONCLUSIONS  In this paper, using the selection of 12,380 SF galaxies from the  COSMOS2020 database and adopting the MAGPHYS+photo-z SED  fitting code, we characterise the intrinsic scatter of galaxy MS over  the redshift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5 < 𝑧 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  We find that the intrinsic MS scatter is larger than the measure-  ment uncertainty by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='6 when IR data is available for  accurately constraining the dust-obscured star formation (Section 3),  with measured MS scatter in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='26-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='47 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  For the COSMOS2020 sample, the inclusion of IR data is the  dominant factor (over 𝑧phot uncertainty), affecting the accuracy of  measuring the scatter on the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Binning the data according to either redshift or look-back time,  we find a slightly negative correlation between intrinsic MS scatter  and look-back time (Equation 6) but with an upturn at 𝑡lb ≳ 10Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  To connect the intrinsic MS scatter with the feedback mecha-  nism, we present a toy model that uses the Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2010)  SMHM relation (Equation 7), which does a reasonable job of match-  ing the distribution of intrinsic MS scatter over 𝑀∗ and redshift  (Figure 12), although with less agreement at the highest redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  14  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Left panel: we compare our results with SHARK models and show the sensitivity of the intrinsic MS scatter to the chosen boundary of sSFR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', ±1  and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='75 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' The trends in SHARK show rough qualitative agreement with our observational results but with slight differences in the normalization, which  is sensitive to the adopted sSFR cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Right panel: we overlap the results from the SHARK (dot-dashed lines) and EAGLE (solid bands) simulations (Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Matthee & Schaye 2019) with our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' For both panels, we have modified our bins to be consistent with the bins used in Matthee & Schaye (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  At 𝑀∗ > 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='8𝑀⊙, where we are mostly mass-complete, we find differing trends between the observational results and the EAGLE simulations, and also note  there are large differences in the trends inferred from SHARK and EAGLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We obtain error bars of intrinsic MS scatter by bootstrap resampling the data by their  uncertainties 100 times and remeasuring the intrinsic scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  We compare our results to other observational studies of the  MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Differing from a non-evolving, mass-independent scatter,  our results are qualitatively similar to the ‘U-shaped’ intrinsic MS  scatter distribution with stellar mass and redshifts found in Davies  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  We compare the intrinsic MS scatter to some theoretical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  The consistent upturn trend in SHARK models suggests the agreement  of the feedback mechanism from past AGN activity for galaxies with  𝑀∗ > 1010𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' These comparisons highlight the significant influ-  ence that sSFR cuts can have on the measured value of the ‘intrinsic’  MS scatter and that particular care needs to be taken with such com-  parisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We also find that the behaviour of intrinsic MS scatter  diverges significantly between our study and EAGLE Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  In the future, the most significant gain in our understanding of the  evolution in the MS scatter will come from deeper surveys in rest-  frame IR to enable accurate characterization of the MS scatter at both  low stellar masses and higher redshifts, where our current sample is  severely limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' There is a weak agreement between observation  data and theory in Equation 7 for high-𝑧 and low mass cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' In  particular, better sampling in these regimes will provide a clear test  of whether our toy model linking the MS scatter to the HMSM  relation is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Alternatively, we also plan to explore less-  restrictive selection criteria in the IR bands to push our sample to  include more low-𝑀∗ and/or high-𝑧 sources from existing surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors appreciate the referee, Antonios Katsianis, who provided  valuable and insightful comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Parts of this  research were supported by the Australian Research Council Centre  of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO  3D), through project number CE170100013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We thank the COSMOS  team for making the data products that we used in this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' is  supported by the Australian Research Council through the Discovery  Early Career Researcher Award (DECRA) Fellowship DE220100766  funded by the Australian Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We also thank Jorryt Matthee  for correspondence regarding his study using the EAGLE survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' This  research was conducted on Ngunnawal Indigenous land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  DATA AVAILABILITY  The observational data used in this paper are publicly available  through catalog and imaging data releases from the COSMOS survey  team (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Other data products can be made available  upon reasonable request to the first author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+page_content=', 2008, MNRAS, 388, 1595  da Cunha E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=', 2015, ApJ, 806, 110  APPENDIX A: sSFR SELECTION VS U-V-J CUT  We show a comparison between the conventional U-V-J (in rest-  frame) selection from Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2012) and our adopted sSFR  selection (see Figure A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Since the sSFR is computed by multi-  band SED fitting, we expect this technique will be more accurate in  excluding quenched galaxies than a color-color cut based on U, V,  and J bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' We find that 75% of galaxies below our sSFR cut are  excluded by the U-V-J cut, so the majority of ‘quenched’ galaxies in  our samples are also designated as ‘passive’ in the U-V-J diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  However, only 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='17% galaxies that were eliminated by U-V-J selec-  tion lie below our sSFR cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Hence, the bulk of galaxies excluded  by the colour-colour cut for our sample are SF galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' presumably,  they are dusty SF galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Therefore, we adopt an sSFR cut rather  than U-V-J cut to remove quenched galaxies for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  16  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Left panel: U-V-J diagram lot for the 13,071 𝜒2 selection galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' Right panel: in addition to Figure 4, we show the galaxies removed by U-V-J cut  from Whitaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content=' 64 galaxies are excluded by the sSFR cut, and 929 galaxies are excluded by the U-V-J cut, whereas only 48 of them are marked as  quenched galaxies jointly by both selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='  MNRAS 000, 1–15 (2022)  3  2  1  U  U-V-J cut  Encloses 68%  Encloses 95%  removed by sSFR cut  Encloses 99%  0  1  2  3  468% enclosed curve  95% enclosed curve  8  sSFR cut: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='57Zphot - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='60  removed by UVl cut  log(sSFR/yr-1)  10  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
+page_content='0  Zphot' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtA0T4oBgHgl3EQfCv_p/content/2301.01995v1.pdf'}
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+arXiv:2301.00581v1  [cs.IT]  2 Jan 2023
+1
+Bent Partitions, Vectorial Dual-Bent Functions and Partial Difference Sets†
+Jiaxin Wang, Fang-Wei Fu, Yadi Wei
+Abstract
+Bent partitions of V (p)
+n
+are quite powerful in constructing bent functions, vectorial bent functions and
+generalized bent functions, where V (p)
+n
+is an n-dimensional vector space over Fp, n is an even positive
+integer and p is a prime. It is known that partial spreads is a class of bent partitions. In [4], [18], two
+classes of bent partitions whose forms are similar to partial spreads were presented. In [3], more bent
+partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 were presented from (pre)semifields, including the bent partitions given
+in [4], [18]. In this paper, we investigate the relations between bent partitions and vectorial dual-bent
+functions. For any prime p, we show that one can generate certain bent partitions (called bent partitions
+satisfying Condition C) from certain vectorial dual-bent functions (called vectorial dual-bent functions
+satisfying Condition A). In particular, when p is an odd prime, we show that bent partitions satisfying
+Condition C one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an
+alternative proof that Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 are bent partitions in terms of vectorial dual-bent functions.
+We present a secondary construction of vectorial dual-bent functions, which can be used to generate
+more bent partitions. We show that any ternary weakly regular bent function f : V (3)
+n
+→ F3 (n even) of
+2-form can generate a bent partition. When such f is weakly regular but not regular, the generated bent
+partition by f is not coming from a normal bent partition, which answers an open problem proposed in
+[4]. We give a sufficient condition on constructing partial difference sets from bent partitions, and when
+p is an odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of
+partial difference sets.
+Index Terms
+Bent partitions; bent functions; vectorial bent functions; vectorial dual-bent functions; semifields;
+partial difference sets
+Jiaxin Wang, Fang-Wei Fu and Yadi Wei are with Chern Institute of Mathematics and LPMC, Nankai University, Tianjin
+300071, China, Emails: wjiaxin@mail.nankai.edu.cn, fwfu@nankai.edu.cn, wydecho@mail.nankai.edu.cn.
+†This research is supported by the National Key Research and Development Program of China (Grant Nos. 2018YFA0704703
+and 2022YFA1005001), the National Natural Science Foundation of China (Grant Nos. 12141108, 61971243, 12226336), the
+Natural Science Foundation of Tianjin (20JCZDJC00610), the Fundamental Research Funds for the Central Universities of China
+(Nankai University), and the Nankai Zhide Foundation.
+January 3, 2023
+DRAFT
+
+2
+I. INTRODUCTION
+Boolean bent functions were introduced by Rothaus [21] and were generalized to p-ary bent
+functions by Kumar, Scholtz and Welch [15], where p is an arbitrary prime. Due to applications
+of p-ary bent functions in cryptography, coding theory, sequence and combinatorics, they have
+been extensively studied. We refer to surveys [5], [17] and a book [19] on p-ary bent functions
+and their generalizations such as vectorial bent functions and generalized bent functions.
+In [10], C¸ es¸melio˘glu et al. introduced vectorial dual-bent functions, which is a special class
+of vectorial bent functions. In [7], [8], [22], vectorial dual-bent functions were used to construct
+partial difference sets. In particular, Wang and Fu [22] showed that for certain vectorial dual-bent
+functions F : V (p)
+n
+→ V (p)
+s
+(where V (p)
+n
+is an n-dimensional vector space over the prime field
+Fp), the preimage set of any subset of V (p)
+s
+for F forms a partial difference set.
+Very recently, bent partitions of V (p)
+n
+were introduced [4], [18], which are quite powerful in
+constructing bent functions, vectorial bent functions and generalized bent functions. The well-
+known partial spreads is a class of bent partitions. In [18], Meidl and Pirsic for the first time
+presented two classes of bent partitions for p = 2 different from partial spreads. In [4], Anbar and
+Meidl generalized the contributions in [18] to the case of p being odd and gave the corresponding
+two classes of bent partitions for odd p. In [3], Anbar, Kalaycı and Meidl presented more bent
+partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 from (pre)semifields, including the bent partitions given in [4],
+[18]. In [2], Anbar, Kalaycı and Meidl showed that any union of elements in the bent partition
+given in [4], [18] forms a partial difference set. In terms of constructing partial difference sets,
+certain vectorial dual-bent functions and certain bent partitions seem to play the same role.
+Therefore, it is interesting to investigate the relations between vectorial dual-bent functions
+and bent partitions. In this paper, we show that by using certain vectorial dual-bent functions
+(called vectorial dual-bent functions satisfying Condition A), we can construct bent partitions
+of V (p)
+n
+with certain properties (called bent partitions satisfying Condition C) for any prime
+p. Particularly, when p is an odd prime, we prove that bent partitions of V (p)
+n
+with Condition
+C one-to-one correspond to vectorial dual-bent functions satisfying Condition A. In terms of
+vectorial dual-bent functions, we provide an alternative proof that Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 given
+in [3] are bent partitions. We provide a secondary construction of vectorial dual-bent functions,
+which can be used to generate more bent partitions. We prove that any ternary weakly regular
+bent function f : V (3)
+n
+→ F3 (n even) of 2-form can generate a bent partition. In the special case
+January 3, 2023
+DRAFT
+
+3
+that f is weakly regular but not regular, the generated bent partition by f is not coming from
+a normal bent partition, which answers an open problem proposed in [4]. By using vectorial
+dual-bent functions as the link between bent partitions and partial difference sets, we give a
+sufficient condition on constructing partial difference sets from bent partitions. When p is an
+odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of
+partial difference sets.
+The rest of the paper is organized as follows. In Section II, we state some needed results on
+vectorial dual-bent functions and bent partitions. In Section III, we present relations between
+certain bent partitions and certain vectorial dual-bent functions. In Section IV, we give a sec-
+ondary construction of vectorial dual-bent functions, which can be used to generate more bent
+partitions. In Section V, we present relations between certain bent partitions and certain partial
+difference sets. In Section VI, we make a conclusion.
+II. PRELIMINARIES
+In this section, we state some basic results on vectorial dual-bent functions and bent partitions.
+First, we fix some notations used throughout this paper.
+• p is a prime.
+• ζp = e
+2π√−1
+p
+is a complex primitive p-th root of unity. Note that ζ2 = −1.
+• Fpn is the finite field with pn elements.
+• Fn
+p is the vector space of the n-tuples over Fp.
+• V (p)
+n
+is an n-dimensional vector space over Fp.
+• ⟨·⟩n denotes a (non-degenerate) inner product of V (p)
+n . In this paper, when V (p)
+n
+= Fpn,
+let ⟨a, b⟩n = Trn
+1(ab), where a, b ∈ Fpn, Trn
+k(·) denotes the trace function from Fpn to
+Fpk, k | n; when V (p)
+n
+= Fn
+p, let ⟨a, b⟩n = a · b = �n
+i=1 aibi, where a = (a1, . . . , an), b =
+(b1, . . . , bn) ∈ Fn
+p; when V (p)
+n
+= V (p)
+n1 ×· · ·×V (p)
+nm (n = �m
+i=1 ni), let ⟨a, b⟩n = �m
+i=1⟨ai, bi⟩ni,
+where a = (a1, . . . , am), b = (b1, . . . , bm) ∈ V (p)
+n .
+• For any set A ⊆ V (p)
+n
+and u ∈ V (p)
+n , let χu(A) = �
+x∈A χu(x), where χu denotes the
+character χu(x) = ζ⟨u,x⟩n
+p
+.
+A. Vectorial dual-bent functions
+A function F : V (p)
+n
+→ V (p)
+s
+is called a vectorial p-ary function, or simply p-ary function
+when s = 1. The Walsh transform of a p-ary function f : V (p)
+n
+→ Fp is the complex valued
+January 3, 2023
+DRAFT
+
+4
+function defined by
+Wf(a) =
+�
+x∈V (p)
+n
+ζf(x)−⟨a,x⟩n
+p
+, a ∈ V (p)
+n .
+(1)
+A p-ary function f : V (p)
+n
+→ Fp is called bent if |Wf(a)| = p
+n
+2 for all a ∈ V (p)
+n . Note that
+when f is a Boolean bent function, that is, p = 2, then n must be even since in this case, Wf is
+an integer valued function. A vectorial p-ary function F : V (p)
+n
+→ V (p)
+s
+is called vectorial bent
+if all component functions Fc : V (p)
+n
+→ Fp, c ∈ V (p)
+s
+\{0} defined as Fc(x) = ⟨c, F(x)⟩s are bent.
+It is known that if F : V (p)
+n
+→ V (p)
+s
+is vectorial bent, then s ≤ n
+2 if p = 2, and s ≤ n if p is
+an odd prime. If f : V (p)
+n
+→ Fp is bent, then so are cf, c ∈ F∗
+p, that is, any p-ary bent function
+is vectorial bent. For F : V (p)
+n
+→ V (p)
+s
+, the vectorial bentness of F is independent of the inner
+products of V (p)
+n
+and V (p)
+s
+. The Walsh transform of a p-ary bent function f : V (p)
+n
+→ Fp satisfies
+that for any a ∈ V (p)
+n , when p = 2, we have
+Wf(a) = 2
+n
+2 (−1)f∗(a),
+(2)
+and when p is an odd prime, we have
+Wf(a) =
+
+
+
+±p
+n
+2 ζf∗(a)
+p
+if p ≡ 1 (mod 4) or n is even,
+±
+√
+−1p
+n
+2 ζf∗(a)
+p
+if p ≡ 3 (mod 4) and n is odd,
+(3)
+where f ∗ is a function from V (p)
+n
+to Fp, called the dual of f. A p-ary bent function f : V (p)
+n
+→ Fp
+is called weakly regular if Wf(a) = εfp
+n
+2 ζf∗(a)
+p
+, where εf is a constant independent of a,
+otherwise f is called non-weakly regular. In particular, if εf = 1, f is called regular. The (non-
+)weakly regularity of f is independent of the inner product of V (p)
+n
+and if f is weakly regular,
+εf is independent of the inner product of V (p)
+n . By (2), all Boolean bent functions are regular.
+If f is a p-ary weakly regular bent function, then the dual f ∗ of f is also weakly regular bent
+with (f ∗)∗(x) = f(−x) (see [9]).
+In 2018, C¸ es¸melio˘glu et al. [10] introduced vectorial dual-bent functions.
+Definition 1. A vectorial p-ary bent function F : V (p)
+n
+→ V (p)
+s
+is called vectorial dual-bent
+if there exists a vectorial bent function G : V (p)
+n
+→ V (p)
+s
+such that (Fc)∗ = Gσ(c) for any
+c ∈ V (p)
+s
+\{0}, where (Fc)∗ is the dual of the component function ⟨c, F(x)⟩s and σ is some
+permutation over V (p)
+s
+\{0}. The vectorial bent function G is called a vectorial dual of F and
+denoted by F ∗.
+January 3, 2023
+DRAFT
+
+5
+It is known in [10] that the property of being vectorial dual-bent is independent of the inner
+products of V (p)
+n
+and V (p)
+s
+. Note that for a vectorial dual-bent function, its vectorial dual is not
+unique since being vectorial bent and vectorial dual-bent for a function is a property of the
+vector space consisting of all component functions (see Remark 1 of [10]). For example, if a
+p-ary function f (seen as a vectorial function into V (p)
+1
+, p odd) is vectorial dual-bent under any
+fixed inner product, then its dual f ∗ is unique, but its vectorial dual is not unique since for any
+c ∈ F∗
+p, cf ∗ is a vectorial dual of f. A p-ary function f : V (p)
+n
+→ Fp is called an l-form if
+f(ax) = alf(x) for any a ∈ F∗
+p and x ∈ V (p)
+n , where 1 ≤ l ≤ p − 1 is an integer. By the results
+in [7], [22], we have the following proposition.
+Proposition 1 ( [7], [22]). A p-ary function f with f(0) = 0 is a weakly regular vectorial dual-
+bent function if and only if f is a weakly regular bent function of l-form with gcd(l−1, p−1) = 1.
+In particular, a p-ary function f is a weakly regular vectorial dual-bent function with (cf)∗ = cf ∗
+for any c ∈ F∗
+p if and only if f is a weakly regular bent function of (p − 1)-form.
+In the rest of this subsection, we recall an important class of p-ary bent functions, called
+Maiorana-McFarland bent functions.
+• Let f : Fpn × Fpn → Fp be defined as
+f(x, y) = Trn
+1(αxπ(y)) + g(y),
+where α ∈ F∗
+pn, π is a permutation over Fpn and g : Fpn → Fp is an arbitrary function.
+Then f is bent and is called a Maiorana-McFarland bent function. The dual f ∗ of f is
+f ∗(x, y) = Trn
+1(−π−1(α−1x)y) + g(π−1(α−1x)).
+(4)
+All Maiorana-McFarland bent functions are regular (see [15]).
+B. Bent partitions
+Very recently, the concept of bent partitions of V (p)
+n
+were introduced [4], [18].
+Definition 2. Let n be an even positive integer, K be a positive integer divisible by p.
+• Let Γ = {A1, . . . , AK} be a partition of V (p)
+n . Assume that every function f for which every
+i ∈ Fp has exactly K
+p of sets Aj in Γ in its preimage, is a p-ary bent function. Then Γ is
+called a bent partition of V (p)
+n
+of depth K and every such bent function f is called a bent
+function constructed from bent partition Γ.
+January 3, 2023
+DRAFT
+
+6
+• Let Γ = {U, A1, . . . , AK} be a partition of V (p)
+n . Assume that every function f with the
+following properties is bent:
+(1) Every c ∈ Fp has exactly K
+p of the sets A1, . . . , AK in its preimage set;
+(2) f(x) = c0 for all x ∈ U and some fixed c0 ∈ Fp.
+Then we call Γ a normal bent partition of V (p)
+n
+of depth K.
+Bent partitions are very powerful in constructing bent functions, vectorial bent function and
+generalized bent functions. In this paper, we focus on the relations between bent partitions and
+vectorial bent functions.
+Proposition 2 ( [4]). Let Γ = {A1, . . . , Aps} be a bent partition of V (p)
+n . Then every function
+F : V (p)
+n
+→ V (p)
+s
+such that every element i ∈ V (p)
+s
+has the elements of exactly one of the sets
+Aj, 1 ≤ j ≤ ps, in its preimage, is a vectorial bent function.
+It is known that partial spreads is a class of bent partitions (for instance see Section 2 of [4]). In
+[4], [18], two classes of explicit bent partitions different from partial spreads were presented. In
+[3], bent partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 were presented from certain (pre)semifields, including
+the bent partitions given in [4], [18]. We will recall bent partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 given
+in [3]. First, we need to introduce some basic knowledge on (pre)semifields.
+Definition 3. Let ◦ be a binary operation defined on (V (p)
+n , +) such that
+(i) x ◦ y = 0 implies x = 0 or y = 0,
+(ii) (x + y) ◦ z = (x ◦ z) + (y ◦ z), (z ◦ (x + y) = (z ◦ x) + (z ◦ y), respectively),
+for all x, y, z ∈ V (p)
+n . Then (V (p)
+n , +, ◦) is called a right (left, respectively) prequasifield. If
+(V (p)
+n , +, ◦) is a right and a left prequasifield, then it is called a presemifield. If (V (p)
+n , +, ◦) is
+a presemifield for which there is an element e ̸= 0 such that e ◦ x = x ◦ e = x for all x ∈ V (p)
+n ,
+then it is called a semifield.
+Let P = (Fpn, +, ◦) be a presemifield. Then one can obtain presemifields P • = (Fpn, +, •)
+and P ⋆ = (Fpn, +, ⋆) from P, where • and ⋆ are given by
+x • y = y ◦ x for all x, y ∈ Fpn,
+and
+Trn
+1(z(x ◦ y)) = Trn
+1(x(z ⋆ y)) for all x, y, z ∈ Fpn,
+January 3, 2023
+DRAFT
+
+7
+respectively. The presemifield P ⋆ is called the dual of P. Let s be a positive divisor of n.
+If x ◦ (cy) = c(x ◦ y) holds for any x, y ∈ Fpn, c ∈ Fps, then P is called right Fps-linear.
+Each presemifield P = (Fpn, +, ◦) can induce a semifield P ′ = (Fpn, +, ∗) via the following
+transformation: choose any α ∈ F∗
+pn and give ∗ by
+(x ◦ α) ∗ (α ◦ y) = x ◦ y for all x, y ∈ Fpn.
+By Lemma 2 of [3], if P is right Fps-linear, then P ′ is also right Fps-linear. The finite field Fpn
+is a right Fps-linear semifield (that is, ◦ is the field multiplication). For more right Fps-linear
+(pre)semifields, see Section 3 of [3].
+Now we recall bent partitions Γ1, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2 given in [3].
+• Let n, s be positive integers satisfying s | n and gcd(pn−1, ps+p−1) = 1. Set u = ps+p−1,
+and let d be an integer with du ≡ 1 mod (pn − 1). Let P = (Fpn, +, ◦) be a (pre)semifield
+such that its dual P ⋆ = (Fpn, +, ⋆) is right Fps-linear. For given x ∈ Fpn, if x = 0, then let
+ηx = 0, and if x ̸= 0, then let ηx be given by x ⋆ η−1
+x
+= 1.
+• Define
+Ut = {(x, t ◦ xu) : x ∈ F∗
+pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Γ1 = {Ai, i ∈ Fps},
+(5)
+where Ai = �
+t∈Fpn:Trn
+s (t)=i Ut if i ̸= i0, Ai0 = �
+t∈Fpn:Trn
+s (t)=i0 Ut
+� U.
+• Define
+U•
+t = {(x, xu ◦ t) : x ∈ F∗
+pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Γ•
+1 = {A•
+i , i ∈ Fps},
+(6)
+where A•
+i = �
+t∈Fpn:Trn
+s (t)=i U•
+t if i ̸= i0, A•
+i0 = �
+t∈Fpn:Trn
+s (t)=i0 U•
+t
+� U.
+• Define
+Vt = {(t ◦ xd, x) : x ∈ F∗
+pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Γ2 = {Bi, i ∈ Fps},
+(7)
+January 3, 2023
+DRAFT
+
+8
+where Bi = �
+t∈Fpn:Trn
+s (t)=i Vi if i ̸= i0, Bi0 = �
+t∈Fpn:Trn
+s (t)=i0 Vi
+� V .
+• Define
+V •
+t = {(xd ◦ t, x) : x ∈ F∗
+pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Γ•
+2 = {B•
+i , i ∈ Fps},
+(8)
+where B•
+i = �
+t∈Fpn:Trn
+s (t)=i V •
+i if i ̸= i0, Bi0 = �
+t∈Fpn:Trn
+s (t)=i0 V •
+i
+� V .
+• Define
+Xt = {(tηd
+x, x) : x ∈ F∗
+pn} if t ∈ Fpn, and X = {(x, 0) : x ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Θ1 = {Si, i ∈ Fps},
+(9)
+where Si = �
+t∈Fpn:Trn
+s (t)=i Xt if i ̸= i0, Si0 = �
+t∈Fpn:Trn
+s (t)=i0 Xt
+� X.
+• Define
+Yt = {(x, tηu
+x) : x ∈ F∗
+pn} if t ∈ Fpn, and Y = {(0, y) : y ∈ Fpn}.
+Let i0 ∈ Fps be an arbitrary element. Define
+Θ2 = {Ti, i ∈ Fps},
+(10)
+where Ti = �
+t∈Fpn:Trn
+s (t)=i Yt if i ̸= i0, Ti0 = �
+t∈Fpn:Trn
+s (t)=i0 Yt
+� Y .
+Remark 1. In the finite field case, that is, ◦ and ⋆ are the field multiplication, then Γ1 = Γ•
+1 =
+Θ2, Γ2 = Γ•
+2 = Θ1, which reduces to the two classes bent partitions given in [4], [18].
+Remark 2. In fact, for the parameter u in the bent partitions Γ1, Γ•
+1, Γ2, Γ•
+2, Θ1, Θ2, one can
+consider the more general form u ≡ pj mod (ps − 1) by the proofs in [3].
+III. RELATIONS BETWEEN CERTAIN BENT PARTITIONS AND CERTAIN VECTORIAL
+DUAL-BENT FUNCTIONS
+Throughout this section, we consider bent partitions and vectorial dual-bent functions satisfy-
+ing the following conditions, respectively.
+Condition C: Let n be an even positive integer, s be a positive integer with s ≤
+n
+2. Let
+Γ = {Ai, i ∈ V (p)
+s
+} be a bent partition of V (p)
+n
+which satisfies that F∗
+pAi = Ai for all i ∈ V (p)
+s
+January 3, 2023
+DRAFT
+
+9
+and all bent functions f constructed from Γ are regular (that is, εf = 1) or weakly regular but
+not regular (that is, εf = −1). We denote by ε = εf for all bent functions f constructed from Γ.
+Condition A: Let n be an even positive integer, s be a positive integer with s ≤
+n
+2. Let
+F : V (p)
+n
+→ V (p)
+s
+be a vectorial dual-bent function with (Fc)∗ = (F ∗)c, c ∈ V (p)
+s
+\{0} for a
+vectorial dual F ∗ of F and all component functions being regular or weakly regular but not
+regular, that is, εFc, c ∈ V (p)
+s
+\{0} are all the same. We denote by ε = εFc for all c ∈ V (p)
+s
+\{0}.
+It is easy to see that the known bent partitions, including partial spreads and Γi, Γ•
+i , Θi, i = 1, 2
+defined by (5)-(10), all satisfy F∗
+pAi = Ai, i ∈ V (p)
+s
+. By the results in [3], [11], [14], all bent
+functions constructed from partial spreads and Γi, Γ•
+i , Θi, i = 1, 2 are regular. Thus, the known
+bent partitions all satisfy Condition C with ε = 1. Moreover, when p = 2, it is easy to see that
+Condition C is trivial for any bent partition of V (2)
+n
+of depth powers of 2. In this section, we
+present relations between bent partitions satisfying Condition C and vectorial dual-bent functions
+satisfying Condition A. First, we need a lemma.
+Lemma 1. Let n be an even positive integer, s be a positive integer with s ≤ n
+2, and F : V (p)
+n
+→
+V (p)
+s
+. Then the following two statements are equivalent.
+(1) F is a vectorial dual-bent function satisfying Condition A.
+(2) There exist pairwise disjoint sets Wi ⊆ V (p)
+n , i ∈ V (p)
+s
+with �
+i∈V (p)
+s
+Wi = V (p)
+n
+and a
+constant ε ∈ {±1} (ε = 1 if p = 2) such that for any nonempty set I ⊆ V (p)
+s
+,
+χu(DF,I) = pn−sδ{0}(u)|I| + εp
+n
+2 −s(psδWI(u) − |I|), u ∈ V (p)
+n ,
+(11)
+where DF,I = {x ∈ V (p)
+n
+: F(x) ∈ I}, WI = �
+i∈I Wi, and for any set S, δS denotes the
+indicator function of S.
+Proof. By Proposition 3 of [22] (Note that although Proposition 3 of [22] only considers the
+case of p being odd, p = 2 also holds), for any u ∈ V (p)
+n , i ∈ V (p)
+s
+we have
+χu(DF,i) = pn−sδ{0}(u) + p−s
+�
+c∈V (p)
+s
+\{0}
+WFc(−u)ζ−⟨c,i⟩s
+p
+,
+(12)
+where DF,i = {x ∈ V (p)
+n
+: F(x) = i}.
+January 3, 2023
+DRAFT
+
+10
+(1) ⇒ (2): If F is a vectorial dual-bent function satisfying Condition A (Note that if p = 2,
+then ε = 1 since all Boolean bent functions are regular), then
+χu(DF,i) = pn−sδ{0}(u) + εp
+n
+2 −s
+�
+c∈V (p)
+s
+\{0}
+ζ(Fc)∗(−u)−⟨c,i⟩s
+p
+= pn−sδ{0}(u) + εp
+n
+2 −s
+�
+c∈V (p)
+s
+\{0}
+ζ(F ∗)c(−u)−⟨c,i⟩s
+p
+= pn−sδ{0}(u) + εp
+n
+2 −s
+�
+c∈V (p)
+s
+\{0}
+ζ⟨c,F ∗(−u)−i⟩s
+p
+= pn−sδ{0}(u) + εp
+n
+2 −s(psδ{0}(F ∗(−u) − i) − 1).
+(13)
+Define Wi = {x ∈ V (p)
+n
+: F ∗(−x) = i}, i ∈ V (p)
+s
+. Then Wi
+� Wj = ∅ for any i ̸= j and
+�
+i∈V (p)
+s
+Wi = V (p)
+n . By (13), for any nonempty set I ⊆ V (p)
+s
+and u ∈ V (p)
+n
+we have
+χu(DF,I) =
+�
+i∈I
+χu(DF,i)
+=
+�
+i∈I
+pn−sδ{0}(u) + εp
+n
+2 −s(psδWi(u) − 1)
+= pn−sδ{0}(u)|I| + εp
+n
+2 −s(psδWI(u) − |I|).
+(2) ⇒ (1): By the assumption on Wi, i ∈ V (p)
+s
+, we have that for any x ∈ V (p)
+n , there exists a
+unique i ∈ V (p)
+s
+such that x ∈ Wi. Define G : V (p)
+n
+→ V (p)
+s
+by
+G(x) = i if − x ∈ Wi.
+By the definition of G, for any u ∈ V (p)
+n , i ∈ V (p)
+s
+we have
+χu(DF,i) = pn−sδ{0}(u) + εp
+n
+2 −s(psδ{0}(G(−u) − i) − 1).
+(14)
+For any c ∈ V (p)
+s
+\{0},
+WFc(−u) =
+�
+x∈V (p)
+n
+ζ⟨c,F (x)⟩s+⟨u,x⟩n
+p
+=
+�
+i∈V (p)
+s
+�
+x∈V (p)
+n
+:F (x)=i
+ζ⟨c,i⟩s+⟨u,x⟩n
+p
+January 3, 2023
+DRAFT
+
+11
+=
+�
+i∈V (p)
+s
+ζ⟨c,i⟩s
+p
+χu(DF,i)
+=
+�
+i∈V (p)
+s
+\{G(−u)}
+ζ⟨c,i⟩s
+p
+(pn−sδ{0}(u) − εp
+n
+2 −s) + (pn−sδ{0}(u) + εp
+n
+2 −s(ps − 1))ζ⟨c,G(−u)⟩s
+p
+= (pn−sδ{0}(u) − εp
+n
+2 −s)
+�
+i∈V (p)
+s
+ζ⟨c,i⟩s
+p
++ εp
+n
+2 ζGc(−u)
+p
+= εp
+n
+2 ζGc(−u)
+p
+.
+(15)
+By (15) and the assumption that ε = 1 if p = 2, F is a vectorial bent function with εFc = ε
+and (Fc)∗ = Gc for any c ∈ V (p)
+s
+\{0}. Since Fc is a weakly regular bent function, we have that
+Gc = (Fc)∗ is also weakly regular bent and G is vectorial bent. Thus, F is vectorial dual-bent
+with εFc = ε and (Fc)∗ = (F ∗)c for any c ∈ V (p)
+s
+\{0}, where F ∗ = G, that is, F satisfies
+Condition A.
+Based on Lemma 1, we have the following theorem.
+Theorem 1. Let F : V (p)
+n
+→ V (p)
+s
+be a vectorial dual-bent function satisfying Condition A.
+Define
+Ai = DF,i, i ∈ V (p)
+s
+,
+where DF,i = {x ∈ V (p)
+n
+: F(x) = i}. Then Γ = {Ai, i ∈ V (p)
+s
+} is a bent partition satisfying
+Condition C.
+Proof. By Lemma 1 and its proof, for any i ∈ V (p)
+s
+and u ∈ V (p)
+n ,
+χu(Ai) = χu(DF,i) = pn−sδ{0}(u) + εp
+n
+2 −s(psδ{0}(F ∗(−u) − i) − 1),
+where ε = 1 if p = 2 since all Boolean bent functions are regular. For any union S of ps−1 sets
+of {Ai : i ∈ V (p)
+s
+}, we have
+χu(S) =
+
+
+
+pn−1δ{0}(u) + εp
+n
+2 −1(p − 1), if AF ∗(−u) ⊆ S,
+pn−1δ{0}(u) − εp
+n
+2 −1, if AF ∗(−u) ⊈ S.
+(16)
+Let f be an arbitrary function such that for every j ∈ Fp, there are exactly ps−1 sets Ai in Γ in
+its preimage. Define g(u) = f(AF ∗(−u)). Note that g is a p-ary function from V (p)
+n
+to Fp. Then
+by (16), we have
+χu(Df,j) =
+
+
+
+pn−1δ{0}(u) + εp
+n
+2 −1(p − 1), if j = g(u),
+pn−1δ{0}(u) − εp
+n
+2 −1, if j ̸= g(u).
+(17)
+January 3, 2023
+DRAFT
+
+12
+By (17), for any u ∈ V (p)
+n
+we have
+Wf(−u) =
+�
+x∈V (p)
+n
+ζf(x)+⟨u,x⟩n
+p
+=
+�
+j∈Fp
+ζj
+p
+�
+x∈V (p)
+n
+:f(x)=j
+ζ⟨u,x⟩n
+p
+=
+�
+j∈Fp
+ζj
+pχu(Df,j)
+=
+�
+j∈Fp\{g(u)}
+ζj
+p(pn−1δ{0}(u) − εp
+n
+2 −1) + ζg(u)
+p
+(pn−1δ{0}(u) + εp
+n
+2 −1(p − 1))
+= (pn−1δ{0}(u) − εp
+n
+2 −1)
+�
+j∈Fp
+ζj
+p + εp
+n
+2 ζg(u)
+p
+= εp
+n
+2 ζg(u)
+p
+.
+(18)
+By (18) and ε = 1 if p = 2, f is a weakly regular bent function with εf = ε and f ∗(x) = g(−x).
+Let
+Wj = {u ∈ V (p)
+n
+: g(u) = j}, j ∈ Fp,
+then Wj, j ∈ Fp are pairwise disjoint and �
+j∈Fp Wj = V (p)
+n . By (17), for any u ∈ V (p)
+n
+and
+nonempty set J ⊆ Fp we have
+χu(Df,J) = pn−1δ{0}(u)|J| + εp
+n
+2 −1(pδWJ(u) − |J|),
+(19)
+where Df,J = {x ∈ V (p)
+n
+: f(x) ∈ J}, WJ = �
+j∈J Wj. By (19) and Lemma 1, f is vectorial dual-
+bent with (cf)∗ = c(βf ∗), c ∈ F∗
+p for some β ∈ F∗
+p (since all vectorial duals of f are cf ∗, c ∈ F∗
+p).
+Let c = 1, we obtain β = 1, that is, f is vectorial dual-bent with (cf)∗ = cf ∗, c ∈ F∗
+p. By
+Proposition 1, f is a (p − 1)-form. In particular, Fc is a (p − 1)-form for any c ∈ F∗
+ps, which
+yields that F(αx) = F(x) for any α ∈ F∗
+p and F∗
+pAi = Ai, i ∈ V (p)
+s
+. Hence, Γ is a bent partition
+satisfying Condition C.
+By Theorem 1, we have the following corollary.
+Corollary 1. Let n be an even positive integer. Let f : V (p)
+n
+→ Fp be a weakly regular bent
+function of (p − 1)-form, then {Df,j, j ∈ Fp} is a bent partition of V (p)
+n , where Df,j = {x ∈
+V (p)
+n
+: f(x) = j}.
+Proof. By Proposition 1, f is a weakly regular vectorial dual-bent function with (cf)∗ = cf ∗.
+Since n is even, εcf = εf for all c ∈ F∗
+p (see Theorem 1 of [6]). Then by Theorem 1, the
+conclusion holds.
+January 3, 2023
+DRAFT
+
+13
+A bent partition Γ = {A1, . . . , AK} of depth K is called coming from a normal bent partition
+if there is U ⊆ Ai for some i such that {U, A1, . . . , Ai−1, Ai\U, Ai+1, . . . , AK} is a normal bent
+partition. In [4], there is an open problem: Do bent partitions exist which are not coming from
+a normal bent partition of depth K > 2? In the following, we provide a positive answer for
+this open problem. By the definition of l-form, a ternary function f is a 2-form if and only if
+f(x) = f(−x). Let n be an even positive integer. If f : V (3)
+n
+→ F3 with f(x) = f(−x) is a
+ternary weakly regular but not regular bent function (that is, εf = −1), then by Corollary 1,
+{Df,0, Df,1, Df,2} is a bent partition of depth 3. There exist such ternary bent functions f, for
+instance see [7], [17]:
+•
+f(x) = Trn
+1(αx2), x ∈ F3n,
+(20)
+where n is even, α ∈ F∗
+3n is a square element if 4 | n, and α ∈ F∗
+3n is a non-square element
+if 4 ∤ n;
+•
+f(x) = Trn
+1(ax
+3n−1
+4
++3m+1), x ∈ F3n,
+(21)
+where n = 2m, m odd, a = α
+3m+1
+4
+for a primitive element α of F3n;
+•
+f(x) = Trn
+1(α(x33k+32k−3k+1 + x2)), x ∈ F3n,
+(22)
+where n = 4k for an arbitrary positive integer k, α ∈ F∗
+32k;
+•
+f(x, y, z) = (g(x) − h(x))z2 + yz + g(x), (x, y, z) ∈ F3n × F3 × F3,
+(23)
+where n is even, g and h are distinct bent functions constructed by (20) or (22) if 4 | n, g
+and h are distinct bent functions constructed by (20) or (21) if 4 ∤ n.
+For any ternary weakly regular but not regular bent function f : V (3)
+n
+→ F3 (n even) with
+f(x) = f(−x), the corresponding bent partition {Df,0, Df,1, Df,2} is not coming from a normal
+bent partition by Theorem 4 (i) of [4], which provides a positive answer for the above open
+problem proposed in [4]. We first recall Theorem 4 (i) of [4] and then give an example to
+illustrate this fact.
+Lemma 2 ( [4]). Let Γ = {U, A1, . . . , AK} be a normal bent partition of V (p)
+n . Then |U| = p
+n
+2
+and |Aj| = pn−p
+n
+2
+K
+, 1 ≤ j ≤ K.
+January 3, 2023
+DRAFT
+
+14
+Example 1. Let f : F34 → F3 be defined by f(x) = Tr4
+1(x2). Then f is ternary weakly regular
+bent with f(x) = f(−x) and εf = −1. By Corollary 1, {Df,0, Df,1, Df,2} is a bent partition.
+By the result of Nyberg [20], for any weakly regular p-ary bent function g : V (p)
+n
+→ Fp with
+n even, we have {|Dg,i|, i ∈ Fp} = {pn−1 + εfp
+n
+2 −1(p − 1), pn−1 − εfp
+n
+2 −1}. For our example,
+|Df,0| = 21, |Df,1| = |Df,2| = 30. By Lemma 2, it is easy to see that {Df,0, Df,1, Df,2} can not
+be from a normal bent partition.
+In the following, based on Theorem 1, we give an alternative proof that Γi, Γ•
+i , Θi, i = 1, 2
+defined by (5)-(10) given in [3] are bent partitions.
+Let s, n be positive integers with s | n, u be an integer with u ≡ pj0 mod (ps − 1) for some
+0 ≤ j0 ≤ s − 1 and gcd(u, pn − 1) = 1, and let d be an integer with du ≡ 1 mod (pn − 1). Let
+P = (Fpn, +, ◦) be a (pre)semifield such that its dual P ⋆ = (Fpn, +, ⋆) is right Fps-linear. For
+given x ∈ Fpn, if x = 0, then let ηx = 0, and if x ̸= 0, then let ηx be given by x ⋆ η−1
+x
+= 1 (For
+convention we set η−1
+0
+= ηpn−2
+0
+= 0). For any α ∈ F∗
+pn and i0 ∈ Fps, define
+F(x, y) = Trn
+s (αa(x, y)) + i0(1 − xpn−1), (x, y) ∈ Fpn × Fpn,
+(24)
+where for given (x, y), if x = 0, then a(x, y) = 0, and if x ̸= 0, then a(x, y) is given by
+a(x, y) ◦ xu = y, and
+F •(x, y) = Trn
+s (αa•(x, y)) + i0(1 − xpn−1), (x, y) ∈ Fpn × Fpn,
+(25)
+where for given (x, y), if x = 0, then a•(x, y) = 0, and if x ̸= 0, then a•(x, y) is given by
+xu ◦ a•(x, y) = y, and
+G(x, y) = Trn
+s (αb(x, y)) + i0(1 − ypn−1), (x, y) ∈ Fpn × Fpn,
+(26)
+where for given (x, y), if y = 0, then b(x, y) = 0, and if y ̸= 0, then b(x, y) is given by
+b(x, y) ◦ yd = x, and
+G•(x, y) = Trn
+s (αb•(x, y)) + i0(1 − ypn−1), (x, y) ∈ Fpn × Fpn,
+(27)
+where for given (x, y), if y = 0, then b•(x, y) = 0, and if y ̸= 0, then b•(x, y) is given by
+yd ◦ b•(x, y) = x, and
+M(x, y) = Trn
+s (αη−u
+x y) + i0(1 − xpn−1), (x, y) ∈ Fpn × Fpn,
+(28)
+and
+N(x, y) = Trn
+s (αxη−d
+y ) + i0(1 − ypn−1), (x, y) ∈ Fpn × Fpn.
+(29)
+January 3, 2023
+DRAFT
+
+15
+Proposition 3. Let F, F •, G, G•, M, N be defined as above. Then they are all vectorial dual-bent
+functions satisfying Condition A with ε = 1.
+Proof. We only prove the result for F and M since the proofs for F •, G, G• are similar to the
+proof for F, and the proof for N is similar to the proof for M.
+• For F:
+For any c ∈ F∗
+ps, we have
+Fc(x, y) = Trn
+1(cαa(x, y)) + Trs
+1(ci0)(1 − xpn−1).
+For any c ∈ F∗
+ps and (w, v) ∈ Fpn × Fpn, we have
+WFcu(w, v) =
+�
+x∈F∗
+pn
+�
+y∈Fpn
+ζTrn
+1 (cuαa(x,y))−Trn
+1 (wx+vy)
+p
++ ζTrs
+1(cui0)
+p
+�
+y∈Fpn
+ζ−Trn
+1 (vy)
+p
+=
+�
+x∈Fpn
+�
+y∈Fpn
+ζTrn
+1 (cuαa(x,y))−Trn
+1 (wx+vy)
+p
++ pn(ζTrs
+1(cui0)
+p
+− 1)δ{0}(v)
+= Wh(w, v) + pn(ζTrs
+1(cui0)
+p
+− 1)δ{0}(v),
+where h(x, y) = Trn
+1(cuαa(x, y)). For given x ∈ Fpn, if x = 0, then let λx = 0, and if
+x ̸= 0, then let λx be given by x ⋆ λ−1
+x
+= α (For convention we set λ−1
+0
+= λpn−2
+0
+= 0). Define
+ρ(x) = λ−d
+x . Then ρ is a permutation over Fpn. For any x ∈ F∗
+pn, set z = ρ−1(c−1x). Then
+λ−d
+z
+= ρ(z) = c−1x. By du ≡ 1 mod (pn − 1), we have λ−1
+z
+= c−uxu. Since z ̸= 0 and P ⋆ is
+right Fps-linear, we have α = z ⋆ λ−1
+z
+= z ⋆ (c−uxu) = c−u(z ⋆ xu), that is, ρ−1(c−1x) ⋆ xu =
+αcu for any x ̸= 0. Thus, when x ̸= 0, Trn
+1(cuαa(x, y)) = Trn
+1(a(x, y)(ρ−1(c−1x) ⋆ xu)) =
+Trn
+1(ρ−1(c−1x)(a(x, y) ◦ xu)) = Trn
+1(ρ−1(c−1x)y). When x = 0, by a(0, y) = ρ−1(0) = 0, we
+have Trn
+1(cuαa(x, y)) = Trn
+1(ρ−1(c−1x)y) = 0. Hence, h(x, y) = Trn
+1(ρ−1(c−1x)y), which is a
+Maiorana-McFarland bent function and by (4),
+Wh(w, v) = pnζ−Trn
+1 (cwρ(v))
+p
+.
+Therefore, for any c ∈ F∗
+ps,
+WFcu(w, v) = pn(ζ−Trn
+1 (cwρ(v))
+p
++ (ζTrs
+1(cui0)
+p
+− 1)δ{0}(v))
+= pnζ−Trn
+1 (cwρ(v))+Trs
+1(cui0)(1−vpn−1)
+p
+.
+(30)
+By (30) and ud ≡ 1 mod (pn − 1), we have that for any c ∈ F∗
+ps, Fc is a regular bent function
+with
+(Fc)∗(x, y) = −Trn
+1(cdxρ(y)) + Trs
+1(ci0)(1 − ypn−1)
+= −Trn
+1(cdpj0(xρ(y))pj0) + Trs
+1(ci0)(1 − ypn−1).
+January 3, 2023
+DRAFT
+
+16
+Since u ≡ pj0 mod (ps − 1) and du ≡ 1 mod (pn − 1), we have d ≡ ps−j0 mod (ps − 1) and
+thus (cd)pj0 = c for any c ∈ F∗
+ps. Therefore, F is a vectorial bent function with εFc = 1 and
+(Fc)∗ = Hc for all c ∈ F∗
+ps, where
+H(x, y) = −Trn
+s ((xρ(y))pj0) + i0(1 − ypn−1) = −(Trn
+s (xρ(y)))pj0 + i0(1 − ypn−1).
+Since Fc is regular bent, we have that (Fc)∗ = Hc is also regular bent and H is vectorial bent.
+Thus, F is vectorial dual-bent with εFc = 1 and (Fc)∗ = (F ∗)c for all c ∈ F∗
+ps, where F ∗ = H,
+that is, F satisfies Condition A.
+• For M:
+For any c ∈ F∗
+ps,
+Mc(x, y) = Trn
+1(cαη−u
+x y) + Trs
+1(ci0)(1 − xpn−1).
+Similar to the discussion for F, for any c ∈ F∗
+ps and (w, v) ∈ Fpn × Fpn we have
+WMc(w, v) = Wg(w, v) + pn(ζTrs
+1(ci0)
+p
+− 1)δ{0}(v),
+where g(x, y) = Trn
+1(cαη−u
+x y). Let π(x) = η−u
+x , then π is a permutation over Fpn. Since g is a
+Maiorana-McFarland bent function, then by (4),
+Wg(w, v) = pnζ−Trn
+1 (wπ−1(c−1α−1v))
+p
+.
+For any given y ∈ F∗
+pn, set π−1(c−1α−1y) = z. Then c−1α−1y = π(z) = η−u
+z . By ud ≡
+1 mod (pn − 1), we have η−1
+z
+= c−dα−dyd. Since z ̸= 0 and P ⋆ is right Fps-linear, we have
+1 = z ⋆ η−1
+z
+= z ⋆ (c−dα−dyd) = c−d(z ⋆ α−dyd), that is, π−1(c−1α−1y) ⋆ α−dyd = cd. For given
+(x, y) ∈ Fpn × Fpn, if y = 0, then let r(x, y) = 0, and if y ̸= 0, then let r(x, y) be given by
+r(x, y)◦α−dyd = x. When v ̸= 0, we have Trn
+1(wπ−1(c−1α−1v)) = Trn
+1(π−1(c−1α−1v)(r(w, v)◦
+α−dvd)) = Trn
+1(r(w, v)(π−1(c−1α−1v) ⋆ α−dvd)) = Trn
+1(cdr(w, v)) = Trn
+1(c(r(w, v))pj0). When
+v = 0, since π−1(0) = 0 and r(w, 0) = 0, we have Trn
+1(wπ−1(c−1α−1v)) = Trn
+1(c(r(w, v))pj0) =
+0. Thus, −Trn
+1(wπ−1(c−1α−1v)) = −Trn
+1(c(r(w, v))pj0) and
+WMc(w, v) = pn(ζ−Trn
+1 (c(r(w,v))pj0 )
+p
++ (ζTrs
+1(ci0)
+p
+− 1)δ{0}(v))
+= pnζ−Trn
+1 (c(r(w,v))pj0 )+Trs
+1(ci0)(1−vpn−1)
+p
+,
+which implies that M is a vectorial dual-bent function with εMc = 1 and (Mc)∗ = (M∗)c for all
+c ∈ F∗
+ps, where
+M∗(x, y) = −Trn
+s ((r(x, y))pj0) + i0(1 − ypn−1),
+January 3, 2023
+DRAFT
+
+17
+that is, M satisfies Condition A.
+By Theorem 1 and Proposition 3, we have that {DF,i, i ∈ Fps}, {DF •,i, i ∈ Fps}, {DG,i, i ∈
+Fps}, {DG•,i, i ∈ Fps}, {DM,i, i ∈ Fps} and {DN,i, i ∈ Fps} are bent partitions. It is easy to
+verify that
+DF,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+Ut, if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+Ut
+�
+U, if i = i0,
+, DF •,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+U •
+t , if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+U •
+t
+�
+U, if i = i0,
+,
+DG,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+Vt, if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+Vt
+�
+V, if i = i0,
+, DG•,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+V •
+t , if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+V •
+t
+�
+V, if i = i0,
+,
+DM,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+Xt, if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+Xt
+�
+X, if i = i0,
+, DN,i =
+
+
+
+
+
+
+
+
+
+
+
+�
+t∈Fpn:T rn
+s (αt)=i
+Yt, if i ̸= i0,
+�
+t∈Fpn:T rn
+s (αt)=i0
+Yt
+�
+Y, if i = i0,
+where
+Ut = {(x, t ◦ xu) : x ∈ F∗
+pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn},
+U•
+t = {(x, xu ◦ t) : x ∈ F∗
+pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn},
+Vt = {(t ◦ xd, x) : x ∈ F∗
+pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn},
+V •
+t = {(xd ◦ t, x) : x ∈ F∗
+pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn},
+Xt = {(tηd
+x, x) : x ∈ F∗
+pn} if t ∈ Fpn, and X = {(x, 0) : x ∈ Fpn},
+Yt = {(x, tηu
+x) : x ∈ F∗
+pn} if t ∈ Fpn, and Y = {(0, y) : y ∈ Fpn}.
+For the above bent partitions from vectorial dual-bent functions F, F •, G, G•, M, N, by set-
+ting α = 1, u = ps + p − 1 with gcd(u, pn − 1) = 1, then we can obtain bent partitions
+Γ1, Γ•
+1, Γ2, Γ•
+2, Θ1, Θ2 defined by (5)-(10) respectively. Thus by the above analysis, we provide
+an alternative derivation that Γ1, Γ•
+1, Γ2, Γ•
+2, Θ1, Θ2 are bent partitions.
+When p is an odd prime, we show that the converse of Theorem 1 also holds.
+Theorem 2. Let p be an odd prime. Let Γ = {Ai, i ∈ V (p)
+s
+} be a bent partition of V (p)
+n
+satisfying
+Condition C. Define F : V (p)
+n
+→ V (p)
+s
+by
+F(x) = i if x ∈ Ai.
+Then F is a vectorial dual-bent function satisfying Condition A.
+January 3, 2023
+DRAFT
+
+18
+Proof. Since F∗
+pAi = Ai for any i ∈ V (p)
+s
+, all bent functions constructed from Γ are (p−1)-form.
+When s = 1, the conclusion follows from Proposition 1. In the following, we consider the case
+of s ≥ 2.
+Let f be an arbitrary bent function constructed from Γ. By Lemma 3.4 of [13], for any
+u ∈ V (p)
+n
+and j ∈ Fp we have
+χu(Df,j) =
+
+
+
+pn−1δ{0}(u) + εp
+n
+2 −1(p − 1), if f ∗(u) = j,
+pn−1δ{0}(u) − εp
+n
+2 −1, if f ∗(u) ̸= j,
+(31)
+where Df,j = {x ∈ V (p)
+n
+: f(x) = j}, j ∈ Fp. For any fixed u ∈ V (p)
+n , since
+{χu(Df,j), j ∈ Fp} = {pn−1δ{0}(u) + εp
+n
+2 −1(p − 1), pn−1δ{0}(u) − εp
+n
+2 −1}
+for any bent function f constructed from Γ, we have that for any fixed u ∈ V (p)
+n , there exists a
+unique G(u) ∈ V (p)
+s
+such that χu(Ai), i ̸= G(u) are all the same and χu(Ai) ̸= χu(AG(u)), i ̸=
+G(u). Note that G is a function from V (p)
+n
+to V (p)
+s
+. Moreover by (31), for any fixed u ∈ V (p)
+n
+we have
+χu(Ai) =
+
+
+
+pn−sδ{0}(u) + εp
+n
+2 −s(ps − 1), if i = G(u),
+pn−sδ{0}(u) − εp
+n
+2 −s, if i ̸= G(u).
+(32)
+Define
+Wi = {u ∈ V (p)
+n
+: G(u) = i}, i ∈ V (p)
+s
+.
+Then obviously Wi, i ∈ V (p)
+s
+are pairwise disjoint and �
+i∈V (p)
+s
+Wi = V (p)
+n . By (32), for any
+u ∈ V (p)
+n
+and nonempty set I ⊆ V (p)
+s
+we have
+χu(DF,I) =
+�
+i∈I
+χu(Ai) = pn−sδ{0}(u)|I| + εp
+n
+2 −s(psδWI(u) − |I|),
+(33)
+where DF,I = {x ∈ V (p)
+n
+: F(x) ∈ I}, WI = �
+i∈I Wi. By (33) and Lemma 1, the conclusion
+holds.
+When p is an odd prime, from Theorems 1 and 2 we obtain a characterization of bent partitions
+satisfying Condition C in terms of vectorial dual-bent functions.
+Theorem 3. Let p be an odd prime. Let Γ = {Ai, i ∈ V (p)
+s
+} be a partition of V (p)
+n , where n is
+even, s ≤ n
+2. Define F : V (p)
+n
+→ V (p)
+s
+as
+F(x) = i if x ∈ Ai.
+Then Γ is a bent partition satisfying Condition C if and only if F is a vectorial dual-bent function
+satisfying Condition A.
+January 3, 2023
+DRAFT
+
+19
+IV. CONSTRUCTING BENT PARTITIONS FROM VECTORIAL DUAL-BENT FUNCTIONS
+In this section, we construct bent partitions from vectorial dual-bent functions.
+The following theorem provides a secondary construction of vectorial dual-bent functions,
+which can be used to generate more bent partitions.
+Theorem 4. Let n, m, s be positive integers for which n is even and s ≤ n
+2, s | m, s ̸= m. For
+any i ∈ Fps, let F(i; x) : V (p)
+n
+→ Fps be a vectorial dual-bent function with ((F(i; x))c)∗ =
+((F(i; x))∗)c and ε(F (i;x))c = ε for any c ∈ F∗
+ps, where (F(i; x))∗ is a vectorial dual of F(i; x)
+and ε ∈ {±1} is a constant independent of i, c. Let α, β ∈ Fpm be linearly independent over Fps.
+Let R be a permutation over Fpm with R(0) = 0 and T : Fps → Fps be an arbitrary function.
+Define H : V (p)
+n
+× Fpm × Fpm → Fps as
+H(x, y1, y2) = F(T rm
+s (αR(y1ypm−2
+2
+)); x) + T rm
+s (βR(y1ypm−2
+2
+)) + T (T rm
+s (αR(y1ypm−2
+2
+))).
+Then H is a vectorial dual-bent function satisfying Condition A and Γ = {Ai, i ∈ Fps} is a bent
+partition satisfying Condition C, where Ai = {(x, y1, y2) ∈ V (p)
+n
+×Fpm ×Fpm : H(x, y1, y2) = i}.
+Proof. Denote
+d(y) = Trm
+s (βR(y1ypm−2
+2
+)), e(y) = Trm
+s ((β − α)R(y1ypm−2
+2
+)), y = (y1, y2) ∈ Fpm × Fpm.
+For any c ∈ F∗
+ps and (a, b) = (a, b1, b2) ∈ V (p)
+n
+× Fpm × Fpm, we have
+WHc(a, b)
+=
+�
+x∈V (p)
+n
+�
+y=(y1,y2)∈Fpm×Fpm
+ζT rs
+1(cF (d(y)−e(y);x))+T rs
+1(cd(y))+T rs
+1(cT (d(y)−e(y)))
+p
+ζ−⟨a,x⟩n−T rm
+1 (b1y1+b2y2)
+p
+=
+�
+i∈Fps
+�
+y=(y1,y2)∈Fpm×Fpm:d(y)−e(y)=i
+�
+x∈V (p)
+n
+ζT rs
+1(cF (i;x))+T rs
+1(cd(y))+T rs
+1(cT (i))
+p
+ζ−⟨a,x⟩n−T rm
+1 (b1y1+b2y2)
+p
+= p−s �
+i∈Fps
+W(F (i;x))c(a)ζT rs
+1(cT (i))
+p
+�
+y=(y1,y2)∈Fpm×Fpm
+ζT rs
+1(cd(y))−T rm
+1 (b1y1+b2y2)
+p
+�
+j∈Fps
+ζT rs
+1(cj(i−(d(y)−e(y))))
+p
+= p−s �
+i∈Fps
+W(F (i;x))c(a)ζT rs
+1(cT (i))
+p
+�
+j∈Fps
+ζT rs
+1(ijc)
+p
+�
+y=(y1,y2)∈Fpm×Fpm
+ζT rs
+1(c((1−j)d(y)+je(y)))−T rm
+1 (b1y1+b2y2)
+p
+.
+By Theorem 3 of [10], for any j ∈ Fps, J(j; y) = (1−j)d(y)+je(y) is a partial spread vectorial
+dual-bent function with ε(J(j;y))c = 1 and ((J(j; y))c)∗ = ((1−j)d∗(y)+je∗(y))c for any c ∈ F∗
+ps,
+January 3, 2023
+DRAFT
+
+20
+where d∗(y) = Trm
+s (βR(−ypm−2
+1
+y2)), e∗(y) = Trm
+s ((β − α)R(−ypm−2
+1
+y2)). Therefore,
+WHc(a, b)
+= pm−s �
+i∈Fps
+W(F (i;x))c(a)ζT rs
+1(cT (i))
+p
+�
+j∈Fps
+ζT rs
+1(ijc)
+p
+ζT rs
+1(c((1−j)d∗(b)+je∗(b)))
+p
+= pm−sζT rs
+1(cd∗(b))
+p
+�
+i∈Fps
+W(F (i;x))c(a)ζT rs
+1(cT (i))
+p
+�
+j∈Fps
+ζT rs
+1(cj(i−(d∗(b)−e∗(b))))
+p
+= pmζT rs
+1(cd∗(b))
+p
+W(F (d∗(b)−e∗(b);x))c(a)ζT rs
+1(cT (d∗(b)−e∗(b)))
+p
+= εp
+n
+2 +mζ
+((F (T rm
+s (αR(−bpm−2
+1
+b2));x))c)∗(a)+T rs
+1(cT rm
+s (βR(−bpm−2
+1
+b2)))+T rs
+1(cT (T rm
+s (αR(−bpm−2
+1
+b2))))
+p
+= εp
+n
+2 +mζ((F (T rm
+s (αR(−bpm−2
+1
+b2));x))∗)c(a)+T rs
+1(cT rm
+s (βR(−bpm−2
+1
+b2)))+T rs
+1(cT (T rm
+s (αR(−bpm−2
+1
+b2))))
+p
+.
+(34)
+Note that ε = 1 if p = 2 since all Boolean bent functions are regular. By (34), H is a vectorial
+bent function with (Hc)∗ = Gc and εHc = ε for any c ∈ F∗
+ps, where
+G(a, b1, b2) = (F(T rm
+s (αR(−bpm−2
+1
+b2)); x))∗(a) + T rm
+s (βR(−bpm−2
+1
+b2)) + T (T rm
+s (αR(−bpm−2
+1
+b2))).
+Since Hc is weakly regular bent, we have that Gc = (Hc)∗ is also weakly regular bent and G is
+vectorial bent. Thus, H is vectorial dual-bent with (Hc)∗ = (H∗)c and εHc = ε for any c ∈ F∗
+ps,
+where H∗ = G, that is, H satisfies Condition A. By Theorem 1, the partition Γ generated from
+H is a bent partition satisfying Condition C.
+The following explicit construction of bent partitions is an immediate result of Proposition 3
+and Theorem 4.
+Theorem 5. Let n, m, s be positive integers with s | n, s | m, s ̸= m, and ui, i ∈ Fps be
+integers for which for any i ∈ Fps, ui ≡ pji mod (ps − 1) for some 0 ≤ ji ≤ s − 1 and
+gcd(ui, pn − 1) = 1. For any i ∈ Fps, let di be an integer with uidi ≡ 1 mod (pn − 1), and
+Pi = (Fpn, +, ◦i) be a (pre)semifield for which its dual P ⋆
+i is right Fps-linear. For any i ∈ Fps,
+let F(i; x1, x2) : Fpn × Fpn → Fps be an arbitrary vectorial dual-bent function constructed by
+Proposition 3 with u = ui, d = di, P = Pi. Let α, β ∈ Fpm be linearly independent over Fps, R
+be a permutation over Fpm with R(0) = 0 and T : Fps → Fps be an arbitrary function. Define
+H : Fpn × Fpn × Fpm × Fpm → Fps as
+H(x1, x2, y1, y2) = F(T rm
+s (αR(y1ypm−2
+2
+)); x1, x2) + T rm
+s (βR(y1ypm−2
+2
+)) + T (T rm
+s (αR(y1ypm−2
+2
+))).
+Then
+Γ = {Ai, i ∈ Fps}
+is a bent partition satisfying Condition C, where
+Ai = {(x1, x2, y1, y2) ∈ Fpn × Fpn × Fpm × Fpm : H(x1, x2, y1, y2) = i}.
+January 3, 2023
+DRAFT
+
+21
+Remark 3. With the same notation as in Theorem 4. Note that in Theorem 4, by setting vectorial
+dual-bent functions H constructed by Theorem 5 as building blocks (that is, as F(i; x)), we can
+obtain more explicit vectorial dual-bent functions which can generate more bent partitions by
+Theorem 4.
+We give an example by using Theorem 5.
+Example 2. Let p = 3, s = 4, n = m = 8. Let α be a primitive element of F38 and β = 1, R be
+the identity map and T = 0. For any i ∈ F34, let
+F(i; x1, x2) =
+
+
+
+Tr8
+4(x−89
+1
+x2), if i ∈ F∗
+34,
+Tr8
+4(x1x−83
+2
+), if i = 0.
+Then
+H(x1, x2, y2, y2) = (Tr8
+4(αy1y6559
+2
+))80(Tr8
+4(x−89
+1
+x2 − x1x−83
+2
+)) + Tr8
+4(x1x−83
+2
++ y1y6559
+2
+),
+and Γ = {DH,i, i ∈ F34} is a bent partition satisfying Condition C, where DH,i = {(x1, x2, y1, y2) ∈
+(F38)4 : H(x1, x2, y1, y2) = i}.
+V. RELATIONS BETWEEN BENT PARTITIONS AND PARTIAL DIFFERENCE SETS
+In this section, by taking vectorial dual-bent functions as the link between bent partitions and
+partial difference sets, we give a sufficient condition on constructing partial difference sets from
+bent partitions. When p is an odd prime, we characterize bent partitions satisfying Condition C
+in terms of partial difference sets.
+Definition 4. Let (G, +) be a finite abelian group of order v and D be a subset of G with k
+elements. Then D is called a (v, k, λ, µ) partial difference set of G, if the expressions d1 − d2,
+for d1 and d2 in D with d1 ̸= d2, represent each nonzero element in D exactly λ times, and
+represent each nonzero element in G \ D exactly µ times. When λ = µ, then D is called a
+(v, k, λ) difference set.
+Note that if D is a partial difference set of G with −D = D, then so are D∪{0}, D \ {0}, G \ D
+(see [16]). There is an important tool to characterize partial difference sets in terms of characters.
+January 3, 2023
+DRAFT
+
+22
+Lemma 3 ( [16]). Let G be an abelian group of order v. Suppose that D is a subset of G with
+k elements which satisfies −D = D and 0 /∈ D. Then D is a (v, k, λ, µ) partial difference set
+if and only if for each non-principal character χ of G,
+χ(D) = β ±
+√
+∆
+2
+,
+where χ(D) = �
+x∈D χ(x), β = λ − µ, γ = k − µ, ∆ = β2 + 4γ.
+When p is an odd prime or s ≥ 2, we give the value distribution of vectorial dual-bent
+functions satisfying Condition A.
+Proposition 4. Let F : V (p)
+n
+→ V (p)
+s
+be a vectorial dual-bent function satisfying Condition A,
+where p is odd or s ≥ 2. Then
+|DF,F (0)| = pn−s + εp
+n
+2 −s(ps − 1), |DF,i| = pn−s − εp
+n
+2 −s if i ̸= F(0).
+Proof. Note that if f is a weakly regular p-ary bent function, then for any a ∈ Fp, f − a is
+a weakly regular bent function with (f − a)∗ = f ∗ − a and εf−a = εf. Since F is a vectorial
+dual-bent function with (Fc)∗ = (F ∗)c, c ∈ V (p)
+s
+\{0}, we have that F(x) − F(0) is a vectorial
+bent function and for any c ∈ V (p)
+s
+\{0},
+((F − F(0))c)∗ = (Fc)∗ − ⟨c, F(0)⟩s = (F ∗)c − ⟨c, F(0)⟩s = (F ∗ − F(0))c,
+which implies that F(x) − F(0) is a vectorial dual-bent function with ((F − F(0))c)∗ = (F ∗ −
+F(0))c and ε(F −F (0))c = ε for any c ∈ V (p)
+s
+\{0}. By the proof of Theorem 1, F(ax) = F(x) for
+any a ∈ F∗
+p and thus F(x) = F(−x). By Corollary 1 of [22] (Note that although Corollary 1 of
+[22] only considers the case of p being odd, the conclusion of Corollary 1 of [22] also holds
+for p = 2, s ≥ 2), we have
+|DF −F (0),0| = pn−s + εp
+n
+2 −s(ps − 1), |DF −F (0),i| = pn−s − εp
+n
+2 −s if i ̸= 0,
+that is,
+|DF,F (0)| = pn−s + εp
+n
+2 −s(ps − 1), |DF,i| = pn−s − εp
+n
+2 −s if i ̸= F(0).
+In the following, we give a characterization of vectorial dual-bent functions satisfying Con-
+dition A in terms of partial difference sets.
+January 3, 2023
+DRAFT
+
+23
+Theorem 6. Let n be an even positive integer, s be a positive integer with s ≤ n
+2, and F :
+V (p)
+n
+→ V (p)
+s
+. The following two statements are equivalent.
+(1) F is a vectorial dual-bent function satisfying Condition A.
+(2) When p = 2, s = 1, then the support supp(F) of F defined as supp(F) = {x ∈ V (2)
+n
+:
+F(x) = 1} is a (2n, 2n−1 ± 2
+n
+2 −1, 2n−2 ± 2
+n
+2 −1) difference set, and when p is odd or s ≥ 2,
+then for any nonempty set I ⊆ V (p)
+s
+, DF,I\{0} is a (pn, k, λ, µ) partial difference set for which
+−DF,I = DF,I and if F(0) ∈ I, then
+k = pn−s|I| + εp
+n
+2 −s(ps − |I|) − 1,
+λ = pn−2s|I|2 + εp
+n
+2 −s(ps − |I|) − 2,
+µ = pn−2s|I|2 + εp
+n
+2 −s|I|,
+(35)
+and if F(0) /∈ I, then
+k = pn−s|I| − εp
+n
+2 −s|I|,
+λ = pn−2s|I|2 + εp
+n
+2 −s(ps − 3|I|),
+µ = pn−2s|I|2 − εp
+n
+2 −s|I|,
+(36)
+where DF,I = {x ∈ V (p)
+n
+: F(x) ∈ I} and ε ∈ {±1} is a constant (ε = 1 if p = 2).
+Proof. It is easy to see that a Boolean function F is a vectorial dual-bent function satisfying
+Condition A if and only if F is bent, that is, Condition A is trivial for any Boolean bent function.
+By the well-known result that a Boolean function F : V (2)
+n
+→ F2 is bent if and only if its support
+supp(F) = {x ∈ V (2)
+n
+: F(x) = 1} is a (2n, 2n−1 ± 2
+n
+2 −1, 2n−2 ± 2
+n
+2 −1) difference set (see [11]),
+the conclusion obviously holds for p = 2, s = 1. In the following, we prove the conclusion for
+p being odd or s ≥ 2.
+(1) ⇒ (2): By the proof of Theorem 1, F(−x) = F(x), that is, −DF,I = DF,I. For any
+u ∈ V (p)
+n \{0}, with the same argument as in the proof of Theorem 2 of [22],
+χu(DF,I) =
+
+
+
+εp
+n
+2 − εp
+n
+2 −s|I|, if F ∗(−u) ∈ I,
+−εp
+n
+2 −s|I|, if F ∗(−u) /∈ I.
+where ε = 1 if p = 2 since all Boolean bent functions are regular.
+If F(0) ∈ I, then |DF,I\{0}| = |DF,I|−1 and χu(DF,I\{0}) = χu(DF,I)−1. By Proposition 4,
+|DF,I\{0}| = (|I|−1)(pn−s−εp
+n
+2 −s)+(pn−s+εp
+n
+2 −s(ps−1)−1) = pn−s|I|+εp
+n
+2 −s(ps−|I|)−1.
+By Lemma 3, DF,I\{0} is a (pn, k, λ, µ) partial difference set, where k, λ, µ are given in (35).
+January 3, 2023
+DRAFT
+
+24
+If F(0) /∈ I, then |DF,I\{0}| = |DF,I| and χu(DF,I\{0}) = χu(DF,I). By Proposition 4,
+|DF,I\{0}| = |I|(pn−s − εp
+n
+2 −s). By Lemma 3, DF,I\{0} is a (pn, k, λ, µ) partial difference set,
+where k, λ, µ are given in (36).
+(2) ⇒ (1): By Lemma 3, for any u ∈ V (p)
+n
+and nonempty set I ⊆ V (p)
+s
+we have
+χu(DF,I) = pn−sδ{0}(u)|I| + εp
+n
+2 − εp
+n
+2 −s|I| or χu(DF,I) = pn−sδ{0}(u)|I| − εp
+n
+2 −s|I|.
+(37)
+For any i ∈ V (p)
+s
+, define Wi = {u ∈ V (p)
+n
+: χu(DF,i) = pn−sδ{0}(u) + εp
+n
+2 − εp
+n
+2 −s}, where
+DF,i = {x ∈ V (p)
+n
+: F(x) = i}. We claim that Wi
+� Wi′ = ∅ for any i ̸= i′ and �
+i∈V (p)
+s
+Wi = V (p)
+n .
+Indeed, if there exist i ̸= i′ such that Wi
+� Wi′ ̸= ∅, that is, there exists u ∈ V (p)
+n
+such that
+χu(DF,i) = χu(DF,i′) = pn−sδ{0}(u) + εp
+n
+2 − εp
+n
+2 −s, then χu(DF,i
+� DF,i′) = 2pn−sδ{0}(u) +
+2εp
+n
+2 − 2εp
+n
+2 −s, which contradicts with (37). Thus, Wi
+� Wi′ = ∅ for any i ̸= i′. If there exists
+u ∈ V (p)
+n
+such that u /∈ Wi for any i ∈ V (p)
+s
+, that is, χu(DF,i) = pn−sδ{0}(u) − εp
+n
+2 −s for
+any i ∈ V (p)
+s
+, then χu(V (p)
+n ) = �
+i∈V (p)
+s
+χu(DF,i) = pnδ{0}(u) − εp
+n
+2 , which contradicts with
+χu(V (p)
+n ) = �
+x∈V (p)
+n
+ζ⟨u,x⟩n
+p
+= pnδ{0}(u). Thus, �
+i∈V (p)
+s
+Wi = V (p)
+n . By the above analysis, we
+have
+χu(DF,I) = pn−sδ{0}(u)|I| + εp
+n
+2 −s(psδWI(u) − |I|),
+(38)
+where WI = �
+i∈I Wi. By (38) and Lemma 1, F is a vectorial dual-bent function satisfying
+Condition A.
+The following theorem provides a sufficient condition on constructing partial difference sets
+from bent partitions.
+Theorem 7. Let n be an even positive integer and s be a positive integer with s ≤ n
+2. Assume
+that Γ = {Ai, i ∈ V (p)
+s
+} is a bent partition of V (p)
+n
+for which the function F : V (p)
+n
+→ V (p)
+s
+defined by
+F(x) = i if x ∈ Ai
+is a vectorial dual-bent function satisfying Condition A. Then when p = 2, s = 1, A0 and A1 are
+(2n, 2n−1 ± 2
+n
+2 −1, 2n−2 ± 2
+n
+2 −1) difference set and (2n, 2n−1 ∓ 2
+n
+2 −1, 2n−2 ∓ 2
+n
+2 −1) difference set,
+respectively, and when p is odd or s ≥ 2, for any nonempty set I ⊆ V (p)
+s
+, AI\{0} = �
+i∈I Ai\{0}
+is a (pn, k, λ, µ) partial difference set, where (k, λ, µ) are given in (35) if 0 ∈ AI and (k, λ, µ)
+are given in (36) if 0 /∈ AI.
+January 3, 2023
+DRAFT
+
+25
+Proof. Note that if D is a (v, k, λ) difference set of a finite abelian group G, then G\D is a
+(v, v − k, v − 2k + λ) difference set of G (for instance see [12]). Then the result follows from
+Theorem 6.
+Remark 4. By Proposition 3, the bent partition Γ1 (resp. Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2) satisfies the
+condition in Theorem 7. By Theorem 7, any union of sets from Γ1 (resp, Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2)
+forms a partial difference set. Thus, the results given in Corollary 15 of [1] on constructing
+partial difference sets from Γ1 (resp. Γ2, Γ•
+1, Γ•
+2, Θ1, Θ2) (which includes the results given in
+Theorem 2 of [2] on constructing partial difference sets from Γ1, resp. Γ2, in the finite field)
+can also be illustrated by our results.
+Since the bent partitions constructed in Theorem 5 satisfy the condition in Theorem 7, we
+have the following corollary from Theorem 7.
+Corollary 2. Let Γ = {Ai, i ∈ Fps} be a bent partition constructed by Theorem 5. Then when
+p = 2, s = 1, A0 and A1 are (2n, 2n−1 ± 2
+n
+2 −1, 2n−2 ± 2
+n
+2 −1) difference set and (2n, 2n−1 ∓
+2
+n
+2 −1, 2n−2 ∓ 2
+n
+2 −1) difference set, respectively, and when p is odd or s ≥ 2, for any nonempty
+set I ⊆ Fps, AI\{0} = �
+i∈I Ai\{0} is a (pn, k, λ, µ) partial difference set, where (k, λ, µ) are
+given in (35) with ε = 1 if 0 ∈ AI and (k, λ, µ) are given in (36) with ε = 1 if 0 /∈ AI.
+We give an example by Corollary 2.
+Example 3. Let Γ = {DH,i, i ∈ F34} be the bent partition constructed in Example 2. By Corol-
+lary 2, DH,i is a (1853020188851841, 22876791923520, 282470988879, 282429005040) partial difference
+set for any i ∈ F∗
+34, DH,0\{0} is a (1853020188851841, 22876834970240, 282472051759, 282430067922)
+partial difference set, (DH,0
+� DH,1)\{0} is a (1853020188851841, 45753626893760, 1129760129761,
+1129719208806) partial difference set.
+When p is an odd prime, we immediately obtain the following characterization of bent
+partitions of V (p)
+n
+satisfying Condition C from Theorems 3 and 6.
+Theorem 8. Let p be an odd prime. Let Γ = {Ai, i ∈ V (p)
+s
+} be a partition of V (p)
+n , where n is
+even and s ≤ n
+2. Then the following two statements are equivalent.
+(1) Γ is a bent partition satisfying Condition C.
+(2) For any nonempty set I ⊆ V (p)
+s
+, AI\{0} = �
+i∈I Ai\{0} is a (pn, k, λ, µ) partial difference
+January 3, 2023
+DRAFT
+
+26
+set with −AI = AI, where (k, λ, µ) are given in (35) if 0 ∈ AI and (k, λ, µ) are given in (36)
+if 0 /∈ AI.
+VI. CONCLUSION
+In this paper, we investigated relations between bent partitions and vectorial dual-bent functions
+(Theorems 1, 2, 3) and gave some new constructions of bent partitions satisfying Condition C
+(Corollary 1, Theorems 4 and 5). We illustrated that for any ternary weakly regular bent function
+f : V (3)
+n
+→ F3 (n even) with f(x) = f(−x) and εf = −1, the generated bent partition by f
+is not coming from a normal bent partition (see Example 1), which answers an open problem
+proposed in [4]. By taking vectorial dual-bent functions as the link between bent partitions and
+partial difference sets, we give a sufficient condition on constructing partial difference sets from
+bent partitions (Theorem 7). When p is an odd prime, we characterized bent partitions satisfying
+Condition C in terms of partial difference sets (Theorem 8).
+REFERENCES
+[1] N.
+Anbar,
+T.
+Kalaycı,
+Amorphic
+association
+schemes
+from
+bent
+partitions,
+Available:
+https://www.researchgate.net/publication/366593699 Amorphic association schemes from bent partitions
+[2] N. Anbar, T. Kalaycı and W. Meidl, Bent partitions and partial difference sets, IEEE Trans. Inf. Theory, vol. 68, no. 10,
+pp. 6894-6903, 2022
+[3] N. Anbar, T. Kalaycı and W. Meidl, Generalized semifield spreads, Des. Codes Cryptogr., Online. DOI: 10.1007/s10623-
+022-01115-2
+[4] N. Anbar and W. Meidl, Bent partitions, Des. Codes Cryptogr., vol. 90, no. 4, pp. 1081-1101, 2022.
+[5] C. Carlet and S. Mesnager, Four decades of research on bent functions, Des. Codes Cryptogr., vol. 78, no. 1, pp. 5-50,
+2016.
+[6] A. C¸ es¸melio˘glu, W. Meidl, A construction of bent functions from plateaued functions, Des. Codes Cryptogr., vol. 66, pp.
+231-242, 2013.
+[7] A. C¸ es¸melio˘glu, W. Meidl, Bent and vectorial bent functions, partial difference sets, and strongly regular graphs, Adv. Math.
+Commun. vol. 12, pp. 691-705, 2018.
+[8] A. C¸ es¸melio˘glu, W. Meidl, I. Pirsic, Vectorial bent functions and partial difference sets, Des. Codes Cryptogr. vol. 89, no.
+10, pp. 2313-2330, 2021.
+[9] A. C¸ es¸melio˘glu, W. Meidl, and A. Pott, On the dual of (non)-weakly regular bent functions and self-dual bent functions,
+Adv. Math. Commun., vol. 7, no. 4, pp. 425-440, 2013.
+[10] A. C¸ es¸melio˘glu, W. Meidl and A. Pott, Vectorial bent functions and their duals, Linear Algebra Appl., vol. 548, pp.
+305-320, 2018.
+[11] J. F. Dillon, Elementary Hadamard difference sets, Ph. D. Thesis, University of Maryland, 1974.
+[12] C. Ding, Codes from Difference Sets, World Scientific, Singapore, 2015.
+[13] J. Y. Hyun, J. Lee and Y. Lee, Ramanujan graphs and expander families constructed from p-ary bent functions, Des. Codes
+Cryptogr. vol. 88, no. 2, pp.453-470, 2020.
+January 3, 2023
+DRAFT
+
+27
+[14] P. Lisonˇek and H. Y. Lu, Bent functions on partial spreads, Des. Codes Cryptogr. vol. 73, no. 1, pp. 209-216, 2014.
+[15] P. V. Kumar, R. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Comb. Theory Ser. A, vol.
+40, no. 1, pp. 90-107, 1985.
+[16] S. L. Ma, A survey of partial difference sets, Des. Codes Cryptogr. vol. 4, no. 4, pp. 221-261, 1994.
+[17] W. Meidl, A survey on p-ary and generalized bent functions, Cryptogr. Commun. vol. 14, no.4, pp. 737-782, 2022.
+[18] W. Meidl and I. Pirsic, Bent and Z2k-Bent functions from spread-like partitions, Des. Codes Cryptogr., vol. 89, no. 1, pp.
+75-89, 2021.
+[19] S. Mesnager, Bent Functions-Fundamentals and Results, Springer, Switzerland, 2016.
+[20] K. Nyberg, Constructions of bent functions and difference sets, In: Advances in cryptology-EUROCRYPT’ 90, Lecture
+Notes in Comput. Sci. 473, Springer, Berlin, pp. 151-160, 1991.
+[21] O. S. Rothaus, On “bent” functions, J. Comb. Theory Ser. A, vol. 20, no. 3, pp. 300-305, 1976.
+[22] J. Wang and F.-W. Fu, New results on vectoril dual-bent functions and partial difference sets, Des. Codes Cryptogr., Online.
+DOI: 10.1007/s10623-022-01103-6
+January 3, 2023
+DRAFT
+
diff --git a/LdAyT4oBgHgl3EQfsvlL/content/tmp_files/load_file.txt b/LdAyT4oBgHgl3EQfsvlL/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d388e80fb919658aa87e9947589329623350b327
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+++ b/LdAyT4oBgHgl3EQfsvlL/content/tmp_files/load_file.txt
@@ -0,0 +1,812 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf,len=811
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='00581v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='IT]  2 Jan 2023  1  Bent Partitions, Vectorial Dual-Bent Functions and Partial Difference Sets†  Jiaxin Wang, Fang-Wei Fu, Yadi Wei  Abstract  Bent partitions of V (p)  n  are quite powerful in constructing bent functions, vectorial bent functions and  generalized bent functions, where V (p)  n  is an n-dimensional vector space over Fp, n is an even positive  integer and p is a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' It is known that partial spreads is a class of bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In [4], [18], two  classes of bent partitions whose forms are similar to partial spreads were presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In [3], more bent  partitions Γ1, Γ2, Γ•  1, Γ•  2, Θ1, Θ2 were presented from (pre)semifields, including the bent partitions given  in [4], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In this paper, we investigate the relations between bent partitions and vectorial dual-bent  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For any prime p, we show that one can generate certain bent partitions (called bent partitions  satisfying Condition C) from certain vectorial dual-bent functions (called vectorial dual-bent functions  satisfying Condition A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In particular, when p is an odd prime, we show that bent partitions satisfying  Condition C one-to-one correspond to vectorial dual-bent functions satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We give an  alternative proof that Γ1, Γ2, Γ•  1, Γ•  2, Θ1, Θ2 are bent partitions in terms of vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  We present a secondary construction of vectorial dual-bent functions, which can be used to generate  more bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We show that any ternary weakly regular bent function f : V (3)  n  → F3 (n even) of  2-form can generate a bent partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' When such f is weakly regular but not regular, the generated bent  partition by f is not coming from a normal bent partition, which answers an open problem proposed in  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We give a sufficient condition on constructing partial difference sets from bent partitions, and when  p is an odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of  partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Index Terms  Bent partitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' bent functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' vectorial bent functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' vectorial dual-bent functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' semifields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  partial difference sets  Jiaxin Wang, Fang-Wei Fu and Yadi Wei are with Chern Institute of Mathematics and LPMC, Nankai University, Tianjin  300071, China, Emails: wjiaxin@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='cn, fwfu@nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='cn, wydecho@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  †This research is supported by the National Key Research and Development Program of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' 2018YFA0704703  and 2022YFA1005001), the National Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' 12141108, 61971243, 12226336), the  Natural Science Foundation of Tianjin (20JCZDJC00610), the Fundamental Research Funds for the Central Universities of China  (Nankai University), and the Nankai Zhide Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' INTRODUCTION  Boolean bent functions were introduced by Rothaus [21] and were generalized to p-ary bent  functions by Kumar, Scholtz and Welch [15], where p is an arbitrary prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Due to applications  of p-ary bent functions in cryptography, coding theory, sequence and combinatorics, they have  been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We refer to surveys [5], [17] and a book [19] on p-ary bent functions  and their generalizations such as vectorial bent functions and generalized bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  In [10], C¸ es¸melio˘glu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' introduced vectorial dual-bent functions, which is a special class  of vectorial bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In [7], [8], [22], vectorial dual-bent functions were used to construct  partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In particular, Wang and Fu [22] showed that for certain vectorial dual-bent  functions F : V (p)  n  → V (p)  s  (where V (p)  n  is an n-dimensional vector space over the prime field  Fp), the preimage set of any subset of V (p)  s  for F forms a partial difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Very recently, bent partitions of V (p)  n  were introduced [4], [18], which are quite powerful in  constructing bent functions, vectorial bent functions and generalized bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' The well-  known partial spreads is a class of bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In [18], Meidl and Pirsic for the first time  presented two classes of bent partitions for p = 2 different from partial spreads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In [4], Anbar and  Meidl generalized the contributions in [18] to the case of p being odd and gave the corresponding  two classes of bent partitions for odd p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In [3], Anbar, Kalaycı and Meidl presented more bent  partitions Γ1, Γ2, Γ•  1, Γ•  2, Θ1, Θ2 from (pre)semifields, including the bent partitions given in [4],  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In [2], Anbar, Kalaycı and Meidl showed that any union of elements in the bent partition  given in [4], [18] forms a partial difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In terms of constructing partial difference sets,  certain vectorial dual-bent functions and certain bent partitions seem to play the same role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Therefore, it is interesting to investigate the relations between vectorial dual-bent functions  and bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In this paper, we show that by using certain vectorial dual-bent functions  (called vectorial dual-bent functions satisfying Condition A), we can construct bent partitions  of V (p)  n  with certain properties (called bent partitions satisfying Condition C) for any prime  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Particularly, when p is an odd prime, we prove that bent partitions of V (p)  n  with Condition  C one-to-one correspond to vectorial dual-bent functions satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In terms of  vectorial dual-bent functions, we provide an alternative proof that Γ1, Γ2, Γ•  1, Γ•  2, Θ1, Θ2 given  in [3] are bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We provide a secondary construction of vectorial dual-bent functions,  which can be used to generate more bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We prove that any ternary weakly regular  bent function f : V (3)  n  → F3 (n even) of 2-form can generate a bent partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In the special case  January 3, 2023  DRAFT  3  that f is weakly regular but not regular, the generated bent partition by f is not coming from  a normal bent partition, which answers an open problem proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By using vectorial  dual-bent functions as the link between bent partitions and partial difference sets, we give a  sufficient condition on constructing partial difference sets from bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' When p is an  odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of  partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In Section II, we state some needed results on  vectorial dual-bent functions and bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In Section III, we present relations between  certain bent partitions and certain vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In Section IV, we give a sec-  ondary construction of vectorial dual-bent functions, which can be used to generate more bent  partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In Section V, we present relations between certain bent partitions and certain partial  difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In Section VI, we make a conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' PRELIMINARIES  In this section, we state some basic results on vectorial dual-bent functions and bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  First, we fix some notations used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  p is a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  ζp = e  2π√−1  p  is a complex primitive p-th root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Note that ζ2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Fpn is the finite field with pn elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Fn  p is the vector space of the n-tuples over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  V (p)  n  is an n-dimensional vector space over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  ⟨·⟩n denotes a (non-degenerate) inner product of V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In this paper, when V (p)  n  = Fpn,  let ⟨a, b⟩n = Trn  1(ab), where a, b ∈ Fpn, Trn  k(·) denotes the trace function from Fpn to  Fpk, k | n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' when V (p)  n  = Fn  p, let ⟨a, b⟩n = a · b = �n  i=1 aibi, where a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , an), b =  (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , bn) ∈ Fn  p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' when V (p)  n  = V (p)  n1 ×· · ·×V (p)  nm (n = �m  i=1 ni), let ⟨a, b⟩n = �m  i=1⟨ai, bi⟩ni,  where a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , am), b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , bm) ∈ V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  For any set A ⊆ V (p)  n  and u ∈ V (p)  n , let χu(A) = �  x∈A χu(x), where χu denotes the  character χu(x) = ζ⟨u,x⟩n  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Vectorial dual-bent functions  A function F : V (p)  n  → V (p)  s  is called a vectorial p-ary function, or simply p-ary function  when s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' The Walsh transform of a p-ary function f : V (p)  n  → Fp is the complex valued  January 3, 2023  DRAFT  4  function defined by  Wf(a) =  �  x∈V (p)  n  ζf(x)−⟨a,x⟩n  p  , a ∈ V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (1)  A p-ary function f : V (p)  n  → Fp is called bent if |Wf(a)| = p  n  2 for all a ∈ V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Note that  when f is a Boolean bent function, that is, p = 2, then n must be even since in this case, Wf is  an integer valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' A vectorial p-ary function F : V (p)  n  → V (p)  s  is called vectorial bent  if all component functions Fc : V (p)  n  → Fp, c ∈ V (p)  s  \\{0} defined as Fc(x) = ⟨c, F(x)⟩s are bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  It is known that if F : V (p)  n  → V (p)  s  is vectorial bent, then s ≤ n  2 if p = 2, and s ≤ n if p is  an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' If f : V (p)  n  → Fp is bent, then so are cf, c ∈ F∗  p, that is, any p-ary bent function  is vectorial bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For F : V (p)  n  → V (p)  s  , the vectorial bentness of F is independent of the inner  products of V (p)  n  and V (p)  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' The Walsh transform of a p-ary bent function f : V (p)  n  → Fp satisfies  that for any a ∈ V (p)  n , when p = 2, we have  Wf(a) = 2  n  2 (−1)f∗(a),  (2)  and when p is an odd prime, we have  Wf(a) =  \uf8f1  \uf8f2  \uf8f3  ±p  n  2 ζf∗(a)  p  if p ≡ 1 (mod 4) or n is even,  ±  √  −1p  n  2 ζf∗(a)  p  if p ≡ 3 (mod 4) and n is odd,  (3)  where f ∗ is a function from V (p)  n  to Fp, called the dual of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' A p-ary bent function f : V (p)  n  → Fp  is called weakly regular if Wf(a) = εfp  n  2 ζf∗(a)  p  , where εf is a constant independent of a,  otherwise f is called non-weakly regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In particular, if εf = 1, f is called regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' The (non-  )weakly regularity of f is independent of the inner product of V (p)  n  and if f is weakly regular,  εf is independent of the inner product of V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By (2), all Boolean bent functions are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  If f is a p-ary weakly regular bent function, then the dual f ∗ of f is also weakly regular bent  with (f ∗)∗(x) = f(−x) (see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  In 2018, C¸ es¸melio˘glu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' [10] introduced vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' A vectorial p-ary bent function F : V (p)  n  → V (p)  s  is called vectorial dual-bent  if there exists a vectorial bent function G : V (p)  n  → V (p)  s  such that (Fc)∗ = Gσ(c) for any  c ∈ V (p)  s  \\{0}, where (Fc)∗ is the dual of the component function ⟨c, F(x)⟩s and σ is some  permutation over V (p)  s  \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' The vectorial bent function G is called a vectorial dual of F and  denoted by F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  5  It is known in [10] that the property of being vectorial dual-bent is independent of the inner  products of V (p)  n  and V (p)  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Note that for a vectorial dual-bent function, its vectorial dual is not  unique since being vectorial bent and vectorial dual-bent for a function is a property of the  vector space consisting of all component functions (see Remark 1 of [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For example, if a  p-ary function f (seen as a vectorial function into V (p)  1  , p odd) is vectorial dual-bent under any  fixed inner product, then its dual f ∗ is unique, but its vectorial dual is not unique since for any  c ∈ F∗  p, cf ∗ is a vectorial dual of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' A p-ary function f : V (p)  n  → Fp is called an l-form if  f(ax) = alf(x) for any a ∈ F∗  p and x ∈ V (p)  n , where 1 ≤ l ≤ p − 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By the results  in [7], [22], we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proposition 1 ( [7], [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' A p-ary function f with f(0) = 0 is a weakly regular vectorial dual-  bent function if and only if f is a weakly regular bent function of l-form with gcd(l−1, p−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  In particular, a p-ary function f is a weakly regular vectorial dual-bent function with (cf)∗ = cf ∗  for any c ∈ F∗  p if and only if f is a weakly regular bent function of (p − 1)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  In the rest of this subsection, we recall an important class of p-ary bent functions, called  Maiorana-McFarland bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let f : Fpn × Fpn → Fp be defined as  f(x, y) = Trn  1(αxπ(y)) + g(y),  where α ∈ F∗  pn, π is a permutation over Fpn and g : Fpn → Fp is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Then f is bent and is called a Maiorana-McFarland bent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' The dual f ∗ of f is  f ∗(x, y) = Trn  1(−π−1(α−1x)y) + g(π−1(α−1x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (4)  All Maiorana-McFarland bent functions are regular (see [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Bent partitions  Very recently, the concept of bent partitions of V (p)  n  were introduced [4], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let n be an even positive integer, K be a positive integer divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let Γ = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , AK} be a partition of V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Assume that every function f for which every  i ∈ Fp has exactly K  p of sets Aj in Γ in its preimage, is a p-ary bent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then Γ is  called a bent partition of V (p)  n  of depth K and every such bent function f is called a bent  function constructed from bent partition Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  6  Let Γ = {U, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , AK} be a partition of V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Assume that every function f with the  following properties is bent:  (1) Every c ∈ Fp has exactly K  p of the sets A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , AK in its preimage set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (2) f(x) = c0 for all x ∈ U and some fixed c0 ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Then we call Γ a normal bent partition of V (p)  n  of depth K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Bent partitions are very powerful in constructing bent functions, vectorial bent function and  generalized bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In this paper, we focus on the relations between bent partitions and  vectorial bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proposition 2 ( [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let Γ = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , Aps} be a bent partition of V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then every function  F : V (p)  n  → V (p)  s  such that every element i ∈ V (p)  s  has the elements of exactly one of the sets  Aj, 1 ≤ j ≤ ps, in its preimage, is a vectorial bent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  It is known that partial spreads is a class of bent partitions (for instance see Section 2 of [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In  [4], [18], two classes of explicit bent partitions different from partial spreads were presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In  [3], bent partitions Γ1, Γ2, Γ•  1, Γ•  2, Θ1, Θ2 were presented from certain (pre)semifields, including  the bent partitions given in [4], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We will recall bent partitions Γ1, Γ2, Γ•  1, Γ•  2, Θ1, Θ2 given  in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' First, we need to introduce some basic knowledge on (pre)semifields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let ◦ be a binary operation defined on (V (p)  n , +) such that  (i) x ◦ y = 0 implies x = 0 or y = 0,  (ii) (x + y) ◦ z = (x ◦ z) + (y ◦ z), (z ◦ (x + y) = (z ◦ x) + (z ◦ y), respectively),  for all x, y, z ∈ V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then (V (p)  n , +, ◦) is called a right (left, respectively) prequasifield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' If  (V (p)  n , +, ◦) is a right and a left prequasifield, then it is called a presemifield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' If (V (p)  n , +, ◦) is  a presemifield for which there is an element e ̸= 0 such that e ◦ x = x ◦ e = x for all x ∈ V (p)  n ,  then it is called a semifield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let P = (Fpn, +, ◦) be a presemifield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then one can obtain presemifields P • = (Fpn, +, •)  and P ⋆ = (Fpn, +, ⋆) from P, where • and ⋆ are given by  x • y = y ◦ x for all x, y ∈ Fpn,  and  Trn  1(z(x ◦ y)) = Trn  1(x(z ⋆ y)) for all x, y, z ∈ Fpn,  January 3, 2023  DRAFT  7  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' The presemifield P ⋆ is called the dual of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let s be a positive divisor of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  If x ◦ (cy) = c(x ◦ y) holds for any x, y ∈ Fpn, c ∈ Fps, then P is called right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Each presemifield P = (Fpn, +, ◦) can induce a semifield P ′ = (Fpn, +, ∗) via the following  transformation: choose any α ∈ F∗  pn and give ∗ by  (x ◦ α) ∗ (α ◦ y) = x ◦ y for all x, y ∈ Fpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  By Lemma 2 of [3], if P is right Fps-linear, then P ′ is also right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' The finite field Fpn  is a right Fps-linear semifield (that is, ◦ is the field multiplication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For more right Fps-linear  (pre)semifields, see Section 3 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Now we recall bent partitions Γ1, Γ2, Γ•  1, Γ•  2, Θ1, Θ2 given in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let n, s be positive integers satisfying s | n and gcd(pn−1, ps+p−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Set u = ps+p−1,  and let d be an integer with du ≡ 1 mod (pn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let P = (Fpn, +, ◦) be a (pre)semifield  such that its dual P ⋆ = (Fpn, +, ⋆) is right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For given x ∈ Fpn, if x = 0, then let  ηx = 0, and if x ̸= 0, then let ηx be given by x ⋆ η−1  x  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Define  Ut = {(x, t ◦ xu) : x ∈ F∗  pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define  Γ1 = {Ai, i ∈ Fps},  (5)  where Ai = �  t∈Fpn:Trn  s (t)=i Ut if i ̸= i0, Ai0 = �  t∈Fpn:Trn  s (t)=i0 Ut  � U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Define  U•  t = {(x, xu ◦ t) : x ∈ F∗  pn} if t ∈ Fpn, and U = {(0, y) : y ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define  Γ•  1 = {A•  i , i ∈ Fps},  (6)  where A•  i = �  t∈Fpn:Trn  s (t)=i U•  t if i ̸= i0, A•  i0 = �  t∈Fpn:Trn  s (t)=i0 U•  t  � U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Define  Vt = {(t ◦ xd, x) : x ∈ F∗  pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define  Γ2 = {Bi, i ∈ Fps},  (7)  January 3, 2023  DRAFT  8  where Bi = �  t∈Fpn:Trn  s (t)=i Vi if i ̸= i0, Bi0 = �  t∈Fpn:Trn  s (t)=i0 Vi  � V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Define  V •  t = {(xd ◦ t, x) : x ∈ F∗  pn} if t ∈ Fpn, and V = {(x, 0) : x ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define  Γ•  2 = {B•  i , i ∈ Fps},  (8)  where B•  i = �  t∈Fpn:Trn  s (t)=i V •  i if i ̸= i0, Bi0 = �  t∈Fpn:Trn  s (t)=i0 V •  i  � V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Define  Xt = {(tηd  x, x) : x ∈ F∗  pn} if t ∈ Fpn, and X = {(x, 0) : x ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define  Θ1 = {Si, i ∈ Fps},  (9)  where Si = �  t∈Fpn:Trn  s (t)=i Xt if i ̸= i0, Si0 = �  t∈Fpn:Trn  s (t)=i0 Xt  � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Define  Yt = {(x, tηu  x) : x ∈ F∗  pn} if t ∈ Fpn, and Y = {(0, y) : y ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let i0 ∈ Fps be an arbitrary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define  Θ2 = {Ti, i ∈ Fps},  (10)  where Ti = �  t∈Fpn:Trn  s (t)=i Yt if i ̸= i0, Ti0 = �  t∈Fpn:Trn  s (t)=i0 Yt  � Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In the finite field case, that is, ◦ and ⋆ are the field multiplication, then Γ1 = Γ•  1 =  Θ2, Γ2 = Γ•  2 = Θ1, which reduces to the two classes bent partitions given in [4], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In fact, for the parameter u in the bent partitions Γ1, Γ•  1, Γ2, Γ•  2, Θ1, Θ2, one can  consider the more general form u ≡ pj mod (ps − 1) by the proofs in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' RELATIONS BETWEEN CERTAIN BENT PARTITIONS AND CERTAIN VECTORIAL  DUAL-BENT FUNCTIONS  Throughout this section, we consider bent partitions and vectorial dual-bent functions satisfy-  ing the following conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Condition C: Let n be an even positive integer, s be a positive integer with s ≤  n  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let  Γ = {Ai, i ∈ V (p)  s  } be a bent partition of V (p)  n  which satisfies that F∗  pAi = Ai for all i ∈ V (p)  s  January 3, 2023  DRAFT  9  and all bent functions f constructed from Γ are regular (that is, εf = 1) or weakly regular but  not regular (that is, εf = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We denote by ε = εf for all bent functions f constructed from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Condition A: Let n be an even positive integer, s be a positive integer with s ≤  n  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let  F : V (p)  n  → V (p)  s  be a vectorial dual-bent function with (Fc)∗ = (F ∗)c, c ∈ V (p)  s  \\{0} for a  vectorial dual F ∗ of F and all component functions being regular or weakly regular but not  regular, that is, εFc, c ∈ V (p)  s  \\{0} are all the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We denote by ε = εFc for all c ∈ V (p)  s  \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  It is easy to see that the known bent partitions, including partial spreads and Γi, Γ•  i , Θi, i = 1, 2  defined by (5)-(10), all satisfy F∗  pAi = Ai, i ∈ V (p)  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By the results in [3], [11], [14], all bent  functions constructed from partial spreads and Γi, Γ•  i , Θi, i = 1, 2 are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Thus, the known  bent partitions all satisfy Condition C with ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Moreover, when p = 2, it is easy to see that  Condition C is trivial for any bent partition of V (2)  n  of depth powers of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In this section, we  present relations between bent partitions satisfying Condition C and vectorial dual-bent functions  satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' First, we need a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let n be an even positive integer, s be a positive integer with s ≤ n  2, and F : V (p)  n  →  V (p)  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then the following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (1) F is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (2) There exist pairwise disjoint sets Wi ⊆ V (p)  n , i ∈ V (p)  s  with �  i∈V (p)  s  Wi = V (p)  n  and a  constant ε ∈ {±1} (ε = 1 if p = 2) such that for any nonempty set I ⊆ V (p)  s  ,  χu(DF,I) = pn−sδ{0}(u)|I| + εp  n  2 −s(psδWI(u) − |I|), u ∈ V (p)  n ,  (11)  where DF,I = {x ∈ V (p)  n  : F(x) ∈ I}, WI = �  i∈I Wi, and for any set S, δS denotes the  indicator function of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Proposition 3 of [22] (Note that although Proposition 3 of [22] only considers the  case of p being odd, p = 2 also holds), for any u ∈ V (p)  n , i ∈ V (p)  s  we have  χu(DF,i) = pn−sδ{0}(u) + p−s  �  c∈V (p)  s  \\{0}  WFc(−u)ζ−⟨c,i⟩s  p  ,  (12)  where DF,i = {x ∈ V (p)  n  : F(x) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  10  (1) ⇒ (2): If F is a vectorial dual-bent function satisfying Condition A (Note that if p = 2,  then ε = 1 since all Boolean bent functions are regular), then  χu(DF,i) = pn−sδ{0}(u) + εp  n  2 −s  �  c∈V (p)  s  \\{0}  ζ(Fc)∗(−u)−⟨c,i⟩s  p  = pn−sδ{0}(u) + εp  n  2 −s  �  c∈V (p)  s  \\{0}  ζ(F ∗)c(−u)−⟨c,i⟩s  p  = pn−sδ{0}(u) + εp  n  2 −s  �  c∈V (p)  s  \\{0}  ζ⟨c,F ∗(−u)−i⟩s  p  = pn−sδ{0}(u) + εp  n  2 −s(psδ{0}(F ∗(−u) − i) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (13)  Define Wi = {x ∈ V (p)  n  : F ∗(−x) = i}, i ∈ V (p)  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then Wi  � Wj = ∅ for any i ̸= j and  �  i∈V (p)  s  Wi = V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By (13), for any nonempty set I ⊆ V (p)  s  and u ∈ V (p)  n  we have  χu(DF,I) =  �  i∈I  χu(DF,i)  =  �  i∈I  pn−sδ{0}(u) + εp  n  2 −s(psδWi(u) − 1)  = pn−sδ{0}(u)|I| + εp  n  2 −s(psδWI(u) − |I|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (2) ⇒ (1): By the assumption on Wi, i ∈ V (p)  s  , we have that for any x ∈ V (p)  n , there exists a  unique i ∈ V (p)  s  such that x ∈ Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define G : V (p)  n  → V (p)  s  by  G(x) = i if − x ∈ Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  By the definition of G, for any u ∈ V (p)  n , i ∈ V (p)  s  we have  χu(DF,i) = pn−sδ{0}(u) + εp  n  2 −s(psδ{0}(G(−u) − i) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (14)  For any c ∈ V (p)  s  \\{0},  WFc(−u) =  �  x∈V (p)  n  ζ⟨c,F (x)⟩s+⟨u,x⟩n  p  =  �  i∈V (p)  s  �  x∈V (p)  n  :F (x)=i  ζ⟨c,i⟩s+⟨u,x⟩n  p  January 3, 2023  DRAFT  11  =  �  i∈V (p)  s  ζ⟨c,i⟩s  p  χu(DF,i)  =  �  i∈V (p)  s  \\{G(−u)}  ζ⟨c,i⟩s  p  (pn−sδ{0}(u) − εp  n  2 −s) + (pn−sδ{0}(u) + εp  n  2 −s(ps − 1))ζ⟨c,G(−u)⟩s  p  = (pn−sδ{0}(u) − εp  n  2 −s)  �  i∈V (p)  s  ζ⟨c,i⟩s  p  + εp  n  2 ζGc(−u)  p  = εp  n  2 ζGc(−u)  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (15)  By (15) and the assumption that ε = 1 if p = 2, F is a vectorial bent function with εFc = ε  and (Fc)∗ = Gc for any c ∈ V (p)  s  \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Since Fc is a weakly regular bent function, we have that  Gc = (Fc)∗ is also weakly regular bent and G is vectorial bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Thus, F is vectorial dual-bent  with εFc = ε and (Fc)∗ = (F ∗)c for any c ∈ V (p)  s  \\{0}, where F ∗ = G, that is, F satisfies  Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Based on Lemma 1, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let F : V (p)  n  → V (p)  s  be a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Define  Ai = DF,i, i ∈ V (p)  s  ,  where DF,i = {x ∈ V (p)  n  : F(x) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then Γ = {Ai, i ∈ V (p)  s  } is a bent partition satisfying  Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Lemma 1 and its proof, for any i ∈ V (p)  s  and u ∈ V (p)  n ,  χu(Ai) = χu(DF,i) = pn−sδ{0}(u) + εp  n  2 −s(psδ{0}(F ∗(−u) − i) − 1),  where ε = 1 if p = 2 since all Boolean bent functions are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For any union S of ps−1 sets  of {Ai : i ∈ V (p)  s  }, we have  χu(S) =  \uf8f1  \uf8f2  \uf8f3  pn−1δ{0}(u) + εp  n  2 −1(p − 1), if AF ∗(−u) ⊆ S,  pn−1δ{0}(u) − εp  n  2 −1, if AF ∗(−u) ⊈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (16)  Let f be an arbitrary function such that for every j ∈ Fp, there are exactly ps−1 sets Ai in Γ in  its preimage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define g(u) = f(AF ∗(−u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Note that g is a p-ary function from V (p)  n  to Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then  by (16), we have  χu(Df,j) =  \uf8f1  \uf8f2  \uf8f3  pn−1δ{0}(u) + εp  n  2 −1(p − 1), if j = g(u),  pn−1δ{0}(u) − εp  n  2 −1, if j ̸= g(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (17)  January 3, 2023  DRAFT  12  By (17), for any u ∈ V (p)  n  we have  Wf(−u) =  �  x∈V (p)  n  ζf(x)+⟨u,x⟩n  p  =  �  j∈Fp  ζj  p  �  x∈V (p)  n  :f(x)=j  ζ⟨u,x⟩n  p  =  �  j∈Fp  ζj  pχu(Df,j)  =  �  j∈Fp\\{g(u)}  ζj  p(pn−1δ{0}(u) − εp  n  2 −1) + ζg(u)  p  (pn−1δ{0}(u) + εp  n  2 −1(p − 1))  = (pn−1δ{0}(u) − εp  n  2 −1)  �  j∈Fp  ζj  p + εp  n  2 ζg(u)  p  = εp  n  2 ζg(u)  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (18)  By (18) and ε = 1 if p = 2, f is a weakly regular bent function with εf = ε and f ∗(x) = g(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let  Wj = {u ∈ V (p)  n  : g(u) = j}, j ∈ Fp,  then Wj, j ∈ Fp are pairwise disjoint and �  j∈Fp Wj = V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By (17), for any u ∈ V (p)  n  and  nonempty set J ⊆ Fp we have  χu(Df,J) = pn−1δ{0}(u)|J| + εp  n  2 −1(pδWJ(u) − |J|),  (19)  where Df,J = {x ∈ V (p)  n  : f(x) ∈ J}, WJ = �  j∈J Wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By (19) and Lemma 1, f is vectorial dual-  bent with (cf)∗ = c(βf ∗), c ∈ F∗  p for some β ∈ F∗  p (since all vectorial duals of f are cf ∗, c ∈ F∗  p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let c = 1, we obtain β = 1, that is, f is vectorial dual-bent with (cf)∗ = cf ∗, c ∈ F∗  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By  Proposition 1, f is a (p − 1)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In particular, Fc is a (p − 1)-form for any c ∈ F∗  ps, which  yields that F(αx) = F(x) for any α ∈ F∗  p and F∗  pAi = Ai, i ∈ V (p)  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Hence, Γ is a bent partition  satisfying Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  By Theorem 1, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let n be an even positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let f : V (p)  n  → Fp be a weakly regular bent  function of (p − 1)-form, then {Df,j, j ∈ Fp} is a bent partition of V (p)  n , where Df,j = {x ∈  V (p)  n  : f(x) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Proposition 1, f is a weakly regular vectorial dual-bent function with (cf)∗ = cf ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Since n is even, εcf = εf for all c ∈ F∗  p (see Theorem 1 of [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then by Theorem 1, the  conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  13  A bent partition Γ = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , AK} of depth K is called coming from a normal bent partition  if there is U ⊆ Ai for some i such that {U, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , Ai−1, Ai\\U, Ai+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , AK} is a normal bent  partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In [4], there is an open problem: Do bent partitions exist which are not coming from  a normal bent partition of depth K > 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In the following, we provide a positive answer for  this open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By the definition of l-form, a ternary function f is a 2-form if and only if  f(x) = f(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let n be an even positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' If f : V (3)  n  → F3 with f(x) = f(−x) is a  ternary weakly regular but not regular bent function (that is, εf = −1), then by Corollary 1,  {Df,0, Df,1, Df,2} is a bent partition of depth 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' There exist such ternary bent functions f, for  instance see [7], [17]: f(x) = Trn  1(αx2), x ∈ F3n,  (20)  where n is even, α ∈ F∗  3n is a square element if 4 | n, and α ∈ F∗  3n is a non-square element  if 4 ∤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' f(x) = Trn  1(ax  3n−1  4  +3m+1), x ∈ F3n,  (21)  where n = 2m, m odd, a = α  3m+1  4  for a primitive element α of F3n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' f(x) = Trn  1(α(x33k+32k−3k+1 + x2)), x ∈ F3n,  (22)  where n = 4k for an arbitrary positive integer k, α ∈ F∗  32k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' f(x, y, z) = (g(x) − h(x))z2 + yz + g(x), (x, y, z) ∈ F3n × F3 × F3,  (23)  where n is even, g and h are distinct bent functions constructed by (20) or (22) if 4 | n, g  and h are distinct bent functions constructed by (20) or (21) if 4 ∤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  For any ternary weakly regular but not regular bent function f : V (3)  n  → F3 (n even) with  f(x) = f(−x), the corresponding bent partition {Df,0, Df,1, Df,2} is not coming from a normal  bent partition by Theorem 4 (i) of [4], which provides a positive answer for the above open  problem proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We first recall Theorem 4 (i) of [4] and then give an example to  illustrate this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Lemma 2 ( [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let Γ = {U, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' , AK} be a normal bent partition of V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then |U| = p  n  2  and |Aj| = pn−p  n  2  K  , 1 ≤ j ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  14  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let f : F34 → F3 be defined by f(x) = Tr4  1(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then f is ternary weakly regular  bent with f(x) = f(−x) and εf = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Corollary 1, {Df,0, Df,1, Df,2} is a bent partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  By the result of Nyberg [20], for any weakly regular p-ary bent function g : V (p)  n  → Fp with  n even, we have {|Dg,i|, i ∈ Fp} = {pn−1 + εfp  n  2 −1(p − 1), pn−1 − εfp  n  2 −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For our example,  |Df,0| = 21, |Df,1| = |Df,2| = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Lemma 2, it is easy to see that {Df,0, Df,1, Df,2} can not  be from a normal bent partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  In the following, based on Theorem 1, we give an alternative proof that Γi, Γ•  i , Θi, i = 1, 2  defined by (5)-(10) given in [3] are bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let s, n be positive integers with s | n, u be an integer with u ≡ pj0 mod (ps − 1) for some  0 ≤ j0 ≤ s − 1 and gcd(u, pn − 1) = 1, and let d be an integer with du ≡ 1 mod (pn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let  P = (Fpn, +, ◦) be a (pre)semifield such that its dual P ⋆ = (Fpn, +, ⋆) is right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For  given x ∈ Fpn, if x = 0, then let ηx = 0, and if x ̸= 0, then let ηx be given by x ⋆ η−1  x  = 1 (For  convention we set η−1  0  = ηpn−2  0  = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For any α ∈ F∗  pn and i0 ∈ Fps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' define  F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = Trn  s (αa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y)) + i0(1 − xpn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (24)  where for given (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then a(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and if x ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then a(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) is given by  a(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) ◦ xu = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and  F •(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = Trn  s (αa•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y)) + i0(1 − xpn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (25)  where for given (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then a•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and if x ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then a•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) is given by  xu ◦ a•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and  G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = Trn  s (αb(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y)) + i0(1 − ypn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (26)  where for given (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if y = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then b(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and if y ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then b(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) is given by  b(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) ◦ yd = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and  G•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = Trn  s (αb•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y)) + i0(1 − ypn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (27)  where for given (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if y = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then b•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and if y ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then b•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) is given by  yd ◦ b•(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and  M(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = Trn  s (αη−u  x y) + i0(1 − xpn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) ∈ Fpn × Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (28)  and  N(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = Trn  s (αxη−d  y ) + i0(1 − ypn−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) ∈ Fpn × Fpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (29)  January 3, 2023  DRAFT  15  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let F, F •, G, G•, M, N be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then they are all vectorial dual-bent  functions satisfying Condition A with ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We only prove the result for F and M since the proofs for F •, G, G• are similar to the  proof for F, and the proof for N is similar to the proof for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  For F:  For any c ∈ F∗  ps, we have  Fc(x, y) = Trn  1(cαa(x, y)) + Trs  1(ci0)(1 − xpn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  For any c ∈ F∗  ps and (w, v) ∈ Fpn × Fpn, we have  WFcu(w, v) =  �  x∈F∗  pn  �  y∈Fpn  ζTrn  1 (cuαa(x,y))−Trn  1 (wx+vy)  p  + ζTrs  1(cui0)  p  �  y∈Fpn  ζ−Trn  1 (vy)  p  =  �  x∈Fpn  �  y∈Fpn  ζTrn  1 (cuαa(x,y))−Trn  1 (wx+vy)  p  + pn(ζTrs  1(cui0)  p  − 1)δ{0}(v)  = Wh(w, v) + pn(ζTrs  1(cui0)  p  − 1)δ{0}(v),  where h(x, y) = Trn  1(cuαa(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For given x ∈ Fpn, if x = 0, then let λx = 0, and if  x ̸= 0, then let λx be given by x ⋆ λ−1  x  = α (For convention we set λ−1  0  = λpn−2  0  = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define  ρ(x) = λ−d  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then ρ is a permutation over Fpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For any x ∈ F∗  pn, set z = ρ−1(c−1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then  λ−d  z  = ρ(z) = c−1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By du ≡ 1 mod (pn − 1), we have λ−1  z  = c−uxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Since z ̸= 0 and P ⋆ is  right Fps-linear, we have α = z ⋆ λ−1  z  = z ⋆ (c−uxu) = c−u(z ⋆ xu), that is, ρ−1(c−1x) ⋆ xu =  αcu for any x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Thus, when x ̸= 0, Trn  1(cuαa(x, y)) = Trn  1(a(x, y)(ρ−1(c−1x) ⋆ xu)) =  Trn  1(ρ−1(c−1x)(a(x, y) ◦ xu)) = Trn  1(ρ−1(c−1x)y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' When x = 0, by a(0, y) = ρ−1(0) = 0, we  have Trn  1(cuαa(x, y)) = Trn  1(ρ−1(c−1x)y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Hence, h(x, y) = Trn  1(ρ−1(c−1x)y), which is a  Maiorana-McFarland bent function and by (4),  Wh(w, v) = pnζ−Trn  1 (cwρ(v))  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Therefore, for any c ∈ F∗  ps,  WFcu(w, v) = pn(ζ−Trn  1 (cwρ(v))  p  + (ζTrs  1(cui0)  p  − 1)δ{0}(v))  = pnζ−Trn  1 (cwρ(v))+Trs  1(cui0)(1−vpn−1)  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (30)  By (30) and ud ≡ 1 mod (pn − 1), we have that for any c ∈ F∗  ps, Fc is a regular bent function  with  (Fc)∗(x, y) = −Trn  1(cdxρ(y)) + Trs  1(ci0)(1 − ypn−1)  = −Trn  1(cdpj0(xρ(y))pj0) + Trs  1(ci0)(1 − ypn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  16  Since u ≡ pj0 mod (ps − 1) and du ≡ 1 mod (pn − 1), we have d ≡ ps−j0 mod (ps − 1) and  thus (cd)pj0 = c for any c ∈ F∗  ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Therefore, F is a vectorial bent function with εFc = 1 and  (Fc)∗ = Hc for all c ∈ F∗  ps, where  H(x, y) = −Trn  s ((xρ(y))pj0) + i0(1 − ypn−1) = −(Trn  s (xρ(y)))pj0 + i0(1 − ypn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Since Fc is regular bent, we have that (Fc)∗ = Hc is also regular bent and H is vectorial bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Thus, F is vectorial dual-bent with εFc = 1 and (Fc)∗ = (F ∗)c for all c ∈ F∗  ps, where F ∗ = H,  that is, F satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  For M:  For any c ∈ F∗  ps,  Mc(x, y) = Trn  1(cαη−u  x y) + Trs  1(ci0)(1 − xpn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Similar to the discussion for F, for any c ∈ F∗  ps and (w, v) ∈ Fpn × Fpn we have  WMc(w, v) = Wg(w, v) + pn(ζTrs  1(ci0)  p  − 1)δ{0}(v),  where g(x, y) = Trn  1(cαη−u  x y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let π(x) = η−u  x , then π is a permutation over Fpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Since g is a  Maiorana-McFarland bent function, then by (4),  Wg(w, v) = pnζ−Trn  1 (wπ−1(c−1α−1v))  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  For any given y ∈ F∗  pn, set π−1(c−1α−1y) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then c−1α−1y = π(z) = η−u  z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By ud ≡  1 mod (pn − 1), we have η−1  z  = c−dα−dyd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Since z ̸= 0 and P ⋆ is right Fps-linear, we have  1 = z ⋆ η−1  z  = z ⋆ (c−dα−dyd) = c−d(z ⋆ α−dyd), that is, π−1(c−1α−1y) ⋆ α−dyd = cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For given  (x, y) ∈ Fpn × Fpn, if y = 0, then let r(x, y) = 0, and if y ̸= 0, then let r(x, y) be given by  r(x, y)◦α−dyd = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' When v ̸= 0, we have Trn  1(wπ−1(c−1α−1v)) = Trn  1(π−1(c−1α−1v)(r(w, v)◦  α−dvd)) = Trn  1(r(w, v)(π−1(c−1α−1v) ⋆ α−dvd)) = Trn  1(cdr(w, v)) = Trn  1(c(r(w, v))pj0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' When  v = 0, since π−1(0) = 0 and r(w, 0) = 0, we have Trn  1(wπ−1(c−1α−1v)) = Trn  1(c(r(w, v))pj0) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Thus, −Trn  1(wπ−1(c−1α−1v)) = −Trn  1(c(r(w, v))pj0) and  WMc(w, v) = pn(ζ−Trn  1 (c(r(w,v))pj0 )  p  + (ζTrs  1(ci0)  p  − 1)δ{0}(v))  = pnζ−Trn  1 (c(r(w,v))pj0 )+Trs  1(ci0)(1−vpn−1)  p  ,  which implies that M is a vectorial dual-bent function with εMc = 1 and (Mc)∗ = (M∗)c for all  c ∈ F∗  ps, where  M∗(x, y) = −Trn  s ((r(x, y))pj0) + i0(1 − ypn−1),  January 3, 2023  DRAFT  17  that is, M satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  By Theorem 1 and Proposition 3, we have that {DF,i, i ∈ Fps}, {DF •,i, i ∈ Fps}, {DG,i, i ∈  Fps}, {DG•,i, i ∈ Fps}, {DM,i, i ∈ Fps} and {DN,i, i ∈ Fps} are bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' It is easy to  verify that  DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='i =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  t∈Fpn:T rn  s (αt)=i  Ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  �  t∈Fpn:T rn  s (αt)=i0  Ut  �  U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' DF •,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='i =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  t∈Fpn:T rn  s (αt)=i  U •  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  �  t∈Fpn:T rn  s (αt)=i0  U •  t  �  U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  DG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='i =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  t∈Fpn:T rn  s (αt)=i  Vt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  �  t∈Fpn:T rn  s (αt)=i0  Vt  �  V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' DG•,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='i =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  t∈Fpn:T rn  s (αt)=i  V •  t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  �  t∈Fpn:T rn  s (αt)=i0  V •  t  �  V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  DM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='i =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  t∈Fpn:T rn  s (αt)=i  Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  �  t∈Fpn:T rn  s (αt)=i0  Xt  �  X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' DN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='i =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  t∈Fpn:T rn  s (αt)=i  Yt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i ̸= i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  �  t∈Fpn:T rn  s (αt)=i0  Yt  �  Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' if i = i0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  where  Ut = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' t ◦ xu) : x ∈ F∗  pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and U = {(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) : y ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  U•  t = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' xu ◦ t) : x ∈ F∗  pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and U = {(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) : y ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Vt = {(t ◦ xd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x) : x ∈ F∗  pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and V = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' 0) : x ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  V •  t = {(xd ◦ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x) : x ∈ F∗  pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and V = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' 0) : x ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Xt = {(tηd  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x) : x ∈ F∗  pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and X = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' 0) : x ∈ Fpn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Yt = {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' tηu  x) : x ∈ F∗  pn} if t ∈ Fpn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and Y = {(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) : y ∈ Fpn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  For the above bent partitions from vectorial dual-bent functions F, F •, G, G•, M, N, by set-  ting α = 1, u = ps + p − 1 with gcd(u, pn − 1) = 1, then we can obtain bent partitions  Γ1, Γ•  1, Γ2, Γ•  2, Θ1, Θ2 defined by (5)-(10) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Thus by the above analysis, we provide  an alternative derivation that Γ1, Γ•  1, Γ2, Γ•  2, Θ1, Θ2 are bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  When p is an odd prime, we show that the converse of Theorem 1 also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let p be an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let Γ = {Ai, i ∈ V (p)  s  } be a bent partition of V (p)  n  satisfying  Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define F : V (p)  n  → V (p)  s  by  F(x) = i if x ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Then F is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  18  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Since F∗  pAi = Ai for any i ∈ V (p)  s  , all bent functions constructed from Γ are (p−1)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  When s = 1, the conclusion follows from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In the following, we consider the case  of s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let f be an arbitrary bent function constructed from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='4 of [13], for any  u ∈ V (p)  n  and j ∈ Fp we have  χu(Df,j) =  \uf8f1  \uf8f2  \uf8f3  pn−1δ{0}(u) + εp  n  2 −1(p − 1), if f ∗(u) = j,  pn−1δ{0}(u) − εp  n  2 −1, if f ∗(u) ̸= j,  (31)  where Df,j = {x ∈ V (p)  n  : f(x) = j}, j ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For any fixed u ∈ V (p)  n , since  {χu(Df,j), j ∈ Fp} = {pn−1δ{0}(u) + εp  n  2 −1(p − 1), pn−1δ{0}(u) − εp  n  2 −1}  for any bent function f constructed from Γ, we have that for any fixed u ∈ V (p)  n , there exists a  unique G(u) ∈ V (p)  s  such that χu(Ai), i ̸= G(u) are all the same and χu(Ai) ̸= χu(AG(u)), i ̸=  G(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Note that G is a function from V (p)  n  to V (p)  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Moreover by (31), for any fixed u ∈ V (p)  n  we have  χu(Ai) =  \uf8f1  \uf8f2  \uf8f3  pn−sδ{0}(u) + εp  n  2 −s(ps − 1), if i = G(u),  pn−sδ{0}(u) − εp  n  2 −s, if i ̸= G(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (32)  Define  Wi = {u ∈ V (p)  n  : G(u) = i}, i ∈ V (p)  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Then obviously Wi, i ∈ V (p)  s  are pairwise disjoint and �  i∈V (p)  s  Wi = V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By (32), for any  u ∈ V (p)  n  and nonempty set I ⊆ V (p)  s  we have  χu(DF,I) =  �  i∈I  χu(Ai) = pn−sδ{0}(u)|I| + εp  n  2 −s(psδWI(u) − |I|),  (33)  where DF,I = {x ∈ V (p)  n  : F(x) ∈ I}, WI = �  i∈I Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By (33) and Lemma 1, the conclusion  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  When p is an odd prime, from Theorems 1 and 2 we obtain a characterization of bent partitions  satisfying Condition C in terms of vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let p be an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let Γ = {Ai, i ∈ V (p)  s  } be a partition of V (p)  n , where n is  even, s ≤ n  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define F : V (p)  n  → V (p)  s  as  F(x) = i if x ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Then Γ is a bent partition satisfying Condition C if and only if F is a vectorial dual-bent function  satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  19  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' CONSTRUCTING BENT PARTITIONS FROM VECTORIAL DUAL-BENT FUNCTIONS  In this section, we construct bent partitions from vectorial dual-bent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  The following theorem provides a secondary construction of vectorial dual-bent functions,  which can be used to generate more bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let n, m, s be positive integers for which n is even and s ≤ n  2, s | m, s ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For  any i ∈ Fps, let F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x) : V (p)  n  → Fps be a vectorial dual-bent function with ((F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x))c)∗ =  ((F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x))∗)c and ε(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))c = ε for any c ∈ F∗  ps, where (F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x))∗ is a vectorial dual of F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x)  and ε ∈ {±1} is a constant independent of i, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let α, β ∈ Fpm be linearly independent over Fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Let R be a permutation over Fpm with R(0) = 0 and T : Fps → Fps be an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Define H : V (p)  n  × Fpm × Fpm → Fps as  H(x, y1, y2) = F(T rm  s (αR(y1ypm−2  2  ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x) + T rm  s (βR(y1ypm−2  2  )) + T (T rm  s (αR(y1ypm−2  2  ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Then H is a vectorial dual-bent function satisfying Condition A and Γ = {Ai, i ∈ Fps} is a bent  partition satisfying Condition C, where Ai = {(x, y1, y2) ∈ V (p)  n  ×Fpm ×Fpm : H(x, y1, y2) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Denote  d(y) = Trm  s (βR(y1ypm−2  2  )), e(y) = Trm  s ((β − α)R(y1ypm−2  2  )), y = (y1, y2) ∈ Fpm × Fpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  For any c ∈ F∗  ps and (a, b) = (a, b1, b2) ∈ V (p)  n  × Fpm × Fpm, we have  WHc(a, b)  =  �  x∈V (p)  n  �  y=(y1,y2)∈Fpm×Fpm  ζT rs  1(cF (d(y)−e(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))+T rs  1(cd(y))+T rs  1(cT (d(y)−e(y)))  p  ζ−⟨a,x⟩n−T rm  1 (b1y1+b2y2)  p  =  �  i∈Fps  �  y=(y1,y2)∈Fpm×Fpm:d(y)−e(y)=i  �  x∈V (p)  n  ζT rs  1(cF (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))+T rs  1(cd(y))+T rs  1(cT (i))  p  ζ−⟨a,x⟩n−T rm  1 (b1y1+b2y2)  p  = p−s �  i∈Fps  W(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))c(a)ζT rs  1(cT (i))  p  �  y=(y1,y2)∈Fpm×Fpm  ζT rs  1(cd(y))−T rm  1 (b1y1+b2y2)  p  �  j∈Fps  ζT rs  1(cj(i−(d(y)−e(y))))  p  = p−s �  i∈Fps  W(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))c(a)ζT rs  1(cT (i))  p  �  j∈Fps  ζT rs  1(ijc)  p  �  y=(y1,y2)∈Fpm×Fpm  ζT rs  1(c((1−j)d(y)+je(y)))−T rm  1 (b1y1+b2y2)  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  By Theorem 3 of [10], for any j ∈ Fps, J(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y) = (1−j)d(y)+je(y) is a partial spread vectorial  dual-bent function with ε(J(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='y))c = 1 and ((J(j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' y))c)∗ = ((1−j)d∗(y)+je∗(y))c for any c ∈ F∗  ps,  January 3, 2023  DRAFT  20  where d∗(y) = Trm  s (βR(−ypm−2  1  y2)), e∗(y) = Trm  s ((β − α)R(−ypm−2  1  y2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Therefore,  WHc(a, b)  = pm−s �  i∈Fps  W(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))c(a)ζT rs  1(cT (i))  p  �  j∈Fps  ζT rs  1(ijc)  p  ζT rs  1(c((1−j)d∗(b)+je∗(b)))  p  = pm−sζT rs  1(cd∗(b))  p  �  i∈Fps  W(F (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))c(a)ζT rs  1(cT (i))  p  �  j∈Fps  ζT rs  1(cj(i−(d∗(b)−e∗(b))))  p  = pmζT rs  1(cd∗(b))  p  W(F (d∗(b)−e∗(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))c(a)ζT rs  1(cT (d∗(b)−e∗(b)))  p  = εp  n  2 +mζ  ((F (T rm  s (αR(−bpm−2  1  b2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))c)∗(a)+T rs  1(cT rm  s (βR(−bpm−2  1  b2)))+T rs  1(cT (T rm  s (αR(−bpm−2  1  b2))))  p  = εp  n  2 +mζ((F (T rm  s (αR(−bpm−2  1  b2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='x))∗)c(a)+T rs  1(cT rm  s (βR(−bpm−2  1  b2)))+T rs  1(cT (T rm  s (αR(−bpm−2  1  b2))))  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (34)  Note that ε = 1 if p = 2 since all Boolean bent functions are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By (34), H is a vectorial  bent function with (Hc)∗ = Gc and εHc = ε for any c ∈ F∗  ps, where  G(a, b1, b2) = (F(T rm  s (αR(−bpm−2  1  b2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x))∗(a) + T rm  s (βR(−bpm−2  1  b2)) + T (T rm  s (αR(−bpm−2  1  b2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Since Hc is weakly regular bent, we have that Gc = (Hc)∗ is also weakly regular bent and G is  vectorial bent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Thus, H is vectorial dual-bent with (Hc)∗ = (H∗)c and εHc = ε for any c ∈ F∗  ps,  where H∗ = G, that is, H satisfies Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Theorem 1, the partition Γ generated from  H is a bent partition satisfying Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  The following explicit construction of bent partitions is an immediate result of Proposition 3  and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let n, m, s be positive integers with s | n, s | m, s ̸= m, and ui, i ∈ Fps be  integers for which for any i ∈ Fps, ui ≡ pji mod (ps − 1) for some 0 ≤ ji ≤ s − 1 and  gcd(ui, pn − 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For any i ∈ Fps, let di be an integer with uidi ≡ 1 mod (pn − 1), and  Pi = (Fpn, +, ◦i) be a (pre)semifield for which its dual P ⋆  i is right Fps-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For any i ∈ Fps,  let F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x1, x2) : Fpn × Fpn → Fps be an arbitrary vectorial dual-bent function constructed by  Proposition 3 with u = ui, d = di, P = Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let α, β ∈ Fpm be linearly independent over Fps, R  be a permutation over Fpm with R(0) = 0 and T : Fps → Fps be an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Define  H : Fpn × Fpn × Fpm × Fpm → Fps as  H(x1, x2, y1, y2) = F(T rm  s (αR(y1ypm−2  2  ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x1, x2) + T rm  s (βR(y1ypm−2  2  )) + T (T rm  s (αR(y1ypm−2  2  ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Then  Γ = {Ai, i ∈ Fps}  is a bent partition satisfying Condition C, where  Ai = {(x1, x2, y1, y2) ∈ Fpn × Fpn × Fpm × Fpm : H(x1, x2, y1, y2) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  21  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' With the same notation as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Note that in Theorem 4, by setting vectorial  dual-bent functions H constructed by Theorem 5 as building blocks (that is, as F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x)), we can  obtain more explicit vectorial dual-bent functions which can generate more bent partitions by  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  We give an example by using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let p = 3, s = 4, n = m = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let α be a primitive element of F38 and β = 1, R be  the identity map and T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For any i ∈ F34, let  F(i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' x1, x2) =  \uf8f1  \uf8f2  \uf8f3  Tr8  4(x−89  1  x2), if i ∈ F∗  34,  Tr8  4(x1x−83  2  ), if i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Then  H(x1, x2, y2, y2) = (Tr8  4(αy1y6559  2  ))80(Tr8  4(x−89  1  x2 − x1x−83  2  )) + Tr8  4(x1x−83  2  + y1y6559  2  ),  and Γ = {DH,i, i ∈ F34} is a bent partition satisfying Condition C, where DH,i = {(x1, x2, y1, y2) ∈  (F38)4 : H(x1, x2, y1, y2) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' RELATIONS BETWEEN BENT PARTITIONS AND PARTIAL DIFFERENCE SETS  In this section, by taking vectorial dual-bent functions as the link between bent partitions and  partial difference sets, we give a sufficient condition on constructing partial difference sets from  bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' When p is an odd prime, we characterize bent partitions satisfying Condition C  in terms of partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let (G, +) be a finite abelian group of order v and D be a subset of G with k  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then D is called a (v, k, λ, µ) partial difference set of G, if the expressions d1 − d2,  for d1 and d2 in D with d1 ̸= d2, represent each nonzero element in D exactly λ times, and  represent each nonzero element in G \\ D exactly µ times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' When λ = µ, then D is called a  (v, k, λ) difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Note that if D is a partial difference set of G with −D = D, then so are D∪{0}, D \\ {0}, G \\ D  (see [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' There is an important tool to characterize partial difference sets in terms of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  22  Lemma 3 ( [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let G be an abelian group of order v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Suppose that D is a subset of G with  k elements which satisfies −D = D and 0 /∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then D is a (v, k, λ, µ) partial difference set  if and only if for each non-principal character χ of G,  χ(D) = β ±  √  ∆  2  ,  where χ(D) = �  x∈D χ(x), β = λ − µ, γ = k − µ, ∆ = β2 + 4γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  When p is an odd prime or s ≥ 2, we give the value distribution of vectorial dual-bent  functions satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let F : V (p)  n  → V (p)  s  be a vectorial dual-bent function satisfying Condition A,  where p is odd or s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then  |DF,F (0)| = pn−s + εp  n  2 −s(ps − 1), |DF,i| = pn−s − εp  n  2 −s if i ̸= F(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Note that if f is a weakly regular p-ary bent function, then for any a ∈ Fp, f − a is  a weakly regular bent function with (f − a)∗ = f ∗ − a and εf−a = εf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Since F is a vectorial  dual-bent function with (Fc)∗ = (F ∗)c, c ∈ V (p)  s  \\{0}, we have that F(x) − F(0) is a vectorial  bent function and for any c ∈ V (p)  s  \\{0},  ((F − F(0))c)∗ = (Fc)∗ − ⟨c, F(0)⟩s = (F ∗)c − ⟨c, F(0)⟩s = (F ∗ − F(0))c,  which implies that F(x) − F(0) is a vectorial dual-bent function with ((F − F(0))c)∗ = (F ∗ −  F(0))c and ε(F −F (0))c = ε for any c ∈ V (p)  s  \\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By the proof of Theorem 1, F(ax) = F(x) for  any a ∈ F∗  p and thus F(x) = F(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Corollary 1 of [22] (Note that although Corollary 1 of  [22] only considers the case of p being odd, the conclusion of Corollary 1 of [22] also holds  for p = 2, s ≥ 2), we have  |DF −F (0),0| = pn−s + εp  n  2 −s(ps − 1), |DF −F (0),i| = pn−s − εp  n  2 −s if i ̸= 0,  that is,  |DF,F (0)| = pn−s + εp  n  2 −s(ps − 1), |DF,i| = pn−s − εp  n  2 −s if i ̸= F(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  In the following, we give a characterization of vectorial dual-bent functions satisfying Con-  dition A in terms of partial difference sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  23  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let n be an even positive integer, s be a positive integer with s ≤ n  2, and F :  V (p)  n  → V (p)  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' The following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (1) F is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (2) When p = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' s = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then the support supp(F) of F defined as supp(F) = {x ∈ V (2)  n  :  F(x) = 1} is a (2n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' 2n−1 ± 2  n  2 −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' 2n−2 ± 2  n  2 −1) difference set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' and when p is odd or s ≥ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  then for any nonempty set I ⊆ V (p)  s  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='I\\{0} is a (pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' µ) partial difference set for which  −DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='I = DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='I and if F(0) ∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then  k = pn−s|I| + εp  n  2 −s(ps − |I|) − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  λ = pn−2s|I|2 + εp  n  2 −s(ps − |I|) − 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  µ = pn−2s|I|2 + εp  n  2 −s|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (35)  and if F(0) /∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' then  k = pn−s|I| − εp  n  2 −s|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  λ = pn−2s|I|2 + εp  n  2 −s(ps − 3|I|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  µ = pn−2s|I|2 − εp  n  2 −s|I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (36)  where DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='I = {x ∈ V (p)  n  : F(x) ∈ I} and ε ∈ {±1} is a constant (ε = 1 if p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' It is easy to see that a Boolean function F is a vectorial dual-bent function satisfying  Condition A if and only if F is bent, that is, Condition A is trivial for any Boolean bent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  By the well-known result that a Boolean function F : V (2)  n  → F2 is bent if and only if its support  supp(F) = {x ∈ V (2)  n  : F(x) = 1} is a (2n, 2n−1 ± 2  n  2 −1, 2n−2 ± 2  n  2 −1) difference set (see [11]),  the conclusion obviously holds for p = 2, s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' In the following, we prove the conclusion for  p being odd or s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (1) ⇒ (2): By the proof of Theorem 1, F(−x) = F(x), that is, −DF,I = DF,I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' For any  u ∈ V (p)  n \\{0}, with the same argument as in the proof of Theorem 2 of [22],  χu(DF,I) =  \uf8f1  \uf8f2  \uf8f3  εp  n  2 − εp  n  2 −s|I|, if F ∗(−u) ∈ I,  −εp  n  2 −s|I|, if F ∗(−u) /∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  where ε = 1 if p = 2 since all Boolean bent functions are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  If F(0) ∈ I, then |DF,I\\{0}| = |DF,I|−1 and χu(DF,I\\{0}) = χu(DF,I)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Proposition 4,  |DF,I\\{0}| = (|I|−1)(pn−s−εp  n  2 −s)+(pn−s+εp  n  2 −s(ps−1)−1) = pn−s|I|+εp  n  2 −s(ps−|I|)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  By Lemma 3, DF,I\\{0} is a (pn, k, λ, µ) partial difference set, where k, λ, µ are given in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  24  If F(0) /∈ I, then |DF,I\\{0}| = |DF,I| and χu(DF,I\\{0}) = χu(DF,I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Proposition 4,  |DF,I\\{0}| = |I|(pn−s − εp  n  2 −s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Lemma 3, DF,I\\{0} is a (pn, k, λ, µ) partial difference set,  where k, λ, µ are given in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (2) ⇒ (1): By Lemma 3, for any u ∈ V (p)  n  and nonempty set I ⊆ V (p)  s  we have  χu(DF,I) = pn−sδ{0}(u)|I| + εp  n  2 − εp  n  2 −s|I| or χu(DF,I) = pn−sδ{0}(u)|I| − εp  n  2 −s|I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (37)  For any i ∈ V (p)  s  , define Wi = {u ∈ V (p)  n  : χu(DF,i) = pn−sδ{0}(u) + εp  n  2 − εp  n  2 −s}, where  DF,i = {x ∈ V (p)  n  : F(x) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We claim that Wi  � Wi′ = ∅ for any i ̸= i′ and �  i∈V (p)  s  Wi = V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Indeed, if there exist i ̸= i′ such that Wi  � Wi′ ̸= ∅, that is, there exists u ∈ V (p)  n  such that  χu(DF,i) = χu(DF,i′) = pn−sδ{0}(u) + εp  n  2 − εp  n  2 −s, then χu(DF,i  � DF,i′) = 2pn−sδ{0}(u) +  2εp  n  2 − 2εp  n  2 −s, which contradicts with (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Thus, Wi  � Wi′ = ∅ for any i ̸= i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' If there exists  u ∈ V (p)  n  such that u /∈ Wi for any i ∈ V (p)  s  , that is, χu(DF,i) = pn−sδ{0}(u) − εp  n  2 −s for  any i ∈ V (p)  s  , then χu(V (p)  n ) = �  i∈V (p)  s  χu(DF,i) = pnδ{0}(u) − εp  n  2 , which contradicts with  χu(V (p)  n ) = �  x∈V (p)  n  ζ⟨u,x⟩n  p  = pnδ{0}(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Thus, �  i∈V (p)  s  Wi = V (p)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By the above analysis, we  have  χu(DF,I) = pn−sδ{0}(u)|I| + εp  n  2 −s(psδWI(u) − |I|),  (38)  where WI = �  i∈I Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By (38) and Lemma 1, F is a vectorial dual-bent function satisfying  Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  The following theorem provides a sufficient condition on constructing partial difference sets  from bent partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let n be an even positive integer and s be a positive integer with s ≤ n  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Assume  that Γ = {Ai, i ∈ V (p)  s  } is a bent partition of V (p)  n  for which the function F : V (p)  n  → V (p)  s  defined by  F(x) = i if x ∈ Ai  is a vectorial dual-bent function satisfying Condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then when p = 2, s = 1, A0 and A1 are  (2n, 2n−1 ± 2  n  2 −1, 2n−2 ± 2  n  2 −1) difference set and (2n, 2n−1 ∓ 2  n  2 −1, 2n−2 ∓ 2  n  2 −1) difference set,  respectively, and when p is odd or s ≥ 2, for any nonempty set I ⊆ V (p)  s  , AI\\{0} = �  i∈I Ai\\{0}  is a (pn, k, λ, µ) partial difference set, where (k, λ, µ) are given in (35) if 0 ∈ AI and (k, λ, µ)  are given in (36) if 0 /∈ AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  January 3, 2023  DRAFT  25  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Note that if D is a (v, k, λ) difference set of a finite abelian group G, then G\\D is a  (v, v − k, v − 2k + λ) difference set of G (for instance see [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then the result follows from  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Proposition 3, the bent partition Γ1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Γ2, Γ•  1, Γ•  2, Θ1, Θ2) satisfies the  condition in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Theorem 7, any union of sets from Γ1 (resp, Γ2, Γ•  1, Γ•  2, Θ1, Θ2)  forms a partial difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Thus, the results given in Corollary 15 of [1] on constructing  partial difference sets from Γ1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Γ2, Γ•  1, Γ•  2, Θ1, Θ2) (which includes the results given in  Theorem 2 of [2] on constructing partial difference sets from Γ1, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Γ2, in the finite field)  can also be illustrated by our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Since the bent partitions constructed in Theorem 5 satisfy the condition in Theorem 7, we  have the following corollary from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let Γ = {Ai, i ∈ Fps} be a bent partition constructed by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then when  p = 2, s = 1, A0 and A1 are (2n, 2n−1 ± 2  n  2 −1, 2n−2 ± 2  n  2 −1) difference set and (2n, 2n−1 ∓  2  n  2 −1, 2n−2 ∓ 2  n  2 −1) difference set, respectively, and when p is odd or s ≥ 2, for any nonempty  set I ⊆ Fps, AI\\{0} = �  i∈I Ai\\{0} is a (pn, k, λ, µ) partial difference set, where (k, λ, µ) are  given in (35) with ε = 1 if 0 ∈ AI and (k, λ, µ) are given in (36) with ε = 1 if 0 /∈ AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  We give an example by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let Γ = {DH,i, i ∈ F34} be the bent partition constructed in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By Corol-  lary 2, DH,i is a (1853020188851841, 22876791923520, 282470988879, 282429005040) partial difference  set for any i ∈ F∗  34, DH,0\\{0} is a (1853020188851841, 22876834970240, 282472051759, 282430067922)  partial difference set, (DH,0  � DH,1)\\{0} is a (1853020188851841, 45753626893760, 1129760129761,  1129719208806) partial difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  When p is an odd prime, we immediately obtain the following characterization of bent  partitions of V (p)  n  satisfying Condition C from Theorems 3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let p be an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Let Γ = {Ai, i ∈ V (p)  s  } be a partition of V (p)  n , where n is  even and s ≤ n  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' Then the following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (1) Γ is a bent partition satisfying Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  (2) For any nonempty set I ⊆ V (p)  s  , AI\\{0} = �  i∈I Ai\\{0} is a (pn, k, λ, µ) partial difference  January 3, 2023  DRAFT  26  set with −AI = AI, where (k, λ, µ) are given in (35) if 0 ∈ AI and (k, λ, µ) are given in (36)  if 0 /∈ AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' CONCLUSION  In this paper, we investigated relations between bent partitions and vectorial dual-bent functions  (Theorems 1, 2, 3) and gave some new constructions of bent partitions satisfying Condition C  (Corollary 1, Theorems 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' We illustrated that for any ternary weakly regular bent function  f : V (3)  n  → F3 (n even) with f(x) = f(−x) and εf = −1, the generated bent partition by f  is not coming from a normal bent partition (see Example 1), which answers an open problem  proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' By taking vectorial dual-bent functions as the link between bent partitions and  partial difference sets, we give a sufficient condition on constructing partial difference sets from  bent partitions (Theorem 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' When p is an odd prime, we characterized bent partitions satisfying  Condition C in terms of partial difference sets (Theorem 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
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+page_content=' Meidl, A construction of bent functions from plateaued functions, Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
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+page_content=' C¸ es¸melio˘glu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
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+page_content='  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdAyT4oBgHgl3EQfsvlL/content/2301.00581v1.pdf'}
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diff --git a/N9E4T4oBgHgl3EQfjw0A/content/tmp_files/2301.05144v1.pdf.txt b/N9E4T4oBgHgl3EQfjw0A/content/tmp_files/2301.05144v1.pdf.txt
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+Draft version January 13, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+The implications of large binding energies of massive stripped core collapse supernova progenitors on
+the explosion mechanism
+Dmitry Shishkin1 and Noam Soker1
+1Department of Physics, Technion, Haifa, 3200003, Israel; s.dmitry@campus.technion.ac.il; soker@physics.technion.ac.il
+(Dated: January 2023)
+ABSTRACT
+We examine the binding energy of massive stripped-envelope core collapse supernova (SECCSN)
+progenitors with the stellar evolution code mesa, and find that only the jittering jets explosion mech-
+anism can account for explosions where carbon-oxygen cores with masses of ≳ 20M⊙ collapse to leave
+a neutron star (NS) remnant. We calculate the binding energy at core collapse under the assumption
+that the remnant is a NS. Namely, stellar gas above mass coordinate of ≃ 1.5−2.5M⊙ is ejected in the
+explosion. We find that the typical binding energy of the ejecta of stripped-envelope progenitors with
+carbon-oxygen core masses of MCO ≳ 20M⊙ is Ebind ≳ 2 × 1051 erg. Since only jet-driven explosion
+mechanisms can supply such high energies, we conclude that jets must explode such cores. We apply
+our results to SN 2020qlb, which is a SECCSN with a claimed core mass of ≃ 30−50M⊙, and conclude
+that the jittering jets explosion mechanism best account for such an explosion that leaves a NS.
+Keywords: stars: jets – stars: massive – supernovae: general – supernovae: individual: 2020qlb
+1. INTRODUCTION
+The binding energy of a core collapse supernova
+(CCSN) progenitor plays a crucial role in determining
+the explosion outcome, like explosion energy and rem-
+nant mass. Typical explosion energies are estimated to
+be in the range of Eexp ≃ 1050 − 1052 erg (e.g., Yang
+& Chevalier 2015; Utrobin et al. 2015; Gal-Yam 2019;
+Burrows & Vartanyan 2021a). The binding energy of
+the most massive pre-collapse cores have similar or even
+larger values than these typical explosion energies (e.g.,
+Pejcha & Thompson 2015; Bruenn et al. 2016; Chan
+et al. 2020; Wang et al. 2022; Burrows et al. 2020). The
+explosion mechanism should both overcome the binding
+energy and account for the explosion energy (radiation
++ final kinetic energy of the ejecta).
+Two theoretical explosion mechanisms of non-rotating
+(or slowly rotating) pre-collapse cores utilize the grav-
+itational energy of the collapsing core to power CC-
+SNe. These are the delayed-neutrino explosion mech-
+anism (e.g., Bethe & Wilson 1985; Ertl et al. 2016; Bur-
+rows et al. 2020; Bruenn et al. 2020; Bollig et al. 2021;
+Burrows & Vartanyan 2021b; Zha et al. 2023) and the
+jittering-jets explosion mechanism (e.g., Soker 2010; Pa-
+pish & Soker 2011; Gilkis & Soker 2015; Soker 2019,
+2022a). Studies show that the maximum energy that the
+delayed neutrino mechanism can supply to overcome the
+binding energy of the ejecta is ≃ 2 × 1051 erg, resulting
+in maximum explosion energies (after removing binding
+energy) of Eexp ≃ 2 × 1051 erg (e.g., Fryer et al. 2012;
+Ertl et al. 2016; Sukhbold et al. 2016; Gogilashvili et al.
+2021). In addition, the delayed neutrino mechanism has
+problems in producing the observed amount of 56Ni, in
+particular in stripped-envelope CCSNe (SECCSNe; for
+a recent study see Sawada & Suwa 2023)
+The jittering jets explosion mechanism, on the other
+hand, can account for much larger explosion energies
+(e.g., Gilkis et al. 2016; Soker 2022b).
+The mag-
+neto rotational explosion mechanism that works only
+for rapidly rotating pre-collapse cores can also account
+for large explosion energies (e.g., LeBlanc & Wilson
+1970; Khokhlov et al. 1999; L´opez-C´amara et al. 2013;
+Wheeler et al. 2015; Bromberg & Tchekhovskoy 2016;
+Kuroda et al. 2020; Gottlieb et al. 2022c,a; Fujibayashi
+et al. 2022; Powell et al. 2022) because the newly born
+neutron star (NS) launches fixed-axis jets. Whether the
+explosion is by jittering jets (most CCSNe according to
+that model; e.g., Soker 2022b) or by fixed-axis jets, the
+jets operate in a negative feedback cycle, i.e., the jet
+feedback explosion mechanism (e.g., Soker 2016a). As
+well, the jets can influence the direction of later jets (e.g.,
+Papish & Soker 2014; Gottlieb et al. 2022b). Even if the
+stochastically accreted mass has sub-Keplerian angular
+arXiv:2301.05144v1  [astro-ph.HE]  12 Jan 2023
+
+2
+momentum, it might still form an accretion belt that
+can launch jets (e.g., Schreier & Soker 2016; Garain &
+Kim 2022).
+The above discussion implies that CCSNe with large
+kinetic energies of Eexp ≳ 2 × 1051, e.g., SN 2020jfo
+(Ailawadhi et al. 2022 – Eexp
+=
+2.9 × 1051 erg);
+SN 2020qlb (West et al. 2022 – Eexp = 20 × 1051 erg);
+SN 2012au (Pandey et al. 2021 – Eexp = 4.8 − 5.4 ×
+1051 erg) require jets to drive the explosion.
+In this study we examine the interesting case of the
+hydrogen-poor and super-energetic CCSN SN 2020qlb.
+West et al. (2022) estimate the kinetic energy of
+SN 2020qlb as ≃ 20 × 1051 erg and suggest that a mag-
+netar supplies the large amount of energy of the ejecta.
+They also provide fitting parameters for the magnetar
+model (e.g., Maeda et al. 2007; Kasen & Bildsten 2010;
+Woosley 2010; Metzger et al. 2015; Nicholl et al. 2017;
+Gomez et al. 2022) and estimate the progenitor pre-
+explosion mass, namely, about the ejecta plus the rem-
+nant mass, to be Mej + Mrem ≃ few × 10M⊙.
+It seems nonetheless, that there are two reasons why
+the explosion of SN 2020qlb must be driven by jets and
+not by the delayed neutrino mechanism. The first one is
+that the formation of an energetic magnetar must be ac-
+companied by the launching of even more energetic jets
+at the explosion itself and possibly after the explosion
+as well (e.g. Soker 2016b, 2017; Soker & Gilkis 2017;
+Shankar et al. 2021; Soker 2022c,b). The second rea-
+son is the new finding of the present study, where by a
+stellar evolutionary code (section 2) we show (section 3)
+that the binding energies of such massive cores are much
+above what the delayed neutrino mechanism can supply
+(Janka 2012; Soker 2022b). In section 4 we summarize
+our results and discuss their implications in the context
+of the jet feedback explosion mechanism.
+2. NUMERICAL SCHEME
+We use the stellar evolution code mesa (Paxton et al.
+2010, 2013, 2015, 2018, 2019) to simulate the struc-
+ture of 52 CCSN progenitor models, all with initial
+metalicity of z = 0.02.
+We base our numerical rou-
+tine on the ‘20M pre ms to core collapse’ example from
+mesa r22.05.1, but simulate the evolution using ver-
+sion r15140. Starting from the zero age main sequence
+(ZAMS), we evolve the star until center He depletion
+(when helium abundance in the center is either ≃ 1%
+or ≃ 5%), at which point we numerically remove the
+hydrogen-rich envelope, leaving only the He core. We
+then evolve the star until core-collapse. We introduce
+several changes to the example routine to both adhere
+to our requirements, i.e., having at least 21 isotopes
+and having sufficiently high resolution (Shishkin & Soker
+2021), and to fit mesa version r15140. We expand on
+the numerical scheme in appendix A.
+By varying the ZAMS stellar mass, the wind parame-
+ters, and the exact time when we numerically remove
+the envelope we obtain stripped-envelope CCSN pro-
+genitors, i.e., hydrogen-poor stellar progenitors, with
+varying mass of a carbon-oxygen (CO) core, MCO ≃
+4 − 35M⊙. This mass range of SECCSNe corresponds
+to ZAMS stellar masses range of ≃ 20 − 82M⊙. In some
+cases the core contains several solar masses of helium,
+while in other cases the core has a much lower helium
+mass, depending on the above parameters.
+Because the inner part of the core collapses to form
+a NS or a black hole (BH) remnant, only the binding
+energy of the outer core is relevant to our study. We
+calculate the binding energy of the outer core, Ebind(r),
+by integrating over the sum of the internal energy and
+gravitational energy from the surface down to the mass
+coordinate that separates the ejecta and the final rem-
+nant m = Min, i.e., the inner boundary of the ejecta. We
+calculate the binding energy of the ejecta for two values
+of this mass coordinate Min = 1.5 , 2.5M⊙, because we
+consider cases where the remnant is a NS.
+We refer to the latest evolution point we simulate
+in the stellar evolution as collapse. This point of col-
+lapse must adhere to an iron core more massive than
+MFe > 1.5, where we assume collapse is imminent. This
+value of iron core mass is more or less when the iron
+core mass reaches its maximum value (as at the onset
+of collapse the iron disintegrates). About half of simu-
+lations reach infall velocities at the edge of the iron core
+of vfe,infall > 100 km s−1. Other simulations encounter
+numerical difficulties and we had to terminate them at
+somewhat earlier times.
+3. RESULTS
+3.1. Binding Energy towards Collapse
+As the stellar core evolves towards collapse and nu-
+cleosynthesis of heavier elements takes place, the core
+becomes denser and the binding energy of the inner lay-
+ers of the star increases. We demonstrate this for one
+stellar model of carbon-oxygen core mass (mainly oxy-
+gen) of MCO = 13.2M⊙ in Fig 1 where we present the
+star at three times: at center oxygen depletion, at cen-
+ter silicon depletion, and at collapse. We present the
+composition of the main isotopes by lines with differ-
+ent colors and the binding energy Ebind(m) by the black
+lines. Here Ebind(m) is the binding energy (gravitational
++ internal) of the envelope laying above mass coordi-
+nate m. Relevant to this study is the binding energy
+of the ejecta, which is the mass above mass coordinate
+m = Min ≃ 1.5, 2.5M⊙. The mass Min is the baryonic
+
+3
+Figure 1. Abundances and the binding energy as function
+of mass coordinate at three times for a SECCSN (hydrogen-
+poor) progenitor model with carbon-oxygen core mass of
+M collapse
+CO
+= 13.2M⊙, which corresponds to a ZAMS mass of
+MZAMS ≈ 40M⊙. The colored step-like lines are the abun-
+dances according to the inset in the lower panel. The black
+and smoothly varying lines represent Ebind(m), which is the
+binding energy of the envelope laying above mass coordi-
+nate m. The three panels present these quantities at three
+different times: center oxygen depletion (upper panel, bind-
+ing energy by the black solid-line), center silicon depletion
+(middle panel; binding energy by the black dashed-line), core
+collapse (lower panel; binding energy by black dotted-line).
+Note that the middle panel contains the binding energy at
+the three times to allow for comparison. The two vertical
+lines mark the mass coordinates m = Min = 1.5M⊙ and
+m = Min = 2.5M⊙. Helium that appears only at collapse
+results from disintegration of iron.
+mass of the NS remnant (the corresponding final gravi-
+tational masses will be ≃ 1.35, 2.1M⊙). We mark these
+two masses by the vertical lines. We see that at collapse
+the binding energy Ebind(Min) is larger than at earlier
+times.
+In Fig.
+2 we present composition and binding en-
+ergy for a model with a much more massive core of
+MCO = 27M⊙.
+There are two qualitative differences
+between this model and the one we present in Fig. 1.
+The first qualitative difference is that the binding en-
+Figure 2. Similar to Fig. 1 but for a more massive core of
+M collapse
+CO
+= 26.5M⊙, which corresponds to a ZAMS mass of
+MZAMS ≈ 65M⊙. Note that the left vertical axis is scaled
+differently than in Fig. 1.
+ergy at collapse is somewhat smaller than at the earlier
+time that we present in the figure. The explanation to
+the decreasing binding energy shortly before collapse is
+that the envelope expands starting from deep in the oxy-
+gen burning shell and outwards. We find (by drawing
+the density profiles) that moving from the upper to the
+middle panel of Fig. 2 the density from m ≃ 10M⊙ and
+outward decreases, reducing the binding energy. This
+mass coordinate is deep inside the shell where oxygen
+(teal line) burns to S+Si (yellow line).
+The second qualitative difference comes from the much
+higher binding energy of the ejecta of the descendant
+CCSN of the more massive model, i.e., Ebind(Min) ≳ 2×
+1051 erg. The implication is that we do not expect that
+the neutrino driven explosion mechanism can account
+for explosions of such cores. We argue that jets explode
+these cores. We leave the discussion of this point, as
+well as our view that jets also explode cores with lower
+binding energy, to section 4, where we also refer to the
+claim of a very massive core of SN 2020qlb (West et al.
+2022).
+We first find the range of such high-binding-
+energy cores.
+
+0
+3
+OCoreDepletion
+2
+91
+erg
+0
+SiCoreDepletion
+BEatO
+dep.
+BEatSi
+dep
+BEatCollapse
+2
+0
+3
+Collapse
+2
+Helium
+Carbon
+Oxygen
+S+Si
+"Fe Group"
+2
+0
+2
+4
+6
+8
+10
+12Abundance6
+0
+4
+OCoreDepletion
+2
+6
+0
+Si Core Depletion
+BEatOont
+. dep.
+- BEat Si
+dep
+"BEatCollapse
+2
+0
+4
+Collapse
+2
+Helium
+Oxygen
+"Fe Group'
+Carbon
+S+Si
+2
+0
+4
+8
+12
+16
+20
+24Abundance4
+3.2. High-binding-energy cores
+We search for the mass range of cores that have bind-
+ing energies at collapse of Ebind(Min) ≳ 2×1051 erg. We
+present the results in Fig. 3. We present the binding en-
+ergy for an inner ejecta mass coordinate of Min = 2.5M⊙
+(upper panel) and Min = 1.5M⊙ (lower panel).
+We
+focus on the binding energy of these two mass coordi-
+nates Min = 1.5 − 2.5M⊙ as the iron core masses at col-
+lapse falls within this mass range. The horizontal line
+at 2 × 1051 erg is the approximate energy above which
+we do not expect that neutrino heating by itself can ex-
+plode the core. In appendix B we provide linear fits to
+the binding energy at collapse as function of CO core
+mass for these two mass coordinates Min = 1.5, 2.5M⊙
+(table B.1).
+We simulated 52 stripped-hydrogen envelope cases. In
+27 cases the cores reach collapse as we present by the
+red circles in the figure. In 25 cases cases the numeri-
+cal code encountered difficulties and we had to stop the
+simulation before reaching collapse. In these cases we
+extrapolate from an evolutionary time before collapse
+to the collapse time as we explain in appendix B (red
+stars in the figure). We also include 35 in the figure the
+binding energies of models with hydrogen-rich envelope
+that we take from Shishkin & Soker (2022), as we mark
+by open purple circles.
+From Fig. 3 (with a more rigorous derivation in ap-
+pendix B) we draw our conclusion that in cases where
+the inner mass of Min = 2.5M⊙ of the core collapses
+to form a NS, the delayed neutrino mechanism cannot
+explode cores with masses of MCO ≳ 15M⊙ (or maybe
+rarely do so). For Min = 1.5M⊙ we find this limit to be
+MCO ≳ 13M⊙.
+In Fig. 4 we present a more detailed binding energy
+profile of the pre-collapse stripped-envelope models that
+we simulated. When we take the lowest binding energy
+of the inner core at the onset of collapse we find the limit
+of core mass that the neutrino-driven explosion cannot
+account for to be MCO > 20M⊙. Namely, still a large
+range.
+In table B.1 we provide the linear fit parameters for
+the binding energy at collapse for the edge of the iron
+core (gray squares in the figure) and the binding energy
+curve break (black circles). The binding energy curve
+break is point we refer to as separating the inner core
+from the outer core, and is the point of lowest binding
+energy for most simulated cases that reached collapse.
+4. DISCUSSION AND SUMMARY
+We simulated the evolution of 52 massive SECCSN
+progenitor models corresponding to ZAMS masses of
+20 ≲ MZAMS ≲ 82. We removed the entire hydrogen-
+rich envelope, and calculated the binding energy just
+before core collapse.
+The final core mass depends on
+the ZAMS mass and on the mass loss parameter (ap-
+pendix C). We present the structure of the pre-collapse
+progenitor for two cases in Figs. 1 and 2. We find that
+to a fare accuracy we can linearly fit the binding energy
+of these stripped-envelope progenitors to the CO core
+mass MCO (Fig. 3 and table B.1).
+We present our main results in Fig.
+3.
+In those
+figures the horizontal gray line represents the approx-
+imate maximum energy that the neutrino-driven mech-
+anism can supply, Emax
+ν
+= 2 × 1051 erg. We find that
+the binding energy calculated at Min = 1.5M⊙ and
+Min = 2.5M⊙, of progenitors with a carbon-oxygen core
+mass of MCO ≳ 13M⊙ and MCO ≳ 15M⊙, respectively,
+are larger than Emax
+ν
+. Namely,
+Emax
+ν
+≲
+�
+�
+�
+Ebind,1.5
+for
+MCO ≳ 13M⊙
+Ebind,2.5
+for
+MCO ≳ 15M⊙.
+(1)
+The main conclusion is that the delayed neutrino ex-
+plosion mechanism cannot explode stars with a core
+mass of MCO ≳ 13 − 15M⊙. The jittering jets explosion
+mechanism, on the other hand, has no limiting explosion
+energy in these ranges as it is fueled by accretion onto
+the compact remnant (e.g., (Gilkis et al. 2016; Soker &
+Gilkis 2017)).
+Let us apply our results to a specific SECCSN. In a
+recent paper West et al. (2022) deduce that SN 2020qlb
+had an explosion energy of ≃ 20×1051 erg and estimate
+the progenitor pre-explosion mass, ejecta plus remnant
+mass, to be Mej + Mrem ≃ 30 − 50M⊙. According to
+our results the binding energy alone of such cores is
+Ebind > 3 × 1051 erg. We therefore conclude that jets
+must have exploded SN 2020qlb. Jets can also supply
+the kinetic energy of the ejecta. Namely, jet-driven ex-
+plosions might make the magnetar powering less criti-
+cal or not needed at all (although a magnetar might be
+present). Most likely the explosion is via jittering jets.
+The reason is that an explosion driven by a fixed-axis
+jets, like if the core is rapidly rotating, will not expel
+mass from the equatorial region, which it turn is ac-
+creted by the newly formed central object. Therefore,
+the final mass of the remnant will be large and the rem-
+nant will be a BH (see discussion in Soker 2022b).
+On a large scope, our study adds to the growing evi-
+dence pointing to the major roles that jets play in the
+explosion, as well as pre-explosion and post-explosion,
+of CCSNe (for a recent review see Soker 2022b).
+ACKNOWLEDGMENTS
+This research was supported by a grant from the Israel
+Science Foundation (769/20).
+
+5
+Figure 3. The binding energies of the simulated models as a function of the carbon-oxygen core mass nearing collapse (vertical
+axis). The panels show the final binding energy at two mass coordinates: Min = 2.5M⊙ (top) and Min = 1.5M⊙ (bottom). The
+red data points (filled circles and stars at the outer panels) are stripped-envelope (SE) models (SECCSNe), whilst purple empty-
+circles data points are models from Shishkin & Soker (2022) that have hydrogen rich envelopes. Red stars are extrapolated
+data points, as explained in appendix B. The horizontal line at 2 × 1051 erg denotes the binding energy above which we do not
+expect the neutrino delayed explosion mechanism to explode the core.
+Data availability
+The data underlying this article will be shared upon
+reasonable request to the corresponding author.
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+9
+APPENDIX
+A. NUMERICAL PRESCRIPTION DETAILS
+Our numerical scheme files (‘inlists’) are a modified version of the ‘20 pre ms to cc’ mesa version r22.05.1 ‘test suite’
+example. We adapted this example to run on mesa version r15140 and incorporated certain parameters (e.g., over-
+shooting and mesh resolution) according to our previous works (Shishkin & Soker 2021; Shishkin & Soker 2022). The
+full ’inlists’ that we used are available online1. Here we mention some of the important parameters.
+In a similar fashion to our previous works which focus on the convective profile of the inner layers of massive stars
+(Shishkin & Soker 2021; Shishkin & Soker 2022), we use the exponential overshooting prescription (Herwig 2000)
+with symmetrical (both ‘bottom’ and ‘above’) and uniform (all burning regions) settings and f = 0.01, f0 = 0.004
+parameters.
+We chose the Henyey scheme (Henyey et al. 1965) for mixing length theory (MLT, B¨ohm-Vitense 1958) with
+αMLT = 1.5.
+We also enable the Ledoux criterion (Ledoux 1947) and set thermohaline option to ‘Kippenhahn’
+with ‘thermohaline coeff = 1’ alongside ‘alpha semiconvection = 0.01.
+We make use of the ’Dutch’ wind loss scheme (e.g., Vink et al. 2001; Nugis & Lamers 2000), and vary the scaling
+factor (along with the initial mass) to achieve different core masses.
+For the nuclear network we use the 22 isotopes of ‘approx21 cr60 plus co56’ (e.g., Timmes 1999), aimed at
+stellar evolution up to collapse.
+This network includes hydrogen, He3 and He4 up to the heavier isotopes of
+Fe52 , Fe54 , Fe56 , Co56 , Ni56 , Cr60.
+We scale mesh refinement gradually up to ‘max dq’ values of 1d − 4 at the later stages (from the default value of
+1d − 2) to properly resolve the fine burning features close to core collapse.
+The mesa equation of state (EOS) is a blend of the OPAL (Rogers & Nayfonov 2002), SCVH (Saumon et al. 1995),
+FreeEOS (Irwin 2004), HELM (Timmes & Swesty 2000), PC (Potekhin & Chabrier 2010), and Skye (Jermyn et al.
+2021) EOSs. Nuclear reaction rates are from JINA REACLIB (Cyburt et al. 2010), NACRE (Angulo et al. 1999) and
+additional tabulated weak reaction rates Fuller et al. (1985); Oda et al. (1994); Langanke & Mart´ınez-Pinedo (2000).
+Screening is included via the prescription of Chugunov et al. (2007). Thermal neutrino loss rates are from Itoh et al.
+(1996). Radiative opacities are primarily from OPAL (Iglesias & Rogers 1993, 1996), with low-temperature data from
+Ferguson et al. (2005) and the high-temperature, Compton-scattering dominated regime by Poutanen (2017). Electron
+conduction opacities are from Cassisi et al. (2007) and Blouin et al. (2020).
+B. BINDING ENERGY ESTIMATION
+Because of numerical difficulties of stripped-envelope progenitors (specifically some steep gradients) some simulations
+did not reach the phase of core collapse, although they did reach oxygen depletion and/or silicon depletion at the center.
+Time steps became much too short and we had to terminate the simulations before core collapse. In these cases we
+estimated the binding energy at collapse (red-stars in Fig. 3) by extrapolating the binding energy during earlier phases
+using linear fits.
+We made linear fits to the binding energies as function of the CO core masses at three evolutionary phases: oxygen
+depletion, silicon depletion, and core collapse. In Fig. B.1 we present these three fittings by blue, orange, and red
+lines, respectively, for Min = 2.5M⊙ (upper panel) and Min = 1.5M⊙ (lower panel). From these three lines we can find
+the ratio of the binding energy at core collapse to the binding energy at oxygen depletion and to the binding energy
+at silicon depletion. In cases where we did not reach core collapse we use this ratio at the given CO core mass to
+calculate the expected binding energy at core collapse. We mark these energies by red-stars in Fig. B.1 and use them
+in Fig. 3. Error bars attached to the red stars signify the 1σ intervals of the this extrapolation procedure. We note
+that the CO core mass does not change much after oxygen depletion in the non-extended helium phase. The average
+difference between the CO core mass at central oxygen depletion and at core collapse is ∆mcore
+CO = 0.06 ± 0.33M⊙.
+1 Zenodo: Modified inlists to reproduce the models. Also included
+a full simulation list and the simulated models at different time
+points.
+
+10
+Figure B.1. The binding energy of the envelope above mass coordinate Min = 2.5M⊙ (upper panel) and Min = 1.5M⊙ (lower
+panel) as a function of the final carbon-oxygen core mass. The blue circles are at central oxygen depletion (5% oxygen in the
+center), the orange circles are at silicon depletion (5% silicon in the center), and red circles are at core collapse. The three
+respective lines are the linear fit to the points. Red stars are the extrapolated values for the binding energy at collapse based
+on available earlier data points (oxygen depletion or silicon depletion) for the cases that did not reach collapse (see text).
+We fit the binding energy Ebind versus the CO core mass MCO by a linear fit Ebind = aMCO + b. In Table B.1 we
+list the values of the two coefficients for the six lines (three stage for two values of the mass that collapses to form the
+NS). We also list (last column) the number of data points that were used at each fitting.
+
+11
+aMCO + b
+Fit1.5M⊙
+Fit2.5M⊙
+FitBEbreak
+FitFecore
+No. of points
+Collapse
+a [1051erg/M⊙]
+0.122 ± 0.024
+0.137 ± 0.024
+0.117 ± 0.021
+0.114 ± 0.02
+27
+b [1051erg]
+0.537 ± 0.46
+−0.053 ± 0.444
+0.165 ± 0.423
+0.442 ± 0.406
+Sicntr depletion
+a [1051erg/M⊙]
+0.132 ± 0.017
+0.145 ± 0.02
+−−
+−−
+43
+b [1051erg]
+0.159 ± 0.38
+−0.099 ± 0.445
+−−
+−−
+Ocntr depletion
+a [1051erg/M⊙]
+0.15 ± 0.017
+0.175 ± 0.02
+−−
+−−
+47
+b [1051erg]
+0.002 ± 0.394
+−0.447 ± 0.479
+−−
+−−
+Table B.1. The linear fits to the lines in Fig. B.1 and the number of data points for each of the simulation groups: at collapse
+(second row), at center silicon depletion (third row) and center oxygen depletion (bottom row). The third and fourth columns
+are the fits to the binding energy Ebind,1.5, Ebind,2.5 at mass coordinates Min = 1.5M⊙ and Min = 2.5M⊙, respectively. In the
+fifth column we present the linear fit to the variation of the binding energy at the black dots in Fig. 4 with the CO core mass.
+In the fifth column we present the linear fit to the variation of the binding energy at the edge of the iron core (gray squares in
+Fig. 4) with the CO core mass. Linear fits are in units of energy Ebind [1051erg] to CO core mass MCO [M⊙]. Values and errors
+(2σ) are in accordance with Fig. B.1.
+
+12
+C. SIMULATIONS LIST
+In Fig. C.2 we present the simulations that we conducted in a three-parameters space. As input we show the dutch
+wind scaling factor in mesa in the range of 0.5 < αDutch,wind < 1 and the zero age main sequence (ZAMS) mass in the
+range of 20M⊙ < MZAMS < 82M⊙. As an output we present the final CO core mass (mainly oxygen mass) in units of
+solar mass according to the color bar.
+Figure C.2. Wind (Dutch scheme) scaling factors (vertical axis) and the ZAMS masses of the cases (horizontal axis) that we
+simulated, and the final CO core mass (by color bar). We denote with a black X the cases where we extended the He burning
+to a later stage before removing the hydrogen envelope.
+
diff --git a/N9E4T4oBgHgl3EQfjw0A/content/tmp_files/load_file.txt b/N9E4T4oBgHgl3EQfjw0A/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..510afb29835af4deff3f50a2521527a8ecd2b025
--- /dev/null
+++ b/N9E4T4oBgHgl3EQfjw0A/content/tmp_files/load_file.txt
@@ -0,0 +1,930 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf,len=929
+page_content='Draft version January 13, 2023  Typeset using LATEX twocolumn style in AASTeX631  The implications of large binding energies of massive stripped core collapse supernova progenitors on  the explosion mechanism  Dmitry Shishkin1 and Noam Soker1  1Department of Physics, Technion, Haifa, 3200003, Israel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='dmitry@campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='technion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='il;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' soker@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='technion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='il  (Dated: January 2023)  ABSTRACT  We examine the binding energy of massive stripped-envelope core collapse supernova (SECCSN)  progenitors with the stellar evolution code mesa, and find that only the jittering jets explosion mech-  anism can account for explosions where carbon-oxygen cores with masses of ≳ 20M⊙ collapse to leave  a neutron star (NS) remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We calculate the binding energy at core collapse under the assumption  that the remnant is a NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Namely, stellar gas above mass coordinate of ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ is ejected in the  explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We find that the typical binding energy of the ejecta of stripped-envelope progenitors with  carbon-oxygen core masses of MCO ≳ 20M⊙ is Ebind ≳ 2 × 1051 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Since only jet-driven explosion  mechanisms can supply such high energies, we conclude that jets must explode such cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We apply  our results to SN 2020qlb, which is a SECCSN with a claimed core mass of ≃ 30−50M⊙, and conclude  that the jittering jets explosion mechanism best account for such an explosion that leaves a NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Keywords: stars: jets – stars: massive – supernovae: general – supernovae: individual: 2020qlb  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' INTRODUCTION  The binding energy of a core collapse supernova  (CCSN) progenitor plays a crucial role in determining  the explosion outcome, like explosion energy and rem-  nant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Typical explosion energies are estimated to  be in the range of Eexp ≃ 1050 − 1052 erg (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Yang  & Chevalier 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Utrobin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Gal-Yam 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Burrows & Vartanyan 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The binding energy of  the most massive pre-collapse cores have similar or even  larger values than these typical explosion energies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=',  Pejcha & Thompson 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Bruenn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Chan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Burrows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The  explosion mechanism should both overcome the binding  energy and account for the explosion energy (radiation  + final kinetic energy of the ejecta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Two theoretical explosion mechanisms of non-rotating  (or slowly rotating) pre-collapse cores utilize the grav-  itational energy of the collapsing core to power CC-  SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' These are the delayed-neutrino explosion mech-  anism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Bethe & Wilson 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
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+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Bruenn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Bollig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Burrows & Vartanyan 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Zha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2023) and the  jittering-jets explosion mechanism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Soker 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Pa-  pish & Soker 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Gilkis & Soker 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Soker 2019,  2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Studies show that the maximum energy that the  delayed neutrino mechanism can supply to overcome the  binding energy of the ejecta is ≃ 2 × 1051 erg, resulting  in maximum explosion energies (after removing binding  energy) of Eexp ≃ 2 × 1051 erg (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Fryer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Ertl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Sukhbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Gogilashvili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In addition, the delayed neutrino mechanism has  problems in producing the observed amount of 56Ni, in  particular in stripped-envelope CCSNe (SECCSNe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' for  a recent study see Sawada & Suwa 2023)  The jittering jets explosion mechanism, on the other  hand, can account for much larger explosion energies  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Gilkis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Soker 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  The mag-  neto rotational explosion mechanism that works only  for rapidly rotating pre-collapse cores can also account  for large explosion energies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', LeBlanc & Wilson  1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Khokhlov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' L´opez-C´amara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Bromberg & Tchekhovskoy 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Kuroda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Gottlieb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2022c,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Fujibayashi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Powell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2022) because the newly born  neutron star (NS) launches fixed-axis jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Whether the  explosion is by jittering jets (most CCSNe according to  that model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Soker 2022b) or by fixed-axis jets, the  jets operate in a negative feedback cycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', the jet  feedback explosion mechanism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Soker 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' As  well, the jets can influence the direction of later jets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=',  Papish & Soker 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Gottlieb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Even if the  stochastically accreted mass has sub-Keplerian angular  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='05144v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='HE]  12 Jan 2023  2  momentum, it might still form an accretion belt that  can launch jets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Schreier & Soker 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Garain &  Kim 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  The above discussion implies that CCSNe with large  kinetic energies of Eexp ≳ 2 × 1051, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', SN 2020jfo  (Ailawadhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2022 – Eexp  =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='9 × 1051 erg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  SN 2020qlb (West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2022 – Eexp = 20 × 1051 erg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  SN 2012au (Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2021 – Eexp = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='8 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='4 ×  1051 erg) require jets to drive the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  In this study we examine the interesting case of the  hydrogen-poor and super-energetic CCSN SN 2020qlb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' (2022) estimate the kinetic energy of  SN 2020qlb as ≃ 20 × 1051 erg and suggest that a mag-  netar supplies the large amount of energy of the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  They also provide fitting parameters for the magnetar  model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Maeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Kasen & Bildsten 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Woosley 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Metzger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Nicholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2022) and estimate the progenitor pre-  explosion mass, namely, about the ejecta plus the rem-  nant mass, to be Mej + Mrem ≃ few × 10M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  It seems nonetheless, that there are two reasons why  the explosion of SN 2020qlb must be driven by jets and  not by the delayed neutrino mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The first one is  that the formation of an energetic magnetar must be ac-  companied by the launching of even more energetic jets  at the explosion itself and possibly after the explosion  as well (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Soker 2016b, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Soker & Gilkis 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Soker 2022c,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The second rea-  son is the new finding of the present study, where by a  stellar evolutionary code (section 2) we show (section 3)  that the binding energies of such massive cores are much  above what the delayed neutrino mechanism can supply  (Janka 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Soker 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In section 4 we summarize  our results and discuss their implications in the context  of the jet feedback explosion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' NUMERICAL SCHEME  We use the stellar evolution code mesa (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  2010, 2013, 2015, 2018, 2019) to simulate the struc-  ture of 52 CCSN progenitor models, all with initial  metalicity of z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We base our numerical rou-  tine on the ‘20M pre ms to core collapse’ example from  mesa r22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1, but simulate the evolution using ver-  sion r15140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Starting from the zero age main sequence  (ZAMS), we evolve the star until center He depletion  (when helium abundance in the center is either ≃ 1%  or ≃ 5%), at which point we numerically remove the  hydrogen-rich envelope, leaving only the He core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We  then evolve the star until core-collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We introduce  several changes to the example routine to both adhere  to our requirements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', having at least 21 isotopes  and having sufficiently high resolution (Shishkin & Soker  2021), and to fit mesa version r15140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We expand on  the numerical scheme in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  By varying the ZAMS stellar mass, the wind parame-  ters, and the exact time when we numerically remove  the envelope we obtain stripped-envelope CCSN pro-  genitors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', hydrogen-poor stellar progenitors, with  varying mass of a carbon-oxygen (CO) core, MCO ≃  4 − 35M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' This mass range of SECCSNe corresponds  to ZAMS stellar masses range of ≃ 20 − 82M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In some  cases the core contains several solar masses of helium,  while in other cases the core has a much lower helium  mass, depending on the above parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Because the inner part of the core collapses to form  a NS or a black hole (BH) remnant, only the binding  energy of the outer core is relevant to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We  calculate the binding energy of the outer core, Ebind(r),  by integrating over the sum of the internal energy and  gravitational energy from the surface down to the mass  coordinate that separates the ejecta and the final rem-  nant m = Min, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', the inner boundary of the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We  calculate the binding energy of the ejecta for two values  of this mass coordinate Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5 , 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙, because we  consider cases where the remnant is a NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We refer to the latest evolution point we simulate  in the stellar evolution as collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' This point of col-  lapse must adhere to an iron core more massive than  MFe > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5, where we assume collapse is imminent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' This  value of iron core mass is more or less when the iron  core mass reaches its maximum value (as at the onset  of collapse the iron disintegrates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' About half of simu-  lations reach infall velocities at the edge of the iron core  of vfe,infall > 100 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Other simulations encounter  numerical difficulties and we had to terminate them at  somewhat earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' RESULTS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Binding Energy towards Collapse  As the stellar core evolves towards collapse and nu-  cleosynthesis of heavier elements takes place, the core  becomes denser and the binding energy of the inner lay-  ers of the star increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We demonstrate this for one  stellar model of carbon-oxygen core mass (mainly oxy-  gen) of MCO = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='2M⊙ in Fig 1 where we present the  star at three times: at center oxygen depletion, at cen-  ter silicon depletion, and at collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We present the  composition of the main isotopes by lines with differ-  ent colors and the binding energy Ebind(m) by the black  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Here Ebind(m) is the binding energy (gravitational  + internal) of the envelope laying above mass coordi-  nate m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Relevant to this study is the binding energy  of the ejecta, which is the mass above mass coordinate  m = Min ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The mass Min is the baryonic  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Abundances and the binding energy as function  of mass coordinate at three times for a SECCSN (hydrogen-  poor) progenitor model with carbon-oxygen core mass of  M collapse  CO  = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='2M⊙, which corresponds to a ZAMS mass of  MZAMS ≈ 40M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The colored step-like lines are the abun-  dances according to the inset in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The black  and smoothly varying lines represent Ebind(m), which is the  binding energy of the envelope laying above mass coordi-  nate m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The three panels present these quantities at three  different times: center oxygen depletion (upper panel, bind-  ing energy by the black solid-line), center silicon depletion  (middle panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' binding energy by the black dashed-line), core  collapse (lower panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' binding energy by black dotted-line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Note that the middle panel contains the binding energy at  the three times to allow for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The two vertical  lines mark the mass coordinates m = Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ and  m = Min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Helium that appears only at collapse  results from disintegration of iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  mass of the NS remnant (the corresponding final gravi-  tational masses will be ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='35, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We mark these  two masses by the vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We see that at collapse  the binding energy Ebind(Min) is larger than at earlier  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  2 we present composition and binding en-  ergy for a model with a much more massive core of  MCO = 27M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  There are two qualitative differences  between this model and the one we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  The first qualitative difference is that the binding en-  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 1 but for a more massive core of  M collapse  CO  = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙, which corresponds to a ZAMS mass of  MZAMS ≈ 65M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Note that the left vertical axis is scaled  differently than in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  ergy at collapse is somewhat smaller than at the earlier  time that we present in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The explanation to  the decreasing binding energy shortly before collapse is  that the envelope expands starting from deep in the oxy-  gen burning shell and outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We find (by drawing  the density profiles) that moving from the upper to the  middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2 the density from m ≃ 10M⊙ and  outward decreases, reducing the binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' This  mass coordinate is deep inside the shell where oxygen  (teal line) burns to S+Si (yellow line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  The second qualitative difference comes from the much  higher binding energy of the ejecta of the descendant  CCSN of the more massive model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Ebind(Min) ≳ 2×  1051 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The implication is that we do not expect that  the neutrino driven explosion mechanism can account  for explosions of such cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We argue that jets explode  these cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We leave the discussion of this point, as  well as our view that jets also explode cores with lower  binding energy, to section 4, where we also refer to the  claim of a very massive core of SN 2020qlb (West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We first find the range of such high-binding-  energy cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  0  3  OCoreDepletion  2  91  erg  0  SiCoreDepletion  BEatO  dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  BEatSi  dep  BEatCollapse  2  0  3  Collapse  2  Helium  Carbon  Oxygen  S+Si  "Fe Group"  2  0  2  4  6  8  10  12Abundance6  0  4  OCoreDepletion  2  6  0  Si Core Depletion  BEatOont  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  BEat Si  dep  "BEatCollapse  2  0  4  Collapse  2  Helium  Oxygen  "Fe Group\'  Carbon  S+Si  2  0  4  8  12  16  20  24Abundance4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' High-binding-energy cores  We search for the mass range of cores that have bind-  ing energies at collapse of Ebind(Min) ≳ 2×1051 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We  present the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We present the binding en-  ergy for an inner ejecta mass coordinate of Min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙  (upper panel) and Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We  focus on the binding energy of these two mass coordi-  nates Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ as the iron core masses at col-  lapse falls within this mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The horizontal line  at 2 × 1051 erg is the approximate energy above which  we do not expect that neutrino heating by itself can ex-  plode the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In appendix B we provide linear fits to  the binding energy at collapse as function of CO core  mass for these two mass coordinates Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙  (table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We simulated 52 stripped-hydrogen envelope cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In  27 cases the cores reach collapse as we present by the  red circles in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In 25 cases cases the numeri-  cal code encountered difficulties and we had to stop the  simulation before reaching collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In these cases we  extrapolate from an evolutionary time before collapse  to the collapse time as we explain in appendix B (red  stars in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We also include 35 in the figure the  binding energies of models with hydrogen-rich envelope  that we take from Shishkin & Soker (2022), as we mark  by open purple circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 3 (with a more rigorous derivation in ap-  pendix B) we draw our conclusion that in cases where  the inner mass of Min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ of the core collapses  to form a NS, the delayed neutrino mechanism cannot  explode cores with masses of MCO ≳ 15M⊙ (or maybe  rarely do so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' For Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ we find this limit to be  MCO ≳ 13M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 4 we present a more detailed binding energy  profile of the pre-collapse stripped-envelope models that  we simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' When we take the lowest binding energy  of the inner core at the onset of collapse we find the limit  of core mass that the neutrino-driven explosion cannot  account for to be MCO > 20M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Namely, still a large  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  In table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1 we provide the linear fit parameters for  the binding energy at collapse for the edge of the iron  core (gray squares in the figure) and the binding energy  curve break (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The binding energy curve  break is point we refer to as separating the inner core  from the outer core, and is the point of lowest binding  energy for most simulated cases that reached collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' DISCUSSION AND SUMMARY  We simulated the evolution of 52 massive SECCSN  progenitor models corresponding to ZAMS masses of  20 ≲ MZAMS ≲ 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We removed the entire hydrogen-  rich envelope, and calculated the binding energy just  before core collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  The final core mass depends on  the ZAMS mass and on the mass loss parameter (ap-  pendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We present the structure of the pre-collapse  progenitor for two cases in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We find that  to a fare accuracy we can linearly fit the binding energy  of these stripped-envelope progenitors to the CO core  mass MCO (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 3 and table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We present our main results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  In those  figures the horizontal gray line represents the approx-  imate maximum energy that the neutrino-driven mech-  anism can supply, Emax  ν  = 2 × 1051 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We find that  the binding energy calculated at Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ and  Min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙, of progenitors with a carbon-oxygen core  mass of MCO ≳ 13M⊙ and MCO ≳ 15M⊙, respectively,  are larger than Emax  ν  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Namely,  Emax  ν  ≲  �  �  �  Ebind,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5  for  MCO ≳ 13M⊙  Ebind,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5  for  MCO ≳ 15M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  (1)  The main conclusion is that the delayed neutrino ex-  plosion mechanism cannot explode stars with a core  mass of MCO ≳ 13 − 15M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The jittering jets explosion  mechanism, on the other hand, has no limiting explosion  energy in these ranges as it is fueled by accretion onto  the compact remnant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', (Gilkis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Soker &  Gilkis 2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Let us apply our results to a specific SECCSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In a  recent paper West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' (2022) deduce that SN 2020qlb  had an explosion energy of ≃ 20×1051 erg and estimate  the progenitor pre-explosion mass, ejecta plus remnant  mass, to be Mej + Mrem ≃ 30 − 50M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' According to  our results the binding energy alone of such cores is  Ebind > 3 × 1051 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We therefore conclude that jets  must have exploded SN 2020qlb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Jets can also supply  the kinetic energy of the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Namely, jet-driven ex-  plosions might make the magnetar powering less criti-  cal or not needed at all (although a magnetar might be  present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Most likely the explosion is via jittering jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  The reason is that an explosion driven by a fixed-axis  jets, like if the core is rapidly rotating, will not expel  mass from the equatorial region, which it turn is ac-  creted by the newly formed central object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Therefore,  the final mass of the remnant will be large and the rem-  nant will be a BH (see discussion in Soker 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  On a large scope, our study adds to the growing evi-  dence pointing to the major roles that jets play in the  explosion, as well as pre-explosion and post-explosion,  of CCSNe (for a recent review see Soker 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This research was supported by a grant from the Israel  Science Foundation (769/20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  5  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The binding energies of the simulated models as a function of the carbon-oxygen core mass nearing collapse (vertical  axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The panels show the final binding energy at two mass coordinates: Min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ (top) and Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The  red data points (filled circles and stars at the outer panels) are stripped-envelope (SE) models (SECCSNe), whilst purple empty-  circles data points are models from Shishkin & Soker (2022) that have hydrogen rich envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Red stars are extrapolated  data points, as explained in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The horizontal line at 2 × 1051 erg denotes the binding energy above which we do not  expect the neutrino delayed explosion mechanism to explode the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Data availability  The data underlying this article will be shared upon  reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
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+page_content='00359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
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+page_content='00359  9  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' NUMERICAL PRESCRIPTION DETAILS  Our numerical scheme files (‘inlists’) are a modified version of the ‘20 pre ms to cc’ mesa version r22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1 ‘test suite’  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We adapted this example to run on mesa version r15140 and incorporated certain parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', over-  shooting and mesh resolution) according to our previous works (Shishkin & Soker 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Shishkin & Soker 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The  full ’inlists’ that we used are available online1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Here we mention some of the important parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  In a similar fashion to our previous works which focus on the convective profile of the inner layers of massive stars  (Shishkin & Soker 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Shishkin & Soker 2022), we use the exponential overshooting prescription (Herwig 2000)  with symmetrical (both ‘bottom’ and ‘above’) and uniform (all burning regions) settings and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='01, f0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='004  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We chose the Henyey scheme (Henyey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 1965) for mixing length theory (MLT, B¨ohm-Vitense 1958) with  αMLT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We also enable the Ledoux criterion (Ledoux 1947) and set thermohaline option to ‘Kippenhahn’  with ‘thermohaline coeff = 1’ alongside ‘alpha semiconvection = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We make use of the ’Dutch’ wind loss scheme (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Vink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Nugis & Lamers 2000), and vary the scaling  factor (along with the initial mass) to achieve different core masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  For the nuclear network we use the 22 isotopes of ‘approx21 cr60 plus co56’ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=', Timmes 1999), aimed at  stellar evolution up to collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  This network includes hydrogen, He3 and He4 up to the heavier isotopes of  Fe52 , Fe54 , Fe56 , Co56 , Ni56 , Cr60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We scale mesh refinement gradually up to ‘max dq’ values of 1d − 4 at the later stages (from the default value of  1d − 2) to properly resolve the fine burning features close to core collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  The mesa equation of state (EOS) is a blend of the OPAL (Rogers & Nayfonov 2002), SCVH (Saumon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 1995),  FreeEOS (Irwin 2004), HELM (Timmes & Swesty 2000), PC (Potekhin & Chabrier 2010), and Skye (Jermyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  2021) EOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Nuclear reaction rates are from JINA REACLIB (Cyburt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 2010), NACRE (Angulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 1999) and  additional tabulated weak reaction rates Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' (1985);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Oda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Langanke & Mart´ınez-Pinedo (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Screening is included via the prescription of Chugunov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Thermal neutrino loss rates are from Itoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Radiative opacities are primarily from OPAL (Iglesias & Rogers 1993, 1996), with low-temperature data from  Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' (2005) and the high-temperature, Compton-scattering dominated regime by Poutanen (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Electron  conduction opacities are from Cassisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' (2007) and Blouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' BINDING ENERGY ESTIMATION  Because of numerical difficulties of stripped-envelope progenitors (specifically some steep gradients) some simulations  did not reach the phase of core collapse, although they did reach oxygen depletion and/or silicon depletion at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Time steps became much too short and we had to terminate the simulations before core collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In these cases we  estimated the binding energy at collapse (red-stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 3) by extrapolating the binding energy during earlier phases  using linear fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We made linear fits to the binding energies as function of the CO core masses at three evolutionary phases: oxygen  depletion, silicon depletion, and core collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1 we present these three fittings by blue, orange, and red  lines, respectively, for Min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ (upper panel) and Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' From these three lines we can find  the ratio of the binding energy at core collapse to the binding energy at oxygen depletion and to the binding energy  at silicon depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In cases where we did not reach core collapse we use this ratio at the given CO core mass to  calculate the expected binding energy at core collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We mark these energies by red-stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1 and use them  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Error bars attached to the red stars signify the 1σ intervals of the this extrapolation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We note  that the CO core mass does not change much after oxygen depletion in the non-extended helium phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The average  difference between the CO core mass at central oxygen depletion and at core collapse is ∆mcore  CO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='33M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  1 Zenodo: Modified inlists to reproduce the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Also included  a full simulation list and the simulated models at different time  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  10  Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The binding energy of the envelope above mass coordinate Min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ (upper panel) and Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ (lower  panel) as a function of the final carbon-oxygen core mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The blue circles are at central oxygen depletion (5% oxygen in the  center), the orange circles are at silicon depletion (5% silicon in the center), and red circles are at core collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The three  respective lines are the linear fit to the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Red stars are the extrapolated values for the binding energy at collapse based  on available earlier data points (oxygen depletion or silicon depletion) for the cases that did not reach collapse (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  We fit the binding energy Ebind versus the CO core mass MCO by a linear fit Ebind = aMCO + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1 we  list the values of the two coefficients for the six lines (three stage for two values of the mass that collapses to form the  NS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We also list (last column) the number of data points that were used at each fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  11  aMCO + b  Fit1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙  Fit2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙  FitBEbreak  FitFecore  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' of points  Collapse  a [1051erg/M⊙]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='122 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='137 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='117 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='114 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='02  27  b [1051erg]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='537 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='46  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='053 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='444  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='165 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='423  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='442 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='406  Sicntr depletion  a [1051erg/M⊙]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='132 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='145 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='02  −−  −−  43  b [1051erg]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='159 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='38  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='099 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='445  −−  −−  Ocntr depletion  a [1051erg/M⊙]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='175 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='02  −−  −−  47  b [1051erg]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='394  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='447 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='479  −−  −−  Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The linear fits to the lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1 and the number of data points for each of the simulation groups: at collapse  (second row), at center silicon depletion (third row) and center oxygen depletion (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' The third and fourth columns  are the fits to the binding energy Ebind,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5, Ebind,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5 at mass coordinates Min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙ and Min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' In the  fifth column we present the linear fit to the variation of the binding energy at the black dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 4 with the CO core mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  In the fifth column we present the linear fit to the variation of the binding energy at the edge of the iron core (gray squares in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' 4) with the CO core mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Linear fits are in units of energy Ebind [1051erg] to CO core mass MCO [M⊙].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Values and errors  (2σ) are in accordance with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  12  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' SIMULATIONS LIST  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='2 we present the simulations that we conducted in a three-parameters space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' As input we show the dutch  wind scaling factor in mesa in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='5 < αDutch,wind < 1 and the zero age main sequence (ZAMS) mass in the  range of 20M⊙ < MZAMS < 82M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' As an output we present the final CO core mass (mainly oxygen mass) in units of  solar mass according to the color bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='  Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' Wind (Dutch scheme) scaling factors (vertical axis) and the ZAMS masses of the cases (horizontal axis) that we  simulated, and the final CO core mass (by color bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
+page_content=' We denote with a black X the cases where we extended the He burning  to a later stage before removing the hydrogen envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQfjw0A/content/2301.05144v1.pdf'}
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+arXiv:2301.11649v1  [math.OC]  27 Jan 2023
+IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, AUGUST 20XX
+1
+Frequency Energy Multiplier Approach to Uniform
+Exponential Stability Analysis of Semi-discrete
+Scheme for a Schr¨odinger Equation
+under Boundary Feedback
+Bao-Zhu Guo and Fu Zheng
+Abstract—In this paper, we investigate the uniform exponential
+stability of a semi-discrete scheme for a Schr¨odinger equation
+under boundary feedback stabilizing control in the natural state
+space L2(0,1). This study is significant since a time domain energy
+multiplier that allows proving the exponential stability of this
+continuous Schr¨odinger system has not yet found, thus leading
+to a major mathematical challenge to semi-discretization of the
+PDE, an open problem for a long time. Although the powerful
+frequency domain energy multiplier approach has been used in
+proving exponential stability for PDEs since 1980s, its use to
+the uniform exponential stability of the semi-discrete scheme for
+PDEs has not been reported yet. The difficulty associated with
+the uniformity is that due to the parameter of the step size,
+it involves a family of operators in different state spaces that
+need to be considered simultaneously. Based on the Huang-Pr¨uss
+frequency domain criterion for uniform exponential stability of a
+family of C0-semigroups in Hilbert spaces, we solve this problem
+for the first time by proving the uniform boundedness for all the
+resolvents of these operators on the imaginary axis. The proof
+almost exactly follows the procedure for the exponential stability
+of the continuous counterpart, highlighting the advantage of this
+discretization method.
+Index Terms—Schr¨odinger equation, boundary damping, fre-
+quency domain multiplier, semi-discretization, uniform exponen-
+tial stability.
+I. INTRODUCTION
+C
+Ontrol systems described by partial differential equations
+(PDEs) is infinite-dimensional. Being such, its controller
+such as the observer-based feedback control is also infinite-
+dimensional. As a result, the discretization finds itself in
+almost all implementations of PDE control. Among many
+discretization methods is the finite-difference method which
+becames popular due to its simplicity in principle and its
+appeal to engineers. One of the most commonly used dis-
+cretization method is the so-called semi-discrete scheme which
+keeps time continuous while discretizing the spatial variable.
+It has been widely studied in literature. The main advantage
+of the semi-discrete scheme is that it results in an ordinary
+This work was supported by the National Natural Science Foundation
+of China under grants no.61873260, 11871117, 12131008. (Corresponding
+author: Fu Zheng)
+Bao-Zhu Guo is with Department of Mathematics and Physics, North
+China Electric Power University, Beijing 102206, China, and Key Laboratory
+of System and Control, Academy of Mathematics and Systems Science,
+Academia Sinica, Beijing 100190, Email:bzguo@iss.ac.cn
+Fu Zheng is with School of Science of Hainan University, Haikou, Hainan
+570228, E-mail: fuzheng@hainanu.edu.cn
+differential equation system, which control researchers are
+most familiar with. However, it has been acknowledged for a
+long time that the uniform exponential stability with respect to
+the spatial discrete step size cannot be guaranteed for classical
+semi-discrete schemes for PDEs, largely due to presence of
+high frequency spurious components. In addition, some other
+typical important control properties such as uniform observ-
+ability and uniform exact controllability cannot be guaranteed
+either. The reason for this loss is that the spurious modes
+are only weakly damped in the process of semi-discretization.
+A detail account can be found in [24]. For wave equations,
+several remedies such as Tichonoff regularization [7], mixed-
+finite elements [2], [17], high frequency filtering [10], and
+non-uniform meshes [4], have been proposed to circumvent
+this difficulty. Among many these remedies, the numerical
+viscosity damping introduced in [19], [20] is the most popular.
+However, this approach brings a viscosity term artificially
+added into the classical discrete scheme. The coefficients of the
+numerical viscosity damping vary from PDE to PDE. Recently,
+a new natural semi-discrete scheme based on order reduction
+finite difference method was introduced in [13] and has been
+applied to different systems [8], [23]. This approach has the
+critical advantages that it guarantees the uniform exponential
+stability. In addition, as a natural semi-discrete scheme, it al-
+lows one to prove the uniform exponential stability in a manner
+parallel to its continuous PDE counterpart. Nevertheless, all
+the previous papers on this scheme involved construction of
+Lyapunov functional which the proof heavenly relies on, both
+for semi-discrete scheme and the continuous counterpart.
+Construction of a suitable Lyapunov functional for a PDE
+relies on a time domain energy multiplier, which is not always
+available and its construction is most often very technical. In
+1980s, a frequency domain energy multiplier approach was
+developed for the exponential stability initially for a single
+PDE ([15]). The approach is based on a frequency domain
+characterization for exponential stability of C0-semigroup in
+Hilbert space. Originally developed independently in [9] and
+[18], the result of was proved later in [16] to be valid for
+uniform exponential stability of a family of C0-semigroups in
+Hilbert spaces as well. Uniform admissibility and observability
+for the finite element space semi-discretizations of abstract
+Schr¨odinger system and second order infinite dimensional
+vibrating systems have also been developed [5], [6].
+In this paper, we investigate the uniform exponential sta-
+
+IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, AUGUST 20XX
+2
+bility of an order reduction semi-discrete scheme for a
+Schr¨odinger equation under boundary control by the frequency
+domain multiplier approach. It is significant because one
+cannot find a suitable time domain Lyapunov functional both
+for the continuous PDE and for its discrete scheme. This
+implies that successful approaches presented in [14], [13], [8],
+[23] cannot be applied here. As a matter of fact, in order to
+apply the Lyapunov method, the paper [14] has to consider
+the Schr¨odinger system in the high order state space H1(0,1),
+whereas our state space is the standard space L2(0,1). The
+problem in L2(0,1) has been open for quite a long time. Fairly
+speaking, this paper brings a new way to the proof of the
+uniform exponential stability of the semi-discrete scheme for
+PDEs. It is also worthy pointing out that the proofs for both
+continuous PDE and for the discrete counterpart are again
+analogous, demonstrating the advantage of the order reduction
+semi-discretization approach.
+We proceed as follows. In the next section, Section II, we
+prove the exponential stability of the continuous PDE by the
+frequency domain multiplier method. Although it is the sim-
+plest PDE ever studied in the literature, it helps in constructing
+a frequency domain multiplier for its semi-discrete counter-
+part. In Section III, we design a semi-discretized scheme and
+obtain a family of finite-dimensional systems. In Section IV,
+the uniform exponential stability is developed by the frequency
+domain multiplier approach. We introduce the shadow ele-
+ment to help understand the numerical approximating scheme,
+which plays an important role in the proof of uniform stability.
+Some concluding remarks are included in Section V.
+II. STABILITY OF SCHR ¨ODINGER SYSTEM VIA FREQUENCY
+DOMAIN MULTIPLIER
+Consider the following Schr¨odinger equation under bound-
+ary control:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+wt(x,t) = −iwxx(x,t), t > 0, x ∈ (0,1),
+w(0,t) = 0, t ≥ 0,
+wx(1,t) = u(t), k > 0, t ≥ 0,
+y(t) = w(1,t), t ≥ 0,
+w(x,0) = w0(x), x ∈ [0,1],
+(1)
+where u(·) is the control, y(·) is the measured output and w0(·)
+is the initial state. Under the proportional feedback control:
+u(t) = −kiy(t), k > 0,
+(2)
+the closed-loop system of (1) becomes
+
+
+
+
+
+
+
+
+
+
+
+
+
+wt(x,t) = −iwxx(x,t), t > 0, x ∈ (0,1),
+w(0,t) = 0, t ≥ 0,
+wx(1,t) = −kiw(1,t), k > 0, t ≥ 0,
+w(x,0) = w0(x), x ∈ [0,1],
+(3)
+We consider system (3) in the natural state space L2(0,1).
+Define the system operator of (3) as follows:
+
+
+
+
+
+
+
+Af = −if ′′,∀f ∈ D(A),
+D(A) = { f ∈ L2(0,1)|f ∈ H2(0,1),
+f(0) = 0, f ′(1) = −ik f(1)}.
+(4)
+Then, (3) can be written as an evolution equation in L2(0,1):
+�
+˙w(·,t) = Aw(·,t),
+w(x,0) = w0(x).
+(5)
+It is seen that
+Re⟨Af, f⟩L2(0,1) = Re
+� 1
+0 −ik f ′′(x)f(x)dx = −k|f(1)|2,
+(6)
+which implies that A is dissipative. In addition, the operator
+A is invertible and
+A−1 f(x) = −kx
+� 1
+0 xf(x)dx
+1 + ki
+− i
+� 1
+x (x− τ)f(τ)dτ − i
+� 1
+0 xf(x)dx,
+(7)
+which is bounded in L2(0,1). As a result, A generates a C0-
+semigroup of contractions on L2(0,1) by the Lumer-Phillips
+theorem ([21, Theorem 3.8.4]) and since A−1 is compact, the
+spectrum of A consists of isolated eigenvalues only.
+Furthermore, define the system energy for (3) as
+E(t) = 1
+2
+� 1
+0 |w(x,t)|2dt,
+(8)
+which is non-increasing as a consequence of (6):
+˙E(t) = −k|w(1,t)|2.
+(9)
+We point out that a different version of (3):
+
+
+
+
+
+
+
+
+
+
+
+
+
+wt(x,t) = −iwxx(x,t), t > 0, x ∈ (0,1),
+w(0,t) = 0, t ≥ 0,
+wx(1,t) = −kwt(1,t), k > 0, t ≥ 0,
+w(x,0) = w0(x), x ∈ [0,1],
+(10)
+was investigated in [14], for which one can find a time
+domain energy multiplier, and a Lyapunov functional was
+then constructed to both system (10) and its semi-discrete
+counterpart. However, system (3) is a rather unusual system
+for which a time domain energy multiplier has been not
+found yet. A first exponential stability result of system (3)
+was proved by the Riesz basis approach in [12]. Although
+the Riesz basis is powerful and the result obtained is much
+deeper than the result obtained from the multiplier method; for
+instance the spectrum-determined growth condition is usually
+a consequence of the Riesz basis approach yet this is usually
+not the case with the multiplier method. Unfortunately, the
+Riesz basis is extremely difficult, at least at the moment, to be
+applied for the uniformly exponential stability of semi-discrete
+model for (3) developed in this paper.
+In this paper, we use an alternative powerful method called
+the frequency energy multiplier method, which has been
+
+IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, AUGUST 20XX
+3
+developed for continuous PDEs over the last three decades
+([15]). In stability analysis, we can almost give one-to-one
+correspondence from continuous system to its discrete coun-
+terpart using this method. Our approach is so powerful that
+can be applied to other PDEs as well. For notation simplicity,
+hereafter, we omit without confusion the obvious dependency
+in time and spatial domains. The Cn denotes the n-dimensional
+complex Euclidean space; the N+ stands for the set of the
+positive integer numbers; and R the set of real numbers.
+Since A generates a C0-semigroup of contractions on
+L2(0,1), a well-known result of Hung-Pr¨uss theorem [9], [18]
+states that the C0-semigroup generated by A is exponentially
+stable if and only if it possesses the following two properties:
+1) Every imaginary number belongs to the resolvent set of
+A, that is, iR ⊂ ρ(A).
+2) The inverse operator of iω −A is uniformly bounded for
+all imaginary numbers, that is,
+sup
+ω∈R
+∥(iω − A)−1∥ < ∞.
+(11)
+The property iR ⊂ ρ(A) is stated in the following Lemma
+2.1.
+Lemma 2.1: Let A be defined by (4). Then, iR ⊂ ρ(A).
+Proof. If there exist β ∈ R,β ̸= 0 and a nonzero f ∈ D(A)
+such that iβ f = Af, then
+�
+iβ f(x) = −if ′′(x),
+f ′(1) = −kif(1), f(0) = 0.
+(12)
+Take the inner product with f(·) over [0,1] on both sides of
+the first equation of (12) to obtain
+iβ∥ f∥2 = −k|f(1)|2 + i
+� 1
+0 |f ′(x)|2dx,
+(13)
+which gives f(1) = 0 and hence f ′(1) = 0. This shows that
+(12) has only zero solution, a contradiction.
+Theorem 2.1: Let A be defined by (4). Then, (11) holds
+true. As a consequence, the C0-semigroup eAt generated by A
+is exponentially stable in L2(0,1).
+Proof. We prove by assuming contrary of (11) that there
+exit a sequence ωn → ∞, fn ∈ D(A), ∥ fn∥ = 1 that
+lim
+n→∞∥(iωn − A)fn∥ = 0,
+i.e.,
+iωn fn + if ′′
+n → 0 in L2(0,1).
+(14)
+Since
+Re⟨(iωn − A)fn, fn⟩L2(0,1) = Re⟨−Afn, fn⟩ = k|fn(1)|2 → 0,
+(15)
+by the boundary condition f ′
+n(1) = −ik fn(1), it gives
+f ′
+n(1) → 0.
+(16)
+From (14) and ∥ fn∥ = 1, it follows that
+f ′′n (·)
+ωn
+is bounded in
+L2(0,1). By
+|f ′
+n(x)− f ′
+n(1)| =
+����
+� x
+1 f ′′
+n (s)ds
+���� ≤ ∥ f ′′
+n ∥,
+it follows from (16) and ωn → ∞ that
+f ′
+n(·)
+ωn
+is bounded in L2(0,1).
+(17)
+Since
+Re
+�
+ωn fn + f ′′
+n , xf ′
+n
+ωn
+�
+L2(0,1)
+= |fn(1)|2
+2
+− 1
+2
+� 1
+0 |fn(x)|2dx
++
+1
+2ωn
+|f ′
+n(1)|2 −
+1
+2ωn
+� 1
+0 |f ′
+n(x)|2dx,
+and
+�
+ωn fn + f ′′
+n , xf ′
+n
+ωn
+�
+L2(0,1)
+→ 0,
+we have by (15) and (16) that
+� 1
+0 |fn(x)|2dx+ 1
+ωn
+� 1
+0 |f ′
+n(x)|2dx → 0,
+(18)
+which shows that when ωn > 0, ∥ fn∥2 → 0, which is a
+contradiction to ∥ fn∥ = 1. On the other hand, since from (14)
+and ωn → ∞, we have
+� 1
+0
+����fn(x)+ f ′′
+n (x)
+ωn
+����
+2
+dx
+=
+� 1
+0
+�
+fn(x)+ f ′′
+n (x)
+ωn
+��
+fn(x)+ f ′′n (x)
+ωn
+�
+dx
+=
+� 1
+0
+�
+|fn(x)|2 + |f ′′
+n (x)|2
+ω2n
+�
+dx
++ 1
+ωn
+� 1
+0 [fn(x)f ′′n (x)+ fn(x) f ′′
+n (x)]dx
+=
+� 1
+0
+�
+|fn(x)|2 + |f ′′
+n (x)|2
+ω2n
+�
+dx
++ 1
+ωn
+[fn(x)f ′n(x)+ fn(x) f ′
+n(x)]1
+0
+− 2
+ωn
+� 1
+0 |f ′
+n(x)|2dx → 0,
+(19)
+Substitute f ′
+n(1) = −ik fn(1) and fn(0) = 0 into (19), and use
+(15)-(16) to obtain
+� 1
+0 |fn(x)|2dx+
+� 1
+0
+|f ′′
+n (x)|2
+ω2n
+dx− 2
+ωn
+� 1
+0 |f ′
+n(x)|2dx → 0. (20)
+which shows that when ωn < 0, ∥ fn∥2 → 0, which is also a
+contradiction.
+III. SEMI-DISCRETE SCHEME OF SCHR ¨ODINGER EQUATION
+In this section we apply the order reduction method to derive
+a semi-discrete scheme for (3). To this purpose, we introduce
+an intermediate variable v(x,t) = wx(x,t) to reduce the order
+of the spacial derivative of (3). In this way, the Schr¨odinger
+equation (3) can be rewritten as the following equivalent form:
+
+
+
+
+
+
+
+
+
+
+
+wt(x,t)+ ivx(x,t) = 0,
+v(x,t) = wx(x,t),
+w(0,t) = 0,
+v(1,t) = −kiw(1,t),
+w(x,0) = w0(x).
+(21)
+
+IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, AUGUST 20XX
+4
+The semi-discretization process is similar to [14]. For the
+sake of completeness, we sketch briefly the process. For fixed
+N ∈ N+, consider an equidistant partition of interval [0,1]:
+0 = x0 < x1 < ··· < xj = jh < ··· < xN+1 = 1,
+where h =
+1
+N+1 is the mesh size. Denote the sequence {u j}N+1
+0
+by {u j} j and introduce respectively the average operator and
+the first-order finite difference operator as
+u j+ 1
+2 = u j + u j+1
+2
+,
+δxu j+ 1
+2 = u j+1 − u j
+h
+.
+(22)
+For the solutions v(x,t) and w(x,t) of (21), let {Vj(t)} j and
+{Wj(t)} j be grid functions at grids {xj} j, satisfying
+Vj(t) = v(xj,t),
+Wj(t) = w(xj,t),
+0 ≤ j ≤ N + 1.
+The first equation of system (21) holds at (xj+ 1
+2 ,t), i.e.,
+w′(xj+ 1
+2 ,t)+ ivx(xj+ 1
+2 ,t) = 0,
+where xj+ 1
+2 = (j + 1
+2)h. Hereafter the prime “′” represents
+the derivative with respect to time t. Replace the differential
+operator ∂x with difference operator δx to get
+W ′
+j+ 1
+2 (t)+ iδxVj+ 1
+2 (t) = O(h2).
+(23)
+Similarly, for the second equation of system (21), it has
+Vj+ 1
+2 (t)− δxWj+ 1
+2 (t) = O(h2).
+(24)
+By dropping the infinitesimal terms in (23) and (24), and
+replacing Wj(t) and Vj(t) by wj(t) and vj(t), respectively, we
+arrive at a semi-discretized finite difference scheme of system
+(21) as follows:
+
+
+
+
+
+
+
+
+
+
+
+
+
+w′
+j+ 1
+2 (t)+ iδxvj+ 1
+2 (t) = 0, 0 ≤ j ≤ N,
+vj+ 1
+2 (t) = δxwj+ 1
+2 (t), 0 ≤ j ≤ N,
+vN+1(t) = −kiwN+1(t), t ≥ 0
+w0(t) = 0,
+wj(0) = w0
+j, 0 ≤ j ≤ N + 1,
+(25)
+where vj(t) and wj(t) are grid functions at grids xj (0 ≤ j ≤
+N +1), and w0
+j is the approximation of the initial value w0(xj).
+Remark 3.1: The semi-discretized system (25) is a family
+of differentiation-algebra systems, which is called singular
+systems for which there are huge amount of references related
+to them. See for instance [3], [11], [22] and the references
+therein.
+Now, we eliminate the intermediate variables vj(t) from
+(25). To this purpose, let
+Wh(t) = (w1(t),w2(t),··· ,wN+1(t))⊤
+be unknown variable of (25) and
+Vh(t) = (v0(t),v1(t),··· ,vN(t))⊤
+the auxiliary variable. We write (25) into vectorial form:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+DhW ′
+h(t) = −iMhVh(t)−
+
+
+
+
+
+0
+...
+0
+kh−1wN+1(t)
+
+
+
+
+,
+D⊤
+h Vh(t) = −M⊤
+h Wh(t)+
+
+
+
+
+
+0
+...
+0
+i2−1kwN+1(t)
+
+
+
+
+,
+Wh(0) = (w0
+0,w0
+1,··· ,w0
+N)⊤,
+(26)
+where the matrices Dh and Mh are given by
+Dh = 1
+2
+
+
+
+
+
+
+
+1
+1
+1
+...
+...
+1
+1
+1
+1
+
+
+
+
+
+
+
+(N+1)×(N+1)
+,
+Mh = 1
+h
+
+
+
+
+
+
+
+−1
+1
+−1
+1
+...
+...
+−1
+1
+−1
+
+
+
+
+
+
+
+(N+1)×(N+1)
+.
+(27)
+Obviously, both Dh and Mh are invertible. The differential
+algebraic system (25) or (26) can be written as an evolution
+equation in CN+1:
+�
+W ′
+h(t) = AhWh(t), Wh(t) ∈ Yh = CN+1,
+Wh(0) = (w0
+1,w0
+2,··· ,w0
+N+1)⊤ ∈ Yh,
+(28)
+where Ah is defined by
+AhYh
+= D−1
+h
+�
+iMh
+�
+D⊤
+h
+�−1 �
+M⊤
+h Yh − (0,··· ,0,2−1ikyN+1)⊤�
+− D−1
+h (0,··· ,0,kh−1yN+1)⊤�
+,
+∀Yh = (y1,y2,··· ,yN+1)⊤ ∈ CN+1.
+(29)
+System (28) is naturally discussed in the state space CN+1.
+To relate CN+1 in (28) with the step size, we write Yh = CN+1
+and define a new inner product for Yh:
+�
+Yh,�Yh
+�
+Yh
+= h
+�
+DhYh,Dh�Yh
+�
+,∀Yh,�Yh ∈ Yh,
+where ⟨·,·⟩ is the standard inner product of CN+1. For Yh =
+(y1,y2,··· ,yN+1)⊤ ∈ Yh, we choose the vector
+Zh = (z0,z1,··· ,zN)⊤ ∈ CN+1 satisfying
+D⊤
+h Zh = −M⊤
+h Yh + (0,··· ,0,2−1ikyN+1)⊤.
+(30)
+We call Zh the shadow element of Yh, which can simplify
+significantly the notation in the later proofs.
+The classical semi-discrete scheme is similar with (28)
+where the average operator Dh = IN+1, i.e.,
+�
+W ′
+h(t) =
+ˆ
+AhWh(t), Wh(t) ∈ Yh = CN+1,
+Wh(0) = (w0
+1,w0
+2,··· ,w0
+N+1)⊤ ∈ Yh,
+(31)
+
+IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, AUGUST 20XX
+5
+in which the
+ˆ
+Ah is defined by
+ˆ
+AhYh = iMh
+�
+M⊤
+h Yh − (0,··· ,0,2−1ikyN+1)⊤�
+− (0,··· ,0,kh−1yN+1)⊤.
+(32)
+At the end of this section we explain the significance of
+the discrete scheme (28). We plot two figures in Figures 1
+and 2, respectively. Figure 1 depicts the maximal real parts
+of the eigenvalues of the classical semi-discrete scheme (31)
+with step size h, from which we see that the real parts of the
+eigenvalues approach zero. Figure 2 depicts the maximal real
+parts of the eigenvalues of the order reduction semi-discrete
+scheme (28) with the same step size, from which we see that
+the real parts of the eigenvalues approach a negative number.
+In both figures, we take k = 1.
+N
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Real part of eigenvalue
+-0.12
+-0.1
+-0.08
+-0.06
+-0.04
+-0.02
+0
+Fig. 1.
+Maximal real parts of eigenvalues of the semi-discrete scheme by
+classical method (31)
+N
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Real part of eigenvalue
+-1.954
+-1.952
+-1.95
+-1.948
+-1.946
+-1.944
+-1.942
+-1.94
+Fig. 2.
+Maximal real parts of eigenvalues of the semi-discrete scheme by
+order reduction method (28)
+IV. PROOF OF UNIFORM EXPONENTIAL STABILITY
+This section is devoted to the proof of the uniform expo-
+nential stability of (28). To begin with, we first show that Ah
+is dissipative for every step size h.
+Lemma 4.1: For the matrix Ah defined by (29), there holds
+Re⟨AhYh,Yh⟩Yh = −k|yN+1|2, ∀ Yh ∈ Yh,
+(33)
+which implies that Ah is dissipative for every h ∈ (0,1).
+Proof.
+For
+Yh = (y1,y2,··· ,yN+1) ∈ Yh,
+let
+Zh =
+(z0,z1,··· ,zN) be the shadow element of Yh:
+�
+D⊤
+h Zh = −M⊤
+h Yh + (0,··· ,0,2−1ikyN+1)⊤,
+AhYh = D−1
+h [−iMhZh + (0,··· ,0,kh−1yN+1)⊤].
+(34)
+Set y0 := 0 and zN+1 := −ikyN+1 and introduce �Yh =
+(�y1,�y2,··· ,�yN+1) ∈ Yh such that AhYh = �Yh. Then,
+D⊤
+h Zh + (0,··· ,0,2−1zN+1)⊤ = −M⊤
+h Yh,
+(35)
+which is equivalent to
+zj+ 1
+2 = δxyj+ 1
+2 , j = 0,1,··· ,N,
+(36)
+and
+Dh�Yh = −iMhZh − (0,··· ,0,ih−1zN+1)⊤,
+(37)
+which is equivalent to
+�yj+ 1
+2 = −iδxzj+ 1
+2 , j = 0,1,··· ,N,
+(38)
+where in all (35) to (38), it was assumed that �y0 = 0. Take the
+inner product between AhYh and Yh in Yh by taking (36) and
+(38) into account to obtain
+Re⟨AhYh,Yh⟩Yh = Re
+�
+�Yh,Yh
+�
+Yh
+= h
+2
+�
+Dh�Yh,DhYh
+�
++ h
+2
+�
+DhYh,Dh�Yh
+�
+= h
+2
+N
+∑
+j=0
+�yj+ 1
+2 yj+ 1
+2 + h
+2
+N
+∑
+j=0
+yj+ 1
+2 �yj+ 1
+2 , (using (38))
+= −hi
+2
+N
+∑
+j=0
+δxzj+ 1
+2 yj+ 1
+2 + hi
+2
+N
+∑
+j=0
+yj+ 1
+2 δxzj+ 1
+2
+= −hi
+2
+N
+∑
+j=0
+�
+δxzj+ 1
+2 yj+ 1
+2 (t)+ zj+ 1
+2 δxyj+ 1
+2
+�
++ hi
+2
+N
+∑
+j=0
+�
+yj+ 1
+2 δxzj+ 1
+2 + δxyj+ 1
+2 zj+ 1
+2
+�
+. (using (36))
+(39)
+A simple calculation shows that
+−hi
+2
+N
+∑
+j=0
+�
+δxzj+ 1
+2 yj+ 1
+2 + zj+ 1
+2 δxyj+ 1
+2
+�
++hi
+2
+N
+∑
+j=0
+�
+yj+ 1
+2 δxzj+ 1
+2 + δxyj+ 1
+2 z j+ 1
+2
+�
+= − i
+4
+N
+∑
+j=0
+�
+(zj+1 − zj)(yj+1 + yj)+ (zj+1 + zj)(yj+1 − yj)
+�
++ i
+4
+N
+∑
+j=0
+�
+(yj+1 + yj)(z j+1 − zj)+ (yj+1 − yj)(zj+1(t)+ zj)
+�
+= − i
+2
+N
+∑
+j=0
+[zj+1yj+1 − zjyj]+ i
+2
+N
+∑
+j=0
+[yj+1zj+1 − yjzj]
+= i
+2[z0y0 − zN+1yN+1]+ i
+2[yN+1zN+1 − y0z0]
+= −k|yN+1|2. ( using − ikyN+1 = zN+1 and y0 = 0) (40)
+
+IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, AUGUST 20XX
+6
+The (39) and (40) leads to (33).
+Define the energy of (28) as
+Eh(t) = h
+2
+N
+∑
+j=0
+���wj+ 1
+2 (t)
+���
+2
+= 1
+2 ⟨Wh(t),Wh(t)⟩Yh .
+(41)
+which is the discretization of the continuous energy (8). The
+following Lemma 4.2 is the discrete counterpart of (9), which
+is a consequence of (33).
+Lemma 4.2: The Eh(t) defined by (41) satisfies
+˙Eh(t) = −k|wN+1(t)|2.
+(42)
+The dissipativity of Ah implies that the spectral set σ(Ah)
+of Ah is contained in the closed left half-plane of the complex
+plane C. Actually, we have more stronger result. Precisely, for
+any 0 < h < 1, the spectral set σ(Ah) of Ah is contained in
+the open left half-plane of C. This is the following Lemma
+4.3.
+Lemma 4.3: For every h ∈ (0,1), iR ⊂ ρ(Ah).
+Proof. If there exist β ∈ R and nonzero Yh ∈ Yh such that
+iβYh = AhYh, then it follows from (33) that
+0 = Re⟨iβYh, Yh⟩Yh = Re⟨AhYh, Yh⟩Yh = −k|yN+1|2.
+(43)
+Replacing �Yh by iβYh in (37), we obtain
+
+
+
+βyj+ 1
+2 + δxzj+ 1
+2 = 0,
+0 ≤ j ≤ N,
+zj+ 1
+2 − δxyj+ 1
+2 = 0,
+0 ≤ j ≤ N,
+(44)
+where Zh is the shadow element of Yh defined in (34), y0 =
+0 and zN+1 := −ikyN+1. Hence zN+1 = yN+1 = 0 from (43).
+Setting j = N in (44) yields
+βhyN = 2zN, zN = −2
+hyN.
+It follows that yN = zN = 0 whenever βh2+4 is nonzero. Under
+the condition βh2 +4 ̸= 0, suppose zj+1 = yj+1 = 0 and solve
+(44) to arrive at zj = yj = 0. This gives Yh = 0 by induction,
+which is a contradiction. On the other hand, whenever βh2 +
+4 = 0, it follows from (44) that
+
+
+
+
+
+1
+h(yj+1 + yj) = 1
+2(zj+1 − zj), j = 0,1,··· ,N,
+1
+h(yj+1 − yj) = 1
+2(zj+1 + zj), j = 0,1,··· ,N,
+(45)
+which implies that
+yj+1 = h
+2zj+1, yj = −h
+2zj, j = 0,1,··· ,N.
+(46)
+This, combining with y0 = 0 and yN+1 = 0, gives yj = 0 (j =
+1,2,··· ,N) which is also a contradiction. This completes the
+proof of the lemma.
+The following lemma comes from [13].
+Lemma 4.4: Let {ui}i, {vi}i and {wi}i be the sequences of
+complex numbers. Then,
+1
+4
+N
+∑
+i=0
+(ui+1 − ui)(vi+1 + vi)(wi+1 + wi)
++1
+4
+N
+∑
+i=0
+(ui+1 − ui)(vi+1 − vi)(wi+1 − wi)
++1
+4
+N
+∑
+i=0
+(ui+1 + ui)(vi+1 − vi)(wi+1 + wi)
++1
+4
+N
+∑
+i=0
+(ui+1 + ui)(vi+1 + vi)(wi+1 − wi)
+= uN+1vN+1wN+1 − u0v0w0.
+(47)
+The following uniformly stability criterion which was pre-
+sented in [15] or [1] will be used in the proof of our main
+result Theorem 4.2 later.
+Theorem 4.1: Let h∗ > 0 and let {Sh(t)}h∈(0,h∗) be a family
+of semigroups of contractions on the Hilbert space Hh, and let
+�Ah be the corresponding infinitesimal generators. The family
+{Sh(t)} is uniformly exponentially stable if and only if the
+following two conditions are fulfilled:
+• For every h ∈ (0,h∗), iR ⊂ ρ(�Ah);
+• suph∈(0,h∗),β∈R∥(iβI − �Ah)−1∥ < ∞.
+Now, we are in a position to give the main result of this
+paper.
+Theorem 4.2: For the matrices Ah defined by (29), the
+corresponding family of C0-semigroups Th(t) generated by
+Ah is uniformly exponentially stable, that is, there exist two
+constants M > 0 and ω > 0 independent of h ∈ (0,1) such that
+∥Th(t)∥ ≤ Me−ωt, ∀t ≥ 0.
+(48)
+Proof. The proof is based on Theorem 4.1. Notice that by
+Lemma 4.1, for every h ∈ (0,1), Th(t) is a C0-semigroup of
+contractions. The fact that Ah satisfies the first condition of
+Theorem 4.1 has been claimed by Lemma 4.3. In order to show
+that the family Ah satisfies the second condition of Theorem
+4.1, we prove by contradiction. If the second condition of
+Theorem 4.1 is false, then there exist a sequence βn ∈ R, hn ∈
+(0,1), and Y n
+hn ∈ Yhn,∥Y n
+hn∥Yhn = 1 such that
+∥Un
+hn∥Yhn ≤ n−1, Un
+hn = (iβnIhn − Ahn)Y n
+hn.
+(49)
+By the Cauchy-Schwartz inequality, it follows from (49) and
+(33) that
+Re
+�
+Un
+hn,Y n
+hn
+�
+Yhn
+= −Re
+�
+AhnY n
+hn,Y n
+hn
+�
+Yhn
+= k|yn
+Nn+1|2 ≤ n−1.
+(50)
+Let
+Zn
+hn = (zn
+0,zn
+1,··· ,zn
+Nn)⊤ ∈ Zhn
+be the shadow element of Y n
+hn = (yn
+1,yn
+2,··· ,yn
+Nn+1)⊤(see(30)),
+Un
+hn = (un
+1,un
+2,··· ,un
+Nn+1)⊤ with hn(Nn +1) = 1. Set artificially
+un
+0 = yn
+0 = 0 and zn
+Nn+1 = −ikyn
+Nn+1 to unify the notation of
+
+IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, AUGUST 20XX
+7
+un
+j+ 1
+2 and δxzn
+j+ 1
+2 from j = 0,1,··· ,Nn. Then, it follows from
+(49) that
+� DhnUn
+hn = iβnDhnY n
+hn + iMhnZn
+hn + (0,··· ,0,ih−1zn
+Nn+1)⊤,
+−M⊤
+hnY n
+hn = D⊤
+hnZn
+hn + (0,··· ,0,2−1zn
+Nn+1)⊤,
+(51)
+or in vector form:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+un
+0+ 1
+2
+un
+1+ 1
+2
+...
+un
+Nn+ 1
+2
+
+
+
+
+
+
+
+= iβn
+
+
+
+
+
+
+
+yn
+0+ 1
+2
+yn
+1+ 1
+2
+...
+yn
+Nn+ 1
+2
+
+
+
+
+
+
+
++ i
+
+
+
+
+
+
+
+δxzn
+0+ 1
+2
+δxzn
+1+ 1
+2
+...
+δxzn
+Nn+ 1
+2
+
+
+
+
+
+
+
+,
+
+
+
+
+
+
+
+zn
+0+ 1
+2
+zn
+1+ 1
+2
+...
+zn
+Nn+ 1
+2
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+δxyn
+0+ 1
+2
+δxyn
+1+ 1
+2
+...
+δxyn
+Nn+ 1
+2
+
+
+
+
+
+
+
+.
+(52)
+The proof will be split into three claims and each claim
+corresponds to that in the proof of stability of PDE. Clam 1
+corresponds to ωn → ∞ in the proof of Theorem 2.1.
+Cliam 1: |βn| ≥ C′ > 0 for some constant C′ independent
+of n ∈ N+.
+Suppose by contrary that the sequence {βn} contains a
+subsequence which is still denoted by {βn} itself without
+loss of generality converging to zero. Since ∥Y n
+hn∥Yhn = 1 and
+∥Un
+hn∥Yhn ≤ n−1, it follows from (52) that
+hn
+Nn
+∑
+j=0
+���δxzn
+j+ 1
+2
+���
+2
+= hn
+Nn
+∑
+j=0
+���un
+j+ 1
+2 − iβnyn
+j+ 1
+2
+���
+2
+≤ 2hn
+Nn
+∑
+j=0
+���un
+j+ 1
+2
+���
+2
++ 2β 2
+nhn
+Nn
+∑
+j=0
+���yn
+j+ 1
+2
+���
+2
+= 2∥Un
+hn∥2
+Yhn + 2β 2
+n∥Y n
+hn∥2
+Yhn ≤ 2n−2,
+(53)
+which holds for all sufficiently large n. On the other hand, by
+some simple operations, we get
+|zn
+j − zn
+Nn+1|2 = |zn
+j − zn
+j+1 + zn
+j+1 − zn
+j+2 + zn
+j+2···− zn
+Nn+1|2
+=
+�����
+Nn
+∑
+l= j
+(zn
+l+1 − zn
+l )
+�����
+2
+≤
+�
+Nn
+∑
+l= j
+|1|2
+��
+Nn
+∑
+l= j
+��zn
+l+1 − zn
+l
+��2
+�
+≤ (Nn + 1)
+�
+Nn
+∑
+l=0
+��zn
+l+1 − zn
+l
+��2
+�
+= hn
+Nn
+∑
+j=1
+���δxzn
+j+ 1
+2
+���
+2
+,
+j = 0,1,··· ,Nn,
+(54)
+in which hn(Nn + 1) = 1 is used in the last step, and for j =
+0,1,··· ,Nn
+|zn
+j| ≤ |zn
+j − zn
+Nn+1|+ |zn
+Nn+1| ≤
+�
+�
+�
+�hn
+Nn
+∑
+j=1
+|δxzn
+j+ 1
+2 |2 + |zn
+Nn+1|.
+This inequality, together with zn
+Nn+1 = −ikyn
+Nn+1 and (50)-
+(53), implies that for each j = 0,1,··· ,Nn, |zn
+j|2 = O(n−1).
+Therefore, in light of hn(Nn + 1) = 1 and the second identity
+of (52),
+hn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+≤ hn
+2
+Nn
+∑
+j=0
+���zn
+j+1
+��2 +
+��zn
+j
+��2�
+≤ hn(Nn + 1)
+������
+�
+�
+�
+�hn
+Nn
+∑
+j=1
+|δxzn
+j+ 1
+2 |2 + |zn
+Nn+1|
+������
+2
+≤ hn(Nn + 1)Cn−1 = O(n−1).
+(55)
+Thus, the deducing process from (53) to (55) tells us that
+hn
+Nn
+∑
+j=0
+���δxzn
+j+ 1
+2
+���
+2
+= O(n−2),
+which implies that
+hn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+= O(n−1).
+By noticing the second identity of (52), we have
+hn
+Nn
+∑
+j=0
+���δxyn
+j+ 1
+2
+���
+2
+= hn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+,
+which means, by (55) that
+hn
+Nn
+∑
+j=0
+���δxyn
+j+ 1
+2
+���
+2
+= O(n−1).
+Similarly, repeating the procedure from (53) to (55), for Y n
+hn,
+we obtain
+∥Y n
+hn∥2
+Yhn = hn
+Nn
+∑
+j=0
+���yn
+j+ 1
+2
+���
+2
+= O(n−1/2),
+which leads to a contradiction. Thus, the sequence {βn} cannot
+contain a subsequence converging to zero. We can therefore
+assume that |βn| ≥ C′ > 0 for some constant C′ independent
+of n ∈ N+.
+The second claim is the discrete counterpart of (18) but with
+two extra terms
+h3
+n
+4βn
+Nn
+∑
+j=0
+���δxzn
+j+ 1
+2
+���
+2
++ h3
+n
+4
+Nn
+∑
+j=0
+���δxyn
+j+ 1
+2
+���
+2
+which play important roles in our proofs.
+Claim 2: the following (56) holds true:
+∥Y n
+hn∥2
+Yhn + 1
+βn
+hn∥Σhn �Zn
+hn∥2
+CNn+2
++ h2
+n
+4βn
+hn∥∆hn �Zn
+hn∥2
+CNn+2 + h2
+n
+4 hn∥∆hn�Y n
+hn∥2
+CNn+2
+= hn
+Nn
+∑
+j=0
+���yn
+j+ 1
+2
+���
+2
++ hn
+βn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
++ h3
+n
+4βn
+Nn
+∑
+j=0
+���δxzn
+j+ 1
+2
+���
+2
++ h3
+n
+4
+Nn
+∑
+j=0
+���δxyn
+j+ 1
+2
+���
+2
+= O(n−1),
+(56)
+
+IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, AUGUST 20XX
+8
+where ∥ ·∥CNn+2 denotes the standard norm of CNn+2 and
+�Zn
+hn = (z0,z1,··· ,zNn+1)⊤ =
+�
+(Zn
+hn)⊤,zNn+1
+�⊤
+,
+�Y n
+hn = (0,y1,··· ,yNn+1)⊤ =
+�
+0,(Y n
+hn)⊤�⊤
+,
+Σh = 1
+2
+
+
+
+
+
+1
+1
+...
+...
+1
+1
+1
+1
+
+
+
+
+
+(N+2)×(N+1)
+,
+∆h = 1
+h
+
+
+
+
+
+−1
+1
+−1
+1
+...
+...
+−1
+1
+
+
+
+
+
+(N+2)×(N+1)
+.
+(57)
+Actually, it follows from (49), (52), and ∥Y n
+hn∥Yhn = 1 that
+β −2
+n hn ∑Nn
+j=0
+����δxzn
+j+ 1
+2
+����
+2
+is uniformly bounded with respect to
+n ∈ N+ because
+1
+βn
+
+
+
+
+
+
+
+δxzn
+0+ 1
+2
+δxzn
+1+ 1
+2
+...
+δxzn
+Nn+ 1
+2
+
+
+
+
+
+
+
+= −
+
+
+
+
+
+
+
+yn
+0+ 1
+2
+yn
+1+ 1
+2
+...
+yn
+Nn+ 1
+2
+
+
+
+
+
+
+
+− i
+βn
+
+
+
+
+
+
+
+un
+0+ 1
+2
+un
+1+ 1
+2
+...
+un
+Nn+ 1
+2
+
+
+
+
+
+
+
+.
+By (55) and Claim 1,
+β −2
+n hn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+is also uniformly bounded with respect to n ∈ N+. Let xn
+j = jhn
+for j = 0,1,··· ,Nn + 1 and consider the following estimates:
+�����hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 (βnyn
+j+ 1
+2 + δxzn
+j+ 1
+2 )
+zn
+j+ 1
+2
+βn
+�����
+2
+=
+�����hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 un
+j+ 1
+2
+zn
+j+ 1
+2
+βn
+�����
+2
+≤
+�
+Nn
+∑
+j=0
+���
+�
+hnun
+j+ 1
+2
+���
+�����
+�
+hn
+zn
+j+ 1
+2
+βn
+�����
+�2
+≤
+�
+hn
+Nn
+∑
+j=0
+���un
+j+ 1
+2
+���
+2
+��
+β −2
+n hn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+�
+= ∥Un
+hn∥2
+Yhn
+�
+β −2
+n hn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+�
+= O(n−2),
+(58)
+where (49) and (52) were used. On the other hand, by the
+second identity of (52), we have
+hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 (βnyn
+j+ 1
+2 + δxzn
+j+ 1
+2 )
+zn
+j+ 1
+2
+βn
+= hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 yn
+j+ 1
+2 δxyn
+j+ 1
+2
++β −1
+n hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 δxzn
+j+ 1
+2 zn
+j+ 1
+2 .
+(59)
+Applying Lemma 4.4 to the two terms of the right hand side
+of (59) and noticing xn
+Nn+1 = 1, xn
+0 = 0, xn
+j+1 − xn
+j = hn, it is
+easy to obtain
+2Re
+�
+hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 yn
+j+ 1
+2 δxyn
+j+ 1
+2
+�
+= hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 yn
+j+ 1
+2 δxyn
+j+ 1
+2 + hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 yn
+j+ 1
+2 δxyn
+j+ 1
+2
+= 1
+4
+Nn
+∑
+j=0
+(xn
+j+1 + xn
+j)(yn
+j+1 + yn
+j)(yn
+j+1 − yn
+j)
++1
+4
+Nn
+∑
+j=0
+(xn
+j+1 + xn
+j)(yn
+j+1 − yn
+j)(yn
+j+1 + yn
+j)
+= |yNn+1|2 − hn
+Nn
+∑
+j=0
+���yn
+j+ 1
+2
+���
+2
+− h3
+n
+4
+Nn
+∑
+j=0
+���δxyn
+j+ 1
+2
+���
+2
+,
+(60)
+and
+2Re
+�
+hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 zn
+j+ 1
+2 δxzn
+j+ 1
+2
+�
+= hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 zn
+j+ 1
+2 δxzn
+j+ 1
+2 + hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 zn
+j+ 1
+2 δxzn
+j+ 1
+2
+= 1
+4
+Nn
+∑
+j=0
+(xn
+j+1 + xn
+j)(zn
+j+1 + zn
+j)(zn
+j+1 − zn
+j)
++ 1
+4
+Nn
+∑
+j=0
+(xn
+j+1 + xn
+j)(zn
+j+1 + zn
+j)(zn
+j+1 − zn
+j)
+= |zNn+1|2 − hn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+− h3
+n
+4
+Nn
+∑
+j=0
+���δxzn
+j+ 1
+2
+���
+2
+.
+(61)
+By (59)-(61), it follows that
+hn
+Nn
+∑
+j=0
+���yn
+j+ 1
+2
+���
+2
++ h3
+n
+4
+Nn
+∑
+j=0
+���δxyn
+j+ 1
+2
+���
+2
++ hn
+βn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
++ h3
+n
+4βn
+Nn
+∑
+j=0
+���δxzn
+j+ 1
+2
+���
+2
+= −2Re
+�
+hn
+Nn
+∑
+j=0
+xn
+j+ 1
+2 (βnyn
+j+ 1
+2 + δxzn
+j+ 1
+2 )
+zn
+j+ 1
+2
+βn
+�
++|yNn+1|2 + β −1
+n |zNn+1|2,
+(62)
+which proves (56) by (50), (58) and zNn+1 = −ikyNn+1.
+
+IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, AUGUST 20XX
+9
+The third claim is perfectly the discrete counterpart of (20).
+Claim 3: The following (63) holds true:
+∥Y n
+hn∥2
+Yhn + 1
+β 2n
+hn∥∆hn �Zn
+hn∥2
+CNn+2 − 2
+βn
+hn∥Σhn �Zn
+hn∥2
+CNn+2
+= hn
+Nn
+∑
+j=0
+���yn
+j+ 1
+2
+���
+2
++ hn
+β 2n
+Nn
+∑
+j=0
+���δxzn
+j+ 1
+2
+���
+2
+− 2hn
+βn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+= O(n−2),
+(63)
+in which Σhn and ∆hn are defined in (57) and ∥ · ∥CNn+2
+denotes the standard norm of CNn+2.
+Actually, from (52), we have
+∥Un
+hn∥2
+Yhn
+β 2n
+= hn
+β 2n
+Nn
+∑
+j=0
+���un
+j+ 1
+2
+���
+2
+= hn
+Nn
+∑
+j=0
+�����yn
+j+ 1
+2 +
+δxzn
+j+ 1
+2
+βn
+�����
+2
+= hn
+Nn
+∑
+j=0
+���yn
+j+ 1
+2
+���
+2
++ hn
+β 2n
+Nn
+∑
+j=0
+���δxzn
+j+ 1
+2
+���
+2
++ hn
+βn
+Nn
+∑
+j=0
+(yn
+j+ 1
+2 δxzn
+j+ 1
+2 + yn
+j+ 1
+2 δxzn
+j+ 1
+2 ).
+(64)
+On the other hand, it follows from the second identity of (52)
+that zn
+j+ 1
+2 = δxyn
+j+ 1
+2 and
+hn
+βn
+Nn
+∑
+j=0
+(yn
+j+ 1
+2 δxzn
+j+ 1
+2 + yn
+j+ 1
+2 δxzn
+j+ 1
+2 )+ 2hn
+βn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+= hn
+βn
+Nn
+∑
+j=0
+(yn
+j+ 1
+2 δxzn
+j+ 1
+2 + δxyn
+j+ 1
+2 zn
+j+ 1
+2 )
++ hn
+βn
+Nn
+∑
+j=0
+(δxyn
+j+ 1
+2 zn
+j+ 1
+2 + yn
+j+ 1
+2 δxzn
+j+ 1
+2 )
+=
+1
+2βn
+Nn
+∑
+j=0
+[(yn
+j+1 + yn
+j)(zn
+j+1 − zn
+j)+ (yn
+j+1 − yn
+j)(zn
+j+1 + zn
+j)]
++ 1
+2βn
+Nn
+∑
+j=0
+[(yn
+j+1 + yn
+j)(zn
+j+1 − zn
+j)+ (yn
+j+1 − yn
+j)(zn
+j+1 + zn
+j)]
+= 1
+βn
+Nn
+∑
+j=0
+(yn
+j+1zn
+j+1 − yn
+jzn
+j)+ 1
+βn
+Nn
+∑
+j=0
+(yn
+j+1zn
+j+1 − yn
+jzn
+j)
+= 1
+βn
+[yn
+Nn+1zn
+Nn+1 + yn
+Nn+1zn
+Nn+1 − yn
+0zn
+0 − yn
+0zn
+0] = 0,
+where zn
+Nn+1 = −ikyn
+Nn+1 and yn
+0 = 0 were used in the last step.
+Hence
+hn
+βn
+Nn
+∑
+j=0
+(yn
+j+ 1
+2 δxzn
+j+ 1
+2 + yn
+j+ 1
+2 δxzn
+j+ 1
+2 ) = −2hn
+βn
+Nn
+∑
+j=0
+���zn
+j+ 1
+2
+���
+2
+. (65)
+Plugging (65) into (64) and using ∥Un
+hn∥Yhn ≤ n−1 and Claim
+1, we arrive at (63).
+Finally, if βn > 0, we have ∥Y n
+hn∥2
+Yhn = O(n−1) from (56),
+which is a contradiction to ∥Y n
+hn∥Yhn = 1. When βn < 0,
+∥Y n
+hn∥Yhn = O(n−1) by virtue of (63), which is also a con-
+tradiction. We therefore complete the proof of the theorem.
+V. CONCLUDING REMARKS
+In this paper, the uniform approximation of exponential
+stability of a one-dimensional Schr¨odinger equation is investi-
+gated. We introduce an order reduction space semi-discretized
+finite difference scheme for approximating uniformly the expo-
+nentially stable closed-loop system. Although the scheme has
+been applied to certain PDEs in previous works, they all share
+a common feature that it is possible to find a suitable Lyapunov
+functional for the closed-loop systems, both for the continuous
+system and its discrete counterpart. However, for the system
+considered in this paper, it is a longstanding problem that in
+the natural state space L2(0,1), even for the continuous system,
+the time domain energy multiplier has not been found. This
+makes the convergence of the semi-discrete scheme of this
+PDE be open for a long time. This paper is the first work
+that applies the frequency domain multiplier approach to the
+uniformly exponential convergence of semi-discretized PDE
+system. The convergence of the discrete scheme to continuous
+system is not included because it is a standard procedure
+and can be followed analogously from [14] and many other
+papers. Considering it is difficult to find a time domain energy
+multiplier for many other PDEs, the approach presented in this
+paper has significant potentials in applying to other PDEs.
+ACKNOWLEDGMENTS
+The authors would like to thank Dr. Jiankang Liu and Dr.
+Hanjin Ren for their careful reading and many suggestions on
+the presentation of the initial manuscript of the paper. The
+Figures 1 and 2 were depicted by Dr. Jiangkang Liu.
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+Anal., 33(1999), 407-438.
+[11] Y. Kao, J. Xie and C. Wang, Stabilization of Singular Markovian
+Jump Systems With Generally Uncertain Transition Rates, IEEE Trans.
+Automat. Control, 59(2014), 2604-2610.
+[12] M. Krstic, B.Z. Guo, and A. Smyshlyaev, Boundary controllers and
+observers for the linearized Schr¨odinger equation, SIAM J. Control
+Optim., 49(2011), 1479-1497.
+[13] J. Liu and B.Z. Guo, A new semi-discretized order reduction finite
+difference scheme for uniform approximation of 1-D wave equation,
+SIAM J. Control Optim., 58(2020), 2256-2287.
+[14] J. Liu, R.Q. Hao, and B.Z. Guo, Order reduction-based uniform approxi-
+mation of exponential stability for one-dimensional Schr¨odinger equation,
+Systems Control Lett., 160(2022), Art.105136.
+[15] Z.Y. Liu and S.M. Zheng, Liu, Exponential stability of the semigroup
+associated with a thermoelastic system, Quart. Appl. Math., 51(1993),
+535-545.
+[16] Z.Y. Liu and S.M. Zheng, Uniform exponnential stability and approx-
+imation in control of a thermoelastic system, SIAM J. Control Optim.,
+32(1994), 1226-1246.
+[17] S. Micu and C. Castro, Boundary controllability of a linear semi-discrete
+1-D wave equation derived from a mixed finite element method, Numer.
+Math, 102(2006), 413-462.
+[18] J. Pr¨uss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc.,
+284(1984), 847-857.
+[19] L.T. Tebou and E. ZuaZua, Uniform exponential long time decay for
+the space semi-discretization of a locally damped wave equation via an
+artificial numerical viscosity, Numer. Math, 95(2003), 563-598.
+[20] L.T. Tebou and E. Zuazua, Uniform boundary stabilization of the finite
+difference space discretization of the 1-d wave equation, Adv. Comput.
+Math, 26(2007), 337-365.
+[21] M. Tucsnak and G. Weiss, Observation and Control for Operator
+Semigroups, Birkhauser Verlag, Basel, 2009.
+[22] Y. Wang, Y. Xia, H. Shen and P. Zhou, SMC Design for Robust
+Stabilization of Nonlinear Markovian Jump Singular Systems, IEEE
+Trans. Automat. Control, 63(1)(2018), 219-224.
+[23] F. Zheng and H. Zhou, State reconstruction of the wave equation with
+general viscosity and non-collocated observation and control, J. Math.
+Anal. Appl., 502(2020), Art. 125257.
+[24] E. Zuazua, Propagation, observation, and control of waves approximated
+by finite difference methods, SIAM Rev., 47(2005), 197-243.
+PLACE
+PHOTO
+HERE
+Bao-Zhu Guo received the Ph.D. degree from the
+Chinese University of Hong Kong in Applied Math-
+ematics in 1991. Since 2000, he has been with
+the Academy of Mathematics and Systems Science,
+the Chinese Academy of Sciences, where he is a
+Research Professor in mathematical system theory.
+From 2004-2019, he was a chair professor in School
+of Computer Science and Applied Mathematics,
+University of the Witwatersrand, South Africa. He is
+the author or co-author of the books: “Stability and
+Stabilization of Infinite Dimensional Systems with
+Applications” (Springer-Verlag, 1999); “Active Disturbance Rejection Control
+for Nonlinear Systems: An Introduction” (John Wiley & Sons, 2016); and
+“Control of Wave and Beam PDEs-The Riesz Basis Approach” (Springer-
+Verlag, 2019). His research interests include theory of infinite-dimensional
+systems and active disturbance rejection control.
+Dr. Guo received the One Hundred Talent Program from the Chinese
+Academy of Sciences (1999), and the National Science Fund for Distinguished
+Young Scholars (2003).
+PLACE
+PHOTO
+HERE
+Fu Zheng received the B.Sc. and M.Sc. degrees
+in mathematics from the Yanbian University, Jilin,
+China in 2002 and 2005, and the Ph.D. from Beijing
+Institute of Information and Control, Beijing, China
+in 2011. He held a postdoctoral research position
+in Academy of Mathematics and Systems Science,
+Academia Sinica, Beijing, China, from 2013 to
+2015. From 2012-2021, he was an associate profes-
+sor at Bohai University, Jinzhou, China. At the end
+of 2021, he joined the School of Science, Hainan
+University, Hainan China, where he is currently a
+professor. His research interests include control theory of infinite-dimensional
+systems and numerical solutions to control problems of distributed parameter
+systems.
+
diff --git a/OdFJT4oBgHgl3EQf0y3g/content/tmp_files/load_file.txt b/OdFJT4oBgHgl3EQf0y3g/content/tmp_files/load_file.txt
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@@ -0,0 +1,649 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf,len=648
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='11649v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='OC]  27 Jan 2023  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' X, AUGUST 20XX  1  Frequency Energy Multiplier Approach to Uniform  Exponential Stability Analysis of Semi-discrete  Scheme for a Schr¨odinger Equation  under Boundary Feedback  Bao-Zhu Guo and Fu Zheng  Abstract—In this paper, we investigate the uniform exponential  stability of a semi-discrete scheme for a Schr¨odinger equation  under boundary feedback stabilizing control in the natural state  space L2(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' This study is significant since a time domain energy  multiplier that allows proving the exponential stability of this  continuous Schr¨odinger system has not yet found, thus leading  to a major mathematical challenge to semi-discretization of the  PDE, an open problem for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Although the powerful  frequency domain energy multiplier approach has been used in  proving exponential stability for PDEs since 1980s, its use to  the uniform exponential stability of the semi-discrete scheme for  PDEs has not been reported yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The difficulty associated with  the uniformity is that due to the parameter of the step size,  it involves a family of operators in different state spaces that  need to be considered simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Based on the Huang-Pr¨uss  frequency domain criterion for uniform exponential stability of a  family of C0-semigroups in Hilbert spaces, we solve this problem  for the first time by proving the uniform boundedness for all the  resolvents of these operators on the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The proof  almost exactly follows the procedure for the exponential stability  of the continuous counterpart, highlighting the advantage of this  discretization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Index Terms—Schr¨odinger equation, boundary damping, fre-  quency domain multiplier, semi-discretization, uniform exponen-  tial stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' INTRODUCTION  C  Ontrol systems described by partial differential equations  (PDEs) is infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Being such, its controller  such as the observer-based feedback control is also infinite-  dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' As a result, the discretization finds itself in  almost all implementations of PDE control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Among many  discretization methods is the finite-difference method which  becames popular due to its simplicity in principle and its  appeal to engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' One of the most commonly used dis-  cretization method is the so-called semi-discrete scheme which  keeps time continuous while discretizing the spatial variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  It has been widely studied in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The main advantage  of the semi-discrete scheme is that it results in an ordinary  This work was supported by the National Natural Science Foundation  of China under grants no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='61873260, 11871117, 12131008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' (Corresponding  author: Fu Zheng)  Bao-Zhu Guo is with Department of Mathematics and Physics, North  China Electric Power University, Beijing 102206, China, and Key Laboratory  of System and Control, Academy of Mathematics and Systems Science,  Academia Sinica, Beijing 100190, Email:bzguo@iss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='cn  Fu Zheng is with School of Science of Hainan University, Haikou, Hainan  570228, E-mail: fuzheng@hainanu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='cn  differential equation system, which control researchers are  most familiar with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' However, it has been acknowledged for a  long time that the uniform exponential stability with respect to  the spatial discrete step size cannot be guaranteed for classical  semi-discrete schemes for PDEs, largely due to presence of  high frequency spurious components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In addition, some other  typical important control properties such as uniform observ-  ability and uniform exact controllability cannot be guaranteed  either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The reason for this loss is that the spurious modes  are only weakly damped in the process of semi-discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  A detail account can be found in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' For wave equations,  several remedies such as Tichonoff regularization [7], mixed-  finite elements [2], [17], high frequency filtering [10], and  non-uniform meshes [4], have been proposed to circumvent  this difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Among many these remedies, the numerical  viscosity damping introduced in [19], [20] is the most popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  However, this approach brings a viscosity term artificially  added into the classical discrete scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The coefficients of the  numerical viscosity damping vary from PDE to PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Recently,  a new natural semi-discrete scheme based on order reduction  finite difference method was introduced in [13] and has been  applied to different systems [8], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' This approach has the  critical advantages that it guarantees the uniform exponential  stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In addition, as a natural semi-discrete scheme, it al-  lows one to prove the uniform exponential stability in a manner  parallel to its continuous PDE counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Nevertheless, all  the previous papers on this scheme involved construction of  Lyapunov functional which the proof heavenly relies on, both  for semi-discrete scheme and the continuous counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Construction of a suitable Lyapunov functional for a PDE  relies on a time domain energy multiplier, which is not always  available and its construction is most often very technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In  1980s, a frequency domain energy multiplier approach was  developed for the exponential stability initially for a single  PDE ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The approach is based on a frequency domain  characterization for exponential stability of C0-semigroup in  Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Originally developed independently in [9] and  [18], the result of was proved later in [16] to be valid for  uniform exponential stability of a family of C0-semigroups in  Hilbert spaces as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Uniform admissibility and observability  for the finite element space semi-discretizations of abstract  Schr¨odinger system and second order infinite dimensional  vibrating systems have also been developed [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  In this paper, we investigate the uniform exponential sta-  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' X, AUGUST 20XX  2  bility of an order reduction semi-discrete scheme for a  Schr¨odinger equation under boundary control by the frequency  domain multiplier approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' It is significant because one  cannot find a suitable time domain Lyapunov functional both  for the continuous PDE and for its discrete scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' This  implies that successful approaches presented in [14], [13], [8],  [23] cannot be applied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' As a matter of fact, in order to  apply the Lyapunov method, the paper [14] has to consider  the Schr¨odinger system in the high order state space H1(0,1),  whereas our state space is the standard space L2(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The  problem in L2(0,1) has been open for quite a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Fairly  speaking, this paper brings a new way to the proof of the  uniform exponential stability of the semi-discrete scheme for  PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' It is also worthy pointing out that the proofs for both  continuous PDE and for the discrete counterpart are again  analogous, demonstrating the advantage of the order reduction  semi-discretization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  We proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In the next section, Section II, we  prove the exponential stability of the continuous PDE by the  frequency domain multiplier method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Although it is the sim-  plest PDE ever studied in the literature, it helps in constructing  a frequency domain multiplier for its semi-discrete counter-  part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In Section III, we design a semi-discretized scheme and  obtain a family of finite-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In Section IV,  the uniform exponential stability is developed by the frequency  domain multiplier approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' We introduce the shadow ele-  ment to help understand the numerical approximating scheme,  which plays an important role in the proof of uniform stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Some concluding remarks are included in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' STABILITY OF SCHR ¨ODINGER SYSTEM VIA FREQUENCY  DOMAIN MULTIPLIER  Consider the following Schr¨odinger equation under bound-  ary control:  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  wt(x,t) = −iwxx(x,t), t > 0, x ∈ (0,1),  w(0,t) = 0, t ≥ 0,  wx(1,t) = u(t), k > 0, t ≥ 0,  y(t) = w(1,t), t ≥ 0,  w(x,0) = w0(x), x ∈ [0,1],  (1)  where u(·) is the control, y(·) is the measured output and w0(·)  is the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Under the proportional feedback control:  u(t) = −kiy(t), k > 0,  (2)  the closed-loop system of (1) becomes  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  wt(x,t) = −iwxx(x,t), t > 0, x ∈ (0,1),  w(0,t) = 0, t ≥ 0,  wx(1,t) = −kiw(1,t), k > 0, t ≥ 0,  w(x,0) = w0(x), x ∈ [0,1],  (3)  We consider system (3) in the natural state space L2(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Define the system operator of (3) as follows:  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  Af = −if ′′,∀f ∈ D(A),  D(A) = { f ∈ L2(0,1)|f ∈ H2(0,1),  f(0) = 0, f ′(1) = −ik f(1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (4)  Then, (3) can be written as an evolution equation in L2(0,1):  �  ˙w(·,t) = Aw(·,t),  w(x,0) = w0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (5)  It is seen that  Re⟨Af, f⟩L2(0,1) = Re  � 1  0 −ik f ′′(x)f(x)dx = −k|f(1)|2,  (6)  which implies that A is dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In addition, the operator  A is invertible and  A−1 f(x) = −kx  � 1  0 xf(x)dx  1 + ki  − i  � 1  x (x− τ)f(τ)dτ − i  � 1  0 xf(x)dx,  (7)  which is bounded in L2(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' As a result, A generates a C0-  semigroup of contractions on L2(0,1) by the Lumer-Phillips  theorem ([21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='4]) and since A−1 is compact, the  spectrum of A consists of isolated eigenvalues only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Furthermore, define the system energy for (3) as  E(t) = 1  2  � 1  0 |w(x,t)|2dt,  (8)  which is non-increasing as a consequence of (6):  ˙E(t) = −k|w(1,t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (9)  We point out that a different version of (3):  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  wt(x,t) = −iwxx(x,t), t > 0, x ∈ (0,1),  w(0,t) = 0, t ≥ 0,  wx(1,t) = −kwt(1,t), k > 0, t ≥ 0,  w(x,0) = w0(x), x ∈ [0,1],  (10)  was investigated in [14], for which one can find a time  domain energy multiplier, and a Lyapunov functional was  then constructed to both system (10) and its semi-discrete  counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' However, system (3) is a rather unusual system  for which a time domain energy multiplier has been not  found yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' A first exponential stability result of system (3)  was proved by the Riesz basis approach in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Although  the Riesz basis is powerful and the result obtained is much  deeper than the result obtained from the multiplier method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' for  instance the spectrum-determined growth condition is usually  a consequence of the Riesz basis approach yet this is usually  not the case with the multiplier method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Unfortunately, the  Riesz basis is extremely difficult, at least at the moment, to be  applied for the uniformly exponential stability of semi-discrete  model for (3) developed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  In this paper, we use an alternative powerful method called  the frequency energy multiplier method, which has been  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' X, AUGUST 20XX  3  developed for continuous PDEs over the last three decades  ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In stability analysis, we can almost give one-to-one  correspondence from continuous system to its discrete coun-  terpart using this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Our approach is so powerful that  can be applied to other PDEs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' For notation simplicity,  hereafter, we omit without confusion the obvious dependency  in time and spatial domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The Cn denotes the n-dimensional  complex Euclidean space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' the N+ stands for the set of the  positive integer numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' and R the set of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Since A generates a C0-semigroup of contractions on  L2(0,1), a well-known result of Hung-Pr¨uss theorem [9], [18]  states that the C0-semigroup generated by A is exponentially  stable if and only if it possesses the following two properties:  1) Every imaginary number belongs to the resolvent set of  A, that is, iR ⊂ ρ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  2) The inverse operator of iω −A is uniformly bounded for  all imaginary numbers, that is,  sup  ω∈R  ∥(iω − A)−1∥ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (11)  The property iR ⊂ ρ(A) is stated in the following Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1: Let A be defined by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Then, iR ⊂ ρ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' If there exist β ∈ R,β ̸= 0 and a nonzero f ∈ D(A)  such that iβ f = Af, then  �  iβ f(x) = −if ′′(x),  f ′(1) = −kif(1), f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (12)  Take the inner product with f(·) over [0,1] on both sides of  the first equation of (12) to obtain  iβ∥ f∥2 = −k|f(1)|2 + i  � 1  0 |f ′(x)|2dx,  (13)  which gives f(1) = 0 and hence f ′(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' This shows that  (12) has only zero solution, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1: Let A be defined by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Then, (11) holds  true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' As a consequence, the C0-semigroup eAt generated by A  is exponentially stable in L2(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' We prove by assuming contrary of (11) that there  exit a sequence ωn → ∞, fn ∈ D(A), ∥ fn∥ = 1 that  lim  n→∞∥(iωn − A)fn∥ = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=',  iωn fn + if ′′  n → 0 in L2(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (14)  Since  Re⟨(iωn − A)fn, fn⟩L2(0,1) = Re⟨−Afn, fn⟩ = k|fn(1)|2 → 0,  (15)  by the boundary condition f ′  n(1) = −ik fn(1), it gives  f ′  n(1) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (16)  From (14) and ∥ fn∥ = 1, it follows that  f ′′n (·)  ωn  is bounded in  L2(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' By  |f ′  n(x)− f ′  n(1)| =  ����  � x  1 f ′′  n (s)ds  ���� ≤ ∥ f ′′  n ∥,  it follows from (16) and ωn → ∞ that  f ′  n(·)  ωn  is bounded in L2(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (17)  Since  Re  �  ωn fn + f ′′  n , xf ′  n  ωn  �  L2(0,1)  = |fn(1)|2  2  − 1  2  � 1  0 |fn(x)|2dx  +  1  2ωn  |f ′  n(1)|2 −  1  2ωn  � 1  0 |f ′  n(x)|2dx,  and  �  ωn fn + f ′′  n , xf ′  n  ωn  �  L2(0,1)  → 0,  we have by (15) and (16) that  � 1  0 |fn(x)|2dx+ 1  ωn  � 1  0 |f ′  n(x)|2dx → 0,  (18)  which shows that when ωn > 0, ∥ fn∥2 → 0, which is a  contradiction to ∥ fn∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' since from (14)  and ωn → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' we have  � 1  0  ����fn(x)+ f ′′  n (x)  ωn  ����  2  dx  =  � 1  0  �  fn(x)+ f ′′  n (x)  ωn  ��  fn(x)+ f ′′n (x)  ωn  �  dx  =  � 1  0  �  |fn(x)|2 + |f ′′  n (x)|2  ω2n  �  dx  + 1  ωn  � 1  0 [fn(x)f ′′n (x)+ fn(x) f ′′  n (x)]dx  =  � 1  0  �  |fn(x)|2 + |f ′′  n (x)|2  ω2n  �  dx  + 1  ωn  [fn(x)f ′n(x)+ fn(x) f ′  n(x)]1  0  − 2  ωn  � 1  0 |f ′  n(x)|2dx → 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (19)  Substitute f ′  n(1) = −ik fn(1) and fn(0) = 0 into (19),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' and use  (15)-(16) to obtain  � 1  0 |fn(x)|2dx+  � 1  0  |f ′′  n (x)|2  ω2n  dx− 2  ωn  � 1  0 |f ′  n(x)|2dx → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' (20)  which shows that when ωn < 0, ∥ fn∥2 → 0, which is also a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' SEMI-DISCRETE SCHEME OF SCHR ¨ODINGER EQUATION  In this section we apply the order reduction method to derive  a semi-discrete scheme for (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' To this purpose, we introduce  an intermediate variable v(x,t) = wx(x,t) to reduce the order  of the spacial derivative of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In this way, the Schr¨odinger  equation (3) can be rewritten as the following equivalent form:  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  wt(x,t)+ ivx(x,t) = 0,  v(x,t) = wx(x,t),  w(0,t) = 0,  v(1,t) = −kiw(1,t),  w(x,0) = w0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (21)  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' X, AUGUST 20XX  4  The semi-discretization process is similar to [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' For the  sake of completeness, we sketch briefly the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' For fixed  N ∈ N+, consider an equidistant partition of interval [0,1]:  0 = x0 < x1 < ··· < xj = jh < ··· < xN+1 = 1,  where h =  1  N+1 is the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Denote the sequence {u j}N+1  0  by {u j} j and introduce respectively the average operator and  the first-order finite difference operator as  u j+ 1  2 = u j + u j+1  2  ,  δxu j+ 1  2 = u j+1 − u j  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (22)  For the solutions v(x,t) and w(x,t) of (21), let {Vj(t)} j and  {Wj(t)} j be grid functions at grids {xj} j, satisfying  Vj(t) = v(xj,t),  Wj(t) = w(xj,t),  0 ≤ j ≤ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  The first equation of system (21) holds at (xj+ 1  2 ,t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=',  w′(xj+ 1  2 ,t)+ ivx(xj+ 1  2 ,t) = 0,  where xj+ 1  2 = (j + 1  2)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Hereafter the prime “′” represents  the derivative with respect to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Replace the differential  operator ∂x with difference operator δx to get  W ′  j+ 1  2 (t)+ iδxVj+ 1  2 (t) = O(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (23)  Similarly, for the second equation of system (21), it has  Vj+ 1  2 (t)− δxWj+ 1  2 (t) = O(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (24)  By dropping the infinitesimal terms in (23) and (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' and  replacing Wj(t) and Vj(t) by wj(t) and vj(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' we  arrive at a semi-discretized finite difference scheme of system  (21) as follows:  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  w′  j+ 1  2 (t)+ iδxvj+ 1  2 (t) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' 0 ≤ j ≤ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  vj+ 1  2 (t) = δxwj+ 1  2 (t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' 0 ≤ j ≤ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  vN+1(t) = −kiwN+1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' t ≥ 0  w0(t) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  wj(0) = w0  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' 0 ≤ j ≤ N + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (25)  where vj(t) and wj(t) are grid functions at grids xj (0 ≤ j ≤  N +1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' and w0  j is the approximation of the initial value w0(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1: The semi-discretized system (25) is a family  of differentiation-algebra systems, which is called singular  systems for which there are huge amount of references related  to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' See for instance [3], [11], [22] and the references  therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Now, we eliminate the intermediate variables vj(t) from  (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' To this purpose, let  Wh(t) = (w1(t),w2(t),··· ,wN+1(t))⊤  be unknown variable of (25) and  Vh(t) = (v0(t),v1(t),··· ,vN(t))⊤  the auxiliary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' We write (25) into vectorial form:  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  DhW ′  h(t) = −iMhVh(t)−  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  0  kh−1wN+1(t)  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f8,  D⊤  h Vh(t) = −M⊤  h Wh(t)+  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  0  i2−1kwN+1(t)  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f8,  Wh(0) = (w0  0,w0  1,··· ,w0  N)⊤,  (26)  where the matrices Dh and Mh are given by  Dh = 1  2  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  1  1  1  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  (N+1)×(N+1)  ,  Mh = 1  h  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  −1  1  −1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  −1  1  −1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  (N+1)×(N+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (27)  Obviously, both Dh and Mh are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The differential  algebraic system (25) or (26) can be written as an evolution  equation in CN+1:  �  W ′  h(t) = AhWh(t), Wh(t) ∈ Yh = CN+1,  Wh(0) = (w0  1,w0  2,··· ,w0  N+1)⊤ ∈ Yh,  (28)  where Ah is defined by  AhYh  = D−1  h  �  iMh  �  D⊤  h  �−1 �  M⊤  h Yh − (0,··· ,0,2−1ikyN+1)⊤�  − D−1  h (0,··· ,0,kh−1yN+1)⊤�  ,  ∀Yh = (y1,y2,··· ,yN+1)⊤ ∈ CN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (29)  System (28) is naturally discussed in the state space CN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  To relate CN+1 in (28) with the step size, we write Yh = CN+1  and define a new inner product for Yh:  �  Yh,�Yh  �  Yh  = h  �  DhYh,Dh�Yh  �  ,∀Yh,�Yh ∈ Yh,  where ⟨·,·⟩ is the standard inner product of CN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' For Yh =  (y1,y2,··· ,yN+1)⊤ ∈ Yh, we choose the vector  Zh = (z0,z1,··· ,zN)⊤ ∈ CN+1 satisfying  D⊤  h Zh = −M⊤  h Yh + (0,··· ,0,2−1ikyN+1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (30)  We call Zh the shadow element of Yh, which can simplify  significantly the notation in the later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  The classical semi-discrete scheme is similar with (28)  where the average operator Dh = IN+1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=',  �  W ′  h(t) =  ˆ  AhWh(t), Wh(t) ∈ Yh = CN+1,  Wh(0) = (w0  1,w0  2,··· ,w0  N+1)⊤ ∈ Yh,  (31)  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' X, AUGUST 20XX  5  in which the  ˆ  Ah is defined by  ˆ  AhYh = iMh  �  M⊤  h Yh − (0,··· ,0,2−1ikyN+1)⊤�  − (0,··· ,0,kh−1yN+1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (32)  At the end of this section we explain the significance of  the discrete scheme (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' We plot two figures in Figures 1  and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Figure 1 depicts the maximal real parts  of the eigenvalues of the classical semi-discrete scheme (31)  with step size h, from which we see that the real parts of the  eigenvalues approach zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Figure 2 depicts the maximal real  parts of the eigenvalues of the order reduction semi-discrete  scheme (28) with the same step size, from which we see that  the real parts of the eigenvalues approach a negative number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  In both figures, we take k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  N  0  20  40  60  80  100  120  140  160  180  200  Real part of eigenvalue  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='02  0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Maximal real parts of eigenvalues of the semi-discrete scheme by  classical method (31)  N  0  20  40  60  80  100  120  140  160  180  200  Real part of eigenvalue  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='954  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='952  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='948  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='946  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='944  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='942  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='94  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Maximal real parts of eigenvalues of the semi-discrete scheme by  order reduction method (28)  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' PROOF OF UNIFORM EXPONENTIAL STABILITY  This section is devoted to the proof of the uniform expo-  nential stability of (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' To begin with, we first show that Ah  is dissipative for every step size h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1: For the matrix Ah defined by (29), there holds  Re⟨AhYh,Yh⟩Yh = −k|yN+1|2, ∀ Yh ∈ Yh,  (33)  which implies that Ah is dissipative for every h ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  For  Yh = (y1,y2,··· ,yN+1) ∈ Yh,  let  Zh =  (z0,z1,··· ,zN) be the shadow element of Yh:  �  D⊤  h Zh = −M⊤  h Yh + (0,··· ,0,2−1ikyN+1)⊤,  AhYh = D−1  h [−iMhZh + (0,··· ,0,kh−1yN+1)⊤].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (34)  Set y0 := 0 and zN+1 := −ikyN+1 and introduce �Yh =  (�y1,�y2,··· ,�yN+1) ∈ Yh such that AhYh = �Yh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Then,  D⊤  h Zh + (0,··· ,0,2−1zN+1)⊤ = −M⊤  h Yh,  (35)  which is equivalent to  zj+ 1  2 = δxyj+ 1  2 , j = 0,1,··· ,N,  (36)  and  Dh�Yh = −iMhZh − (0,··· ,0,ih−1zN+1)⊤,  (37)  which is equivalent to  �yj+ 1  2 = −iδxzj+ 1  2 , j = 0,1,··· ,N,  (38)  where in all (35) to (38), it was assumed that �y0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Take the  inner product between AhYh and Yh in Yh by taking (36) and  (38) into account to obtain  Re⟨AhYh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Yh⟩Yh = Re  �  �Yh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Yh  �  Yh  = h  2  �  Dh�Yh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='DhYh  �  + h  2  �  DhYh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Dh�Yh  �  = h  2  N  ∑  j=0  �yj+ 1  2 yj+ 1  2 + h  2  N  ∑  j=0  yj+ 1  2 �yj+ 1  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' (using (38))  = −hi  2  N  ∑  j=0  δxzj+ 1  2 yj+ 1  2 + hi  2  N  ∑  j=0  yj+ 1  2 δxzj+ 1  2  = −hi  2  N  ∑  j=0  �  δxzj+ 1  2 yj+ 1  2 (t)+ zj+ 1  2 δxyj+ 1  2  �  + hi  2  N  ∑  j=0  �  yj+ 1  2 δxzj+ 1  2 + δxyj+ 1  2 zj+ 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' (using (36))  (39)  A simple calculation shows that  −hi  2  N  ∑  j=0  �  δxzj+ 1  2 yj+ 1  2 + zj+ 1  2 δxyj+ 1  2  �  +hi  2  N  ∑  j=0  �  yj+ 1  2 δxzj+ 1  2 + δxyj+ 1  2 z j+ 1  2  �  = − i  4  N  ∑  j=0  �  (zj+1 − zj)(yj+1 + yj)+ (zj+1 + zj)(yj+1 − yj)  �  + i  4  N  ∑  j=0  �  (yj+1 + yj)(z j+1 − zj)+ (yj+1 − yj)(zj+1(t)+ zj)  �  = − i  2  N  ∑  j=0  [zj+1yj+1 − zjyj]+ i  2  N  ∑  j=0  [yj+1zj+1 − yjzj]  = i  2[z0y0 − zN+1yN+1]+ i  2[yN+1zN+1 − y0z0]  = −k|yN+1|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' ( using − ikyN+1 = zN+1 and y0 = 0) (40)  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' X, AUGUST 20XX  6  The (39) and (40) leads to (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Define the energy of (28) as  Eh(t) = h  2  N  ∑  j=0  ���wj+ 1  2 (t)  ���  2  = 1  2 ⟨Wh(t),Wh(t)⟩Yh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (41)  which is the discretization of the continuous energy (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The  following Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2 is the discrete counterpart of (9), which  is a consequence of (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2: The Eh(t) defined by (41) satisfies  ˙Eh(t) = −k|wN+1(t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (42)  The dissipativity of Ah implies that the spectral set σ(Ah)  of Ah is contained in the closed left half-plane of the complex  plane C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Actually, we have more stronger result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Precisely, for  any 0 < h < 1, the spectral set σ(Ah) of Ah is contained in  the open left half-plane of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' This is the following Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='3: For every h ∈ (0,1), iR ⊂ ρ(Ah).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' If there exist β ∈ R and nonzero Yh ∈ Yh such that  iβYh = AhYh, then it follows from (33) that  0 = Re⟨iβYh, Yh⟩Yh = Re⟨AhYh, Yh⟩Yh = −k|yN+1|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (43)  Replacing �Yh by iβYh in (37), we obtain  \uf8f1  \uf8f2  \uf8f3  βyj+ 1  2 + δxzj+ 1  2 = 0,  0 ≤ j ≤ N,  zj+ 1  2 − δxyj+ 1  2 = 0,  0 ≤ j ≤ N,  (44)  where Zh is the shadow element of Yh defined in (34), y0 =  0 and zN+1 := −ikyN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Hence zN+1 = yN+1 = 0 from (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Setting j = N in (44) yields  βhyN = 2zN, zN = −2  hyN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  It follows that yN = zN = 0 whenever βh2+4 is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Under  the condition βh2 +4 ̸= 0, suppose zj+1 = yj+1 = 0 and solve  (44) to arrive at zj = yj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' This gives Yh = 0 by induction,  which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' On the other hand, whenever βh2 +  4 = 0, it follows from (44) that  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  1  h(yj+1 + yj) = 1  2(zj+1 − zj), j = 0,1,··· ,N,  1  h(yj+1 − yj) = 1  2(zj+1 + zj), j = 0,1,··· ,N,  (45)  which implies that  yj+1 = h  2zj+1, yj = −h  2zj, j = 0,1,··· ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (46)  This, combining with y0 = 0 and yN+1 = 0, gives yj = 0 (j =  1,2,··· ,N) which is also a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' This completes the  proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  The following lemma comes from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='4: Let {ui}i, {vi}i and {wi}i be the sequences of  complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Then,  1  4  N  ∑  i=0  (ui+1 − ui)(vi+1 + vi)(wi+1 + wi)  +1  4  N  ∑  i=0  (ui+1 − ui)(vi+1 − vi)(wi+1 − wi)  +1  4  N  ∑  i=0  (ui+1 + ui)(vi+1 − vi)(wi+1 + wi)  +1  4  N  ∑  i=0  (ui+1 + ui)(vi+1 + vi)(wi+1 − wi)  = uN+1vN+1wN+1 − u0v0w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (47)  The following uniformly stability criterion which was pre-  sented in [15] or [1] will be used in the proof of our main  result Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2 later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1: Let h∗ > 0 and let {Sh(t)}h∈(0,h∗) be a family  of semigroups of contractions on the Hilbert space Hh, and let  �Ah be the corresponding infinitesimal generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The family  {Sh(t)} is uniformly exponentially stable if and only if the  following two conditions are fulfilled:  For every h ∈ (0,h∗), iR ⊂ ρ(�Ah);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  suph∈(0,h∗),β∈R∥(iβI − �Ah)−1∥ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Now, we are in a position to give the main result of this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2: For the matrices Ah defined by (29), the  corresponding family of C0-semigroups Th(t) generated by  Ah is uniformly exponentially stable, that is, there exist two  constants M > 0 and ω > 0 independent of h ∈ (0,1) such that  ∥Th(t)∥ ≤ Me−ωt, ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (48)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The proof is based on Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Notice that by  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1, for every h ∈ (0,1), Th(t) is a C0-semigroup of  contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The fact that Ah satisfies the first condition of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1 has been claimed by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' In order to show  that the family Ah satisfies the second condition of Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1, we prove by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' If the second condition of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1 is false, then there exist a sequence βn ∈ R, hn ∈  (0,1), and Y n  hn ∈ Yhn,∥Y n  hn∥Yhn = 1 such that  ∥Un  hn∥Yhn ≤ n−1, Un  hn = (iβnIhn − Ahn)Y n  hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (49)  By the Cauchy-Schwartz inequality, it follows from (49) and  (33) that  Re  �  Un  hn,Y n  hn  �  Yhn  = −Re  �  AhnY n  hn,Y n  hn  �  Yhn  = k|yn  Nn+1|2 ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (50)  Let  Zn  hn = (zn  0,zn  1,··· ,zn  Nn)⊤ ∈ Zhn  be the shadow element of Y n  hn = (yn  1,yn  2,··· ,yn  Nn+1)⊤(see(30)),  Un  hn = (un  1,un  2,··· ,un  Nn+1)⊤ with hn(Nn +1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Set artificially  un  0 = yn  0 = 0 and zn  Nn+1 = −ikyn  Nn+1 to unify the notation of  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' X, AUGUST 20XX  7  un  j+ 1  2 and δxzn  j+ 1  2 from j = 0,1,··· ,Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Then, it follows from  (49) that  � DhnUn  hn = iβnDhnY n  hn + iMhnZn  hn + (0,··· ,0,ih−1zn  Nn+1)⊤,  −M⊤  hnY n  hn = D⊤  hnZn  hn + (0,··· ,0,2−1zn  Nn+1)⊤,  (51)  or in vector form:  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  un  0+ 1  2  un  1+ 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  un  Nn+ 1  2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  = iβn  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  yn  0+ 1  2  yn  1+ 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  yn  Nn+ 1  2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  + i  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  δxzn  0+ 1  2  δxzn  1+ 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  δxzn  Nn+ 1  2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  zn  0+ 1  2  zn  1+ 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  zn  Nn+ 1  2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  δxyn  0+ 1  2  δxyn  1+ 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  δxyn  Nn+ 1  2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (52)  The proof will be split into three claims and each claim  corresponds to that in the proof of stability of PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Clam 1  corresponds to ωn → ∞ in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Cliam 1: |βn| ≥ C′ > 0 for some constant C′ independent  of n ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Suppose by contrary that the sequence {βn} contains a  subsequence which is still denoted by {βn} itself without  loss of generality converging to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Since ∥Y n  hn∥Yhn = 1 and  ∥Un  hn∥Yhn ≤ n−1, it follows from (52) that  hn  Nn  ∑  j=0  ���δxzn  j+ 1  2  ���  2  = hn  Nn  ∑  j=0  ���un  j+ 1  2 − iβnyn  j+ 1  2  ���  2  ≤ 2hn  Nn  ∑  j=0  ���un  j+ 1  2  ���  2  + 2β 2  nhn  Nn  ∑  j=0  ���yn  j+ 1  2  ���  2  = 2∥Un  hn∥2  Yhn + 2β 2  n∥Y n  hn∥2  Yhn ≤ 2n−2,  (53)  which holds for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' by  some simple operations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' we get  |zn  j − zn  Nn+1|2 = |zn  j − zn  j+1 + zn  j+1 − zn  j+2 + zn  j+2···− zn  Nn+1|2  =  �����  Nn  ∑  l= j  (zn  l+1 − zn  l )  �����  2  ≤  �  Nn  ∑  l= j  |1|2  ��  Nn  ∑  l= j  ��zn  l+1 − zn  l  ��2  �  ≤ (Nn + 1)  �  Nn  ∑  l=0  ��zn  l+1 − zn  l  ��2  �  = hn  Nn  ∑  j=1  ���δxzn  j+ 1  2  ���  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  j = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Nn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (54)  in which hn(Nn + 1) = 1 is used in the last step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' and for j =  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Nn  |zn  j| ≤ |zn  j − zn  Nn+1|+ |zn  Nn+1| ≤  �  �  �  �hn  Nn  ∑  j=1  |δxzn  j+ 1  2 |2 + |zn  Nn+1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  This inequality, together with zn  Nn+1 = −ikyn  Nn+1 and (50)-  (53), implies that for each j = 0,1,··· ,Nn, |zn  j|2 = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Therefore, in light of hn(Nn + 1) = 1 and the second identity  of (52),  hn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  ≤ hn  2  Nn  ∑  j=0  ���zn  j+1  ��2 +  ��zn  j  ��2�  ≤ hn(Nn + 1)  ������  �  �  �  �hn  Nn  ∑  j=1  |δxzn  j+ 1  2 |2 + |zn  Nn+1|  ������  2  ≤ hn(Nn + 1)Cn−1 = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (55)  Thus, the deducing process from (53) to (55) tells us that  hn  Nn  ∑  j=0  ���δxzn  j+ 1  2  ���  2  = O(n−2),  which implies that  hn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  By noticing the second identity of (52), we have  hn  Nn  ∑  j=0  ���δxyn  j+ 1  2  ���  2  = hn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  ,  which means, by (55) that  hn  Nn  ∑  j=0  ���δxyn  j+ 1  2  ���  2  = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Similarly, repeating the procedure from (53) to (55), for Y n  hn,  we obtain  ∥Y n  hn∥2  Yhn = hn  Nn  ∑  j=0  ���yn  j+ 1  2  ���  2  = O(n−1/2),  which leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Thus, the sequence {βn} cannot  contain a subsequence converging to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' We can therefore  assume that |βn| ≥ C′ > 0 for some constant C′ independent  of n ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  The second claim is the discrete counterpart of (18) but with  two extra terms  h3  n  4βn  Nn  ∑  j=0  ���δxzn  j+ 1  2  ���  2  + h3  n  4  Nn  ∑  j=0  ���δxyn  j+ 1  2  ���  2  which play important roles in our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Claim 2: the following (56) holds true:  ∥Y n  hn∥2  Yhn + 1  βn  hn∥Σhn �Zn  hn∥2  CNn+2  + h2  n  4βn  hn∥∆hn �Zn  hn∥2  CNn+2 + h2  n  4 hn∥∆hn�Y n  hn∥2  CNn+2  = hn  Nn  ∑  j=0  ���yn  j+ 1  2  ���  2  + hn  βn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  + h3  n  4βn  Nn  ∑  j=0  ���δxzn  j+ 1  2  ���  2  + h3  n  4  Nn  ∑  j=0  ���δxyn  j+ 1  2  ���  2  = O(n−1),  (56)  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' X, AUGUST 20XX  8  where ∥ ·∥CNn+2 denotes the standard norm of CNn+2 and  �Zn  hn = (z0,z1,··· ,zNn+1)⊤ =  �  (Zn  hn)⊤,zNn+1  �⊤  ,  �Y n  hn = (0,y1,··· ,yNn+1)⊤ =  �  0,(Y n  hn)⊤�⊤  ,  Σh = 1  2  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  1  1  1  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f8  (N+2)×(N+1)  ,  ∆h = 1  h  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ed  −1  1  −1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  −1  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f8  (N+2)×(N+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (57)  Actually, it follows from (49), (52), and ∥Y n  hn∥Yhn = 1 that  β −2  n hn ∑Nn  j=0  ����δxzn  j+ 1  2  ����  2  is uniformly bounded with respect to  n ∈ N+ because  1  βn  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  δxzn  0+ 1  2  δxzn  1+ 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  δxzn  Nn+ 1  2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  = −  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  yn  0+ 1  2  yn  1+ 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  yn  Nn+ 1  2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  − i  βn  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  un  0+ 1  2  un  1+ 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  un  Nn+ 1  2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  By (55) and Claim 1,  β −2  n hn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  is also uniformly bounded with respect to n ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Let xn  j = jhn  for j = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Nn + 1 and consider the following estimates:  �����hn  Nn  ∑  j=0  xn  j+ 1  2 (βnyn  j+ 1  2 + δxzn  j+ 1  2 )  zn  j+ 1  2  βn  �����  2  =  �����hn  Nn  ∑  j=0  xn  j+ 1  2 un  j+ 1  2  zn  j+ 1  2  βn  �����  2  ≤  �  Nn  ∑  j=0  ���  �  hnun  j+ 1  2  ���  �����  �  hn  zn  j+ 1  2  βn  �����  �2  ≤  �  hn  Nn  ∑  j=0  ���un  j+ 1  2  ���  2  ��  β −2  n hn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  �  = ∥Un  hn∥2  Yhn  �  β −2  n hn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  �  = O(n−2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (58)  where (49) and (52) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' On the other hand, by the  second identity of (52), we have  hn  Nn  ∑  j=0  xn  j+ 1  2 (βnyn  j+ 1  2 + δxzn  j+ 1  2 )  zn  j+ 1  2  βn  = hn  Nn  ∑  j=0  xn  j+ 1  2 yn  j+ 1  2 δxyn  j+ 1  2  +β −1  n hn  Nn  ∑  j=0  xn  j+ 1  2 δxzn  j+ 1  2 zn  j+ 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (59)  Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='4 to the two terms of the right hand side  of (59) and noticing xn  Nn+1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' xn  0 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' xn  j+1 − xn  j = hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' it is  easy to obtain  2Re  �  hn  Nn  ∑  j=0  xn  j+ 1  2 yn  j+ 1  2 δxyn  j+ 1  2  �  = hn  Nn  ∑  j=0  xn  j+ 1  2 yn  j+ 1  2 δxyn  j+ 1  2 + hn  Nn  ∑  j=0  xn  j+ 1  2 yn  j+ 1  2 δxyn  j+ 1  2  = 1  4  Nn  ∑  j=0  (xn  j+1 + xn  j)(yn  j+1 + yn  j)(yn  j+1 − yn  j)  +1  4  Nn  ∑  j=0  (xn  j+1 + xn  j)(yn  j+1 − yn  j)(yn  j+1 + yn  j)  = |yNn+1|2 − hn  Nn  ∑  j=0  ���yn  j+ 1  2  ���  2  − h3  n  4  Nn  ∑  j=0  ���δxyn  j+ 1  2  ���  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (60)  and  2Re  �  hn  Nn  ∑  j=0  xn  j+ 1  2 zn  j+ 1  2 δxzn  j+ 1  2  �  = hn  Nn  ∑  j=0  xn  j+ 1  2 zn  j+ 1  2 δxzn  j+ 1  2 + hn  Nn  ∑  j=0  xn  j+ 1  2 zn  j+ 1  2 δxzn  j+ 1  2  = 1  4  Nn  ∑  j=0  (xn  j+1 + xn  j)(zn  j+1 + zn  j)(zn  j+1 − zn  j)  + 1  4  Nn  ∑  j=0  (xn  j+1 + xn  j)(zn  j+1 + zn  j)(zn  j+1 − zn  j)  = |zNn+1|2 − hn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  − h3  n  4  Nn  ∑  j=0  ���δxzn  j+ 1  2  ���  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (61)  By (59)-(61), it follows that  hn  Nn  ∑  j=0  ���yn  j+ 1  2  ���  2  + h3  n  4  Nn  ∑  j=0  ���δxyn  j+ 1  2  ���  2  + hn  βn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  + h3  n  4βn  Nn  ∑  j=0  ���δxzn  j+ 1  2  ���  2  = −2Re  �  hn  Nn  ∑  j=0  xn  j+ 1  2 (βnyn  j+ 1  2 + δxzn  j+ 1  2 )  zn  j+ 1  2  βn  �  +|yNn+1|2 + β −1  n |zNn+1|2,  (62)  which proves (56) by (50), (58) and zNn+1 = −ikyNn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' X, AUGUST 20XX  9  The third claim is perfectly the discrete counterpart of (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Claim 3: The following (63) holds true:  ∥Y n  hn∥2  Yhn + 1  β 2n  hn∥∆hn �Zn  hn∥2  CNn+2 − 2  βn  hn∥Σhn �Zn  hn∥2  CNn+2  = hn  Nn  ∑  j=0  ���yn  j+ 1  2  ���  2  + hn  β 2n  Nn  ∑  j=0  ���δxzn  j+ 1  2  ���  2  − 2hn  βn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  = O(n−2),  (63)  in which Σhn and ∆hn are defined in (57) and ∥ · ∥CNn+2  denotes the standard norm of CNn+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Actually, from (52), we have  ∥Un  hn∥2  Yhn  β 2n  = hn  β 2n  Nn  ∑  j=0  ���un  j+ 1  2  ���  2  = hn  Nn  ∑  j=0  �����yn  j+ 1  2 +  δxzn  j+ 1  2  βn  �����  2  = hn  Nn  ∑  j=0  ���yn  j+ 1  2  ���  2  + hn  β 2n  Nn  ∑  j=0  ���δxzn  j+ 1  2  ���  2  + hn  βn  Nn  ∑  j=0  (yn  j+ 1  2 δxzn  j+ 1  2 + yn  j+ 1  2 δxzn  j+ 1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  (64)  On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' it follows from the second identity of (52)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='that zn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='j+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2 = δxyn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='j+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2 and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='hn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='βn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Nn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='∑  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='j=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='(yn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='j+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2 δxzn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='j+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2 + yn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='j+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2 δxzn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='j+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='2 )+ 2hn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='βn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Nn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='∑  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
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+page_content='[yn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Nn+1zn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Nn+1 + yn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Nn+1zn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Nn+1 − yn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='0zn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='0 − yn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='0zn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='0] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  where zn  Nn+1 = −ikyn  Nn+1 and yn  0 = 0 were used in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Hence  hn  βn  Nn  ∑  j=0  (yn  j+ 1  2 δxzn  j+ 1  2 + yn  j+ 1  2 δxzn  j+ 1  2 ) = −2hn  βn  Nn  ∑  j=0  ���zn  j+ 1  2  ���  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' (65)  Plugging (65) into (64) and using ∥Un  hn∥Yhn ≤ n−1 and Claim  1, we arrive at (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Finally, if βn > 0, we have ∥Y n  hn∥2  Yhn = O(n−1) from (56),  which is a contradiction to ∥Y n  hn∥Yhn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' When βn < 0,  ∥Y n  hn∥Yhn = O(n−1) by virtue of (63), which is also a con-  tradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' We therefore complete the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' CONCLUDING REMARKS  In this paper, the uniform approximation of exponential  stability of a one-dimensional Schr¨odinger equation is investi-  gated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' We introduce an order reduction space semi-discretized  finite difference scheme for approximating uniformly the expo-  nentially stable closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Although the scheme has  been applied to certain PDEs in previous works, they all share  a common feature that it is possible to find a suitable Lyapunov  functional for the closed-loop systems, both for the continuous  system and its discrete counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' However, for the system  considered in this paper, it is a longstanding problem that in  the natural state space L2(0,1), even for the continuous system,  the time domain energy multiplier has not been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' This  makes the convergence of the semi-discrete scheme of this  PDE be open for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' This paper is the first work  that applies the frequency domain multiplier approach to the  uniformly exponential convergence of semi-discretized PDE  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The convergence of the discrete scheme to continuous  system is not included because it is a standard procedure  and can be followed analogously from [14] and many other  papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Considering it is difficult to find a time domain energy  multiplier for many other PDEs, the approach presented in this  paper has significant potentials in applying to other PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Jiankang Liu and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Hanjin Ren for their careful reading and many suggestions on  the presentation of the initial manuscript of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' The  Figures 1 and 2 were depicted by Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Jiangkang Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
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+page_content=' Zhou, State reconstruction of the wave equation with  general viscosity and non-collocated observation and control, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
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+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=', 502(2020), Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' 125257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  [24] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Zuazua, Propagation, observation, and control of waves approximated  by finite difference methods, SIAM Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=', 47(2005), 197-243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  PLACE  PHOTO  HERE  Bao-Zhu Guo received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' degree from the  Chinese University of Hong Kong in Applied Math-  ematics in 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Since 2000, he has been with  the Academy of Mathematics and Systems Science,  the Chinese Academy of Sciences, where he is a  Research Professor in mathematical system theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  From 2004-2019, he was a chair professor in School  of Computer Science and Applied Mathematics,  University of the Witwatersrand, South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' He is  the author or co-author of the books: “Stability and  Stabilization of Infinite Dimensional Systems with  Applications” (Springer-Verlag, 1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' “Active Disturbance Rejection Control  for Nonlinear Systems: An Introduction” (John Wiley & Sons, 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' and  “Control of Wave and Beam PDEs-The Riesz Basis Approach” (Springer-  Verlag, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' His research interests include theory of infinite-dimensional  systems and active disturbance rejection control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' Guo received the One Hundred Talent Program from the Chinese  Academy of Sciences (1999), and the National Science Fund for Distinguished  Young Scholars (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='  PLACE  PHOTO  HERE  Fu Zheng received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' degrees  in mathematics from the Yanbian University, Jilin,  China in 2002 and 2005, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' from Beijing  Institute of Information and Control, Beijing, China  in 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' He held a postdoctoral research position  in Academy of Mathematics and Systems Science,  Academia Sinica, Beijing, China, from 2013 to  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' From 2012-2021, he was an associate profes-  sor at Bohai University, Jinzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' At the end  of 2021, he joined the School of Science, Hainan  University, Hainan China, where he is currently a  professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
+page_content=' His research interests include control theory of infinite-dimensional  systems and numerical solutions to control problems of distributed parameter  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQf0y3g/content/2301.11649v1.pdf'}
diff --git a/RdE4T4oBgHgl3EQflQ1W/content/tmp_files/2301.05158v1.pdf.txt b/RdE4T4oBgHgl3EQflQ1W/content/tmp_files/2301.05158v1.pdf.txt
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+Preprint.
+SEMPPL: PREDICTING PSEUDO-LABELS FOR BETTER
+CONTRASTIVE REPRESENTATIONS
+Matko Boˇsnjak, Pierre H. Richemond, Nenad Tomasev, Florian Strub, Jacob C. Walker,
+Felix Hill, Lars Holger Buesing, Razvan Pascanu, Charles Blundell, Jovana Mitrovic
+DeepMind
+{matko, richemond, mitrovic}@deepmind.com
+ABSTRACT
+Learning from large amounts of unsupervised data and a small amount of super-
+vision is an important open problem in computer vision. We propose a new semi-
+supervised learning method, Semantic Positives via Pseudo-Labels (SEMPPL),
+that combines labelled and unlabelled data to learn informative representations.
+Our method extends self-supervised contrastive learning—where representations
+are shaped by distinguishing whether two samples represent the same underlying
+datum (positives) or not (negatives)—with a novel approach to selecting posi-
+tives. To enrich the set of positives, we leverage the few existing ground-truth
+labels to predict the missing ones through a k-nearest neighbours classifier by
+using the learned embeddings of the labelled data. We thus extend the set of
+positives with datapoints having the same pseudo-label and call these semantic
+positives. We jointly learn the representation and predict bootstrapped pseudo-
+labels. This creates a reinforcing cycle. Strong initial representations enable better
+pseudo-label predictions which then improve the selection of semantic positives
+and lead to even better representations. SEMPPL outperforms competing semi-
+supervised methods setting new state-of-the-art performance of 68.5% and 76%
+top-1 accuracy when using a ResNet-50 and training on 1% and 10% of labels on
+ImageNet, respectively. Furthermore, when using selective kernels, SEMPPL sig-
+nificantly outperforms previous state-of-the-art achieving 72.3% and 78.3% top-1
+accuracy on ImageNet with 1% and 10% labels, respectively, which improves
+absolute +7.8% and +6.2% over previous work. SEMPPL also exhibits state-
+of-the-art performance over larger ResNet models as well as strong robustness,
+out-of-distribution and transfer performance.
+1
+INTRODUCTION
+In recent years, self-supervised learning has made significant strides in learning useful visual fea-
+tures from large unlabelled datasets [Oord et al., 2018; Chen et al., 2020a; Mitrovic et al., 2021;
+Grill et al., 2020; Caron et al., 2021]. Moreover, self-supervised representations have matched the
+performance of historical supervised baselines on the ImageNet-1k benchmark [Russakovsky et al.,
+2015] in like-for-like comparisons as well as outperformed supervised learning in many transfer
+settings [Tomasev et al., 2022]. While such results show exciting progress in the field, in many
+real-wold applications often there exists a small amount of ground-truth labelled datapoints making
+the problem of representation learning semi-supervised.
+In this work we propose a novel approach to semi-supervised learning called Semantic Positives
+via Pseudo-Labels (SEMPPL) which incorporates supervised information during the representation
+learning stage within a self-supervised loss. Unlike previous work which uses the available super-
+vision as targets within a cross-entropy objective, we propose to use the supervised information to
+help inform which points should have similar representations. We propose to learn representations
+using a contrastive approach, i.e. we learn the representation of a datapoint (anchor) by maximizing
+the similarity of the embedding of that datapoint with a set of similar points (positives), while simul-
+taneously minimizing the similarity of that embedding with a set of dissimilar points (negatives).
+1
+arXiv:2301.05158v1  [cs.CV]  12 Jan 2023
+
+Preprint.
+As such, the appropriate construction of these sets of positives and negatives is crucial to the success
+of contrastive learning methods. While strategies for sampling negatives have been extensively
+studied in the literature [Schroff et al., 2015; Harwood et al., 2017; Ge et al., 2018; Wang et al.,
+2019a; He et al., 2020; Chen et al., 2020c], the sampling of positives has received far less attention.
+We propose a novel approach to selecting positives which leverages supervised information. Specif-
+ically, we propose using the small amount of available ground-truth labels in order to non-
+parametrically predict the missing labels (pseudo-labels) for the unlabelled data. Note that many
+previous semi-supervised approaches use pseudo-labels as targets within a cross-entropy-based ob-
+jective [Van Engelen & Hoos, 2020; Yang et al., 2021]. In SEMPPL we use pseudo-labels in a very
+different way, i.e. we use them to select positives based on whether two datapoints (we call these
+semantic positives) share the same (pseudo-)label. By maximizing the similarity of a datapoint with
+its semantic positives we expect to learn representations that are more semantically aligned and as a
+consequence encode more abstract, higher-level features which should generalise better. To predict
+informative pseudo-labels, we compare the representations of the unlabelled data with those of the
+labelled subset and use a k-nearest neighbours (k-NN) classifier to impute the missing labels.
+We simultaneously learn the representation, predict pseudo-labels and select semantic positives.
+This creates a virtuous cycle: better representations enable better pseudo-label prediction which in
+turn enables better selection of semantic positives and thus helps us learn better representations. Im-
+portantly, as the prediction of pseudo-labels and selection of semantic positives does not depend on
+the exact form of the contrastive objective employed, SEMPPL is compatible with and complements
+all contrastive losses, e.g. [Chen et al., 2020a;b; Caron et al., 2020; He et al., 2020; Mitrovic et al.,
+2021] and may even be extended to non-contrastive losses [Grill et al., 2020; Chen & He, 2021].
+We evaluate the representations learned with SEMPPL across a varied set of tasks and datasets. In
+particular, SEMPPL sets new state-of-the-art in semi-supervised learning on ImageNet with 1% and
+10% of labels on the standard ResNet-50 (1×) architecture with respectively 68.5% and 76.0% top-
+1 performance and across larger architectures. When combined with Selective Kernels [Li et al.,
+2019b], we achieve 72.3% and 78.3% top-1 performance with 1% and 10% labels, respectively,
+significantly outperforming previous state-of-the-art by absolute +7.8% and +6.2% in top-1 perfor-
+mance. We also outperform previous state-of-the-art on robustness and out-of-distribution (OOD)
+generalisation benchmarks while retaining competitive performance in transfer learning.
+Our main contributions are:
+• We extend contrastive learning to the semi-supervised setting by introducing the idea of es-
+timating pseudo-labels for selecting semantic positives as a key component especially in the
+low-label regime,
+• We propose a novel semi-supervised method SEMPPL that jointly estimates pseudo-labels,
+selects semantic positives and learns representations which creates a virtuous cycle and enables
+us to learn more informative representations,
+• We extensively evaluate SEMPPL and achieve a new state-of-the-art in semi-supervised learn-
+ing, robustness and out-of-distribution generalisation, and competitive performance in transfer.
+2
+SEMANTIC POSITIVES VIA PSEUDO-LABELS
+The selection of appropriate positive and negative examples are the cornerstone of contrastive learn-
+ing. Though the research community has mainly focused on the selection of negatives, positives
+are equally important as they play a vital role in learning semantic similarity. We thus leverage
+labelled information as it encodes semantic information to improve the selection of informative pos-
+itives. Specifically, we expand a self-supervised model to use this labelled data to non-parametrically
+predict pseudo-labels for the remaining unlabelled data. Using both ground-truth labels and the pre-
+dicted pseudo-labels, we expand the set of positives with semantic positives.
+Notations Let D = Dl ∪Du be a dataset consisting of labelled training data Dl = {(xi, yi)}N
+i=1 and
+unlabelled training data Du = {(xj)}M
+j=N+1 with M ≫ N. Let B be a batch of data of size B with
+B = {(xi, yi)}b
+i=1 ∪ {xj}B
+j=b+1 where (xi, yi) ∈ Dl and xj ∈ Du, where the indices i, j and m to
+denote labelled, unlabelled, and all datapoints, respectively. Following established self-supervised
+learning practices [Chen et al., 2020a;b; Caron et al., 2020; Mitrovic et al., 2021; Dwibedi et al.,
+2
+
+Preprint.
+Embedding
+Projection
+Semantic 
+positive
+Loss
+Prediction
+Projection
+Projection
+Projection
+Projection
+K-NN
+Query
+Pseudo 
+labels
+Embedding
+Projection
+Target Network
+Negatives
+Loss
+Target Network
+Online Network
+Online Network
+Embedding
+Projection
+Prediction
+Online Network
+Prediction
+Queue
+Figure 1: Sketch of SEMPPL. (Left) Standard contrastive pipelines. (Middle) Unlabelled data are
+tagged with pseudo-labels by using a k-NN over projected labelled data. (Right) Semantic positives
+are queried from the queue and processed to compute an additional contrastive loss.
+2021; Tomasev et al., 2022], we create different views of the data by applying pairs of randomly
+sampled augmentations a1, a2 ∼ A from the augmentation distribution A proposed in Chen et al.
+[2020a]. For every datapoint xm ∈ D we denote the corresponding augmentations as xa1
+m , xa2
+m .
+Augmentation positives
+We embed one data view xa1
+m via an online encoder network f and embed
+the other data view xa2
+m with a target encoder network ft, i.e. we get latent representations za1
+m =
+f(xa1
+m ) and za2
+m,t = ft(xa2
+m ). Note that the weights of ft are an exponential moving average of the
+weights of f. Next, we pass these latent representations through projection and prediction multi-
+layer perceptrons. Specifically, we use an online projector g and target projector gt, as well as an
+online predictor h, to further transform za1
+m and za2
+m,t; again, the weights of gt are an exponential
+moving average of the weights of g. We then get ˆza1
+m = h(g(za1
+m )) and ˜za2
+m,t = gt(za2
+m,t) and l2-
+normalise these; we use ˆza1
+m , ˜za2
+m,t onward as the normalised latent embeddings.
+In order to learn the representation of ˆza1
+m , we contrast it against the augmentation-based positive
+˜za2
+m,t as well as against negatives. For this, we use the contrastive loss:
+LAUGM = −
+B
+�
+m=1
+log
+ϕ(ˆza1
+m , ˜za2
+m,t)
+ϕ(ˆza1
+m , ˜za2
+m,t) + �
+xn∈N (xm) ϕ(ˆza1
+m , ˜za2
+n,t)
+(1)
+where N(xk) is the set of negatives, randomly uniformly sampled from the current batch, ˜za2
+n,t =
+gt(ft(xn)) the target network projection of the negative sample; ϕ(x1, x2) = τ · exp(⟨x1, x2⟩/τ)
+is the scoring function, τ > 0 is a scalar temperature, and ⟨·, ·⟩ denotes the Euclidean dot product.
+Since the representations we contrast are l2-normalised, the dot product effectively turns into cosine
+similarity.
+Pseudo-label prediction and semantic positives
+Since we have access to a small labelled dataset,
+we can use the label information to select more informative positives beyond just augmentations of
+the original image. Specifically, we can associate images with the same label as positives and we call
+these semantic positives. We want to select semantic positives for all the data, not just the labelled
+subset. For this purpose, we propose to compute pseudo-labels for the unlabelled data and use this
+to select semantic positives. To compute pseudo-labels we compare the current latent embeddings
+of the unlabelled data to those of the labelled data. Specifically, we propose to use a first-in-first-out
+3
+
+020202Preprint.
+queue Q with capacity C for storing labelled embeddings which we use for computing the pseudo-
+labels. At the start of training, we simply initialise the queue with random vectors, and use the queue
+from the first step. For each batch B, we add the target projection of only the labelled data to the
+queue, i.e. Q ← (˜za2
+i,t, yi). To predict a pseudo-label for an unlabelled datapoint xj, we first compute
+the online predictor output ˆza1
+j , before retrieving its k-nearest neighbours {(˜za2
+s,t, ys)}k
+s=1 in cosine
+similarity from the queue Q.1 Finally, we compute the pseudo-label ¯yj of xj as:
+¯yj = mode
+ys
+{(˜za2
+s,t, ys)}k
+s=1
+(2)
+where mode is the mode of the set, tasked with obtaining the most frequent class in the k-nearest
+neighbours. We use the ground-truth labels (for the labelled data) or the computed pseudo-labels
+(for the unlabelled data) to select semantic positives for every datapoint in B. For each xm ∈ B,
+we uniformly sample over all the embeddings in Q that share the same (pseudo-) label as xm to
+get a semantic positive ˜za2,+
+m,t
+∼ U({(˜za2
+l,t, yl) ∈ Q | yl = pl(xm)}), where pl(xm) = ym if xm is
+labelled and pl(xm) = ¯ym if xm is unlabelled. Next, we include these semantic positives within our
+representation learning process through the contrastive objective
+LSEMPOS = −
+B
+�
+m=1
+log
+ϕ(ˆza1
+m , ˜za2,+
+m,t )
+ϕ(ˆza1
+m , ˜za2,+
+m,t ) + �
+xn∈N (xm) ϕ(ˆza1
+m , ˜za2
+n,t)
+(3)
+Taking these two losses (1) and (3) together, we propose to learn representations in our method
+SemPPL by minimising the following total loss
+LSEMPPL = LAUGM + αLSEMPOS
+(4)
+where α controls the ratio between these sub-losses.
+2.1
+IMPLEMENTATION DETAILS
+Architecture We use Residual Networks [He et al., 2016] (v1; pre-activation as customary in the
+literature) for f and ft and use either 50 or 200 layers deep networks and with a width multiplier
+ranging from 1× to 4×. As in [Grill et al., 2020; Tomasev et al., 2022], we use multi-layer per-
+ceptrons with 2 layers of size 4096 and 256, with batch normalisation [Ioffe & Szegedy, 2015] and
+rectified linear activation.
+Self-supervised learning method We use RELICv2 [Tomasev et al., 2022] as our default self-
+supervised training objective due to its competitive performance. Therefore, we add an invariance
+penalty on top of Equation 4 to further enforce the similarity constraints and regularize the learning
+process as detailed in Appendix B. We also explore other self-supervised learning objectives in
+Section 4.
+Algorithm parameters We use a queue of capacity C = 20B, with batch size B = 4096, and
+temperature τ = 0.2 while randomly sampling negatives from the current batch; we take |N(x)| =
+10 negatives in total. For augmentations, we use the standard SIMCLR augmentations [Chen et al.,
+2020a] and the RELICV2 multi-crop and saliency-based masking [Tomasev et al., 2022]; we use 4
+large views and 2 small views for augmentation positives and 3 semantic positives. The semantic
+positives are computed with a k-NN with k = 1 (see the analysis section in Appendix D); we build
+a single k-NN instance per augmentation a queried with all the augmentations where |a| = 4. This
+produces |a|2 = 16 k-NN induced pseudo-labels in total for each unlabelled image among which
+we then perform majority voting to compute the final pseudo-label.
+Optimisation Our networks are optimized with LARS [You et al., 2017]. Our base learning rate
+is 0.3 and we train our models for 300 epochs with a learning rate warm-up period of 10 epochs
+and cosine decay schedule thereafter. We use a weight decay of 10−6 and batch size B = 4096.
+We exclude the biases and batch normalisation parameters both from LARS adaptation and weight
+decay. The exponential moving average parameter for target networks is 0.996. Our pseudo-code
+is described in the appendix along with precise architectural and implementation details. Pretrained
+model checkpoints and code will be made available on GitHub.
+1We use the cosine similarity as the embeddings are normalised.
+4
+
+Preprint.
+3
+EXPERIMENTAL RESULTS
+To evaluate SEMPPL, we pre-train representations using 1% and 10% labelled data from the Ima-
+geNet dataset [Russakovsky et al., 2015] based on the splits from Chen et al. [2020a]
+We then test SEMPPL in semi-supervised classification, robustness and out-of-distribution general-
+isation tasks. Lastly, we probe the transfer capabilities of the representations to other image classi-
+fication datasets. For a complete set of results and experimental details, please see the Appendix A.
+3.1
+SEMI-SUPERVISED LEARNING
+In Table 1, we report top-1 accuracy on the ImageNet test set when either 1% or 10% of the data is
+labelled for the ResNet-50 architecture as well as deeper and wider ResNets.
+SEMPPL achieves top-1 accuracy of 68.5% with 1% of labels, significantly outperforming the previ-
+ous state-of-the-art SimMatch [Zheng et al., 2022] by an absolute +1.3% in ImageNet test accuracy.
+With 10% of label data, our top-1 accuracy on ResNet-50 reaches 76.0%, outperforming the previ-
+ous state-of-the-art PAWS [Assran et al., 2021] in semi-supervised learning. SEMPPL outperforms
+competing representation learning methods across the board, achieving state-of-the-art performance
+on all ResNet-50 2×, ResNet-50 4× and , in both the 1% and 10% labelled settings. SEMPPL does
+not use, and therefore excludes from comparison, distillation from larger networks as in [Chen et al.,
+2020b; Pham et al., 2021].
+Similar to [Chen et al., 2020b], we also tested SEMPPL on ResNets with Selective Kernels (SK) [Li
+et al., 2019b].
+This increases the encoder parameter count to 27.5M.
+We thus achieve a new absolute state-of-the-art of 72.3% and 78.3% top-1 accuracies, respectively,
+when using 1% and 10% of labelled data. Finally, SEMPPL reaches a new state-of-the-art using 76.0
+and 80.5 on 1% and 10% of labels without self-distillation with a ResNet-200 2× + SK architecture.
+For implementation details of the semi-supervised results and additional results, see the Ap-
+pendix A.1.
+Table 1: Top-1 accuracy (in %) for ResNet encoders with different depth and width.
+ResNet-50 1×
+ResNet-50 2×
+ResNet-50 4×
+ResNet-200 2×
+Method
+Top-1
+Top-1
+Top-1
+Top-1
+1%
+10%
+1%
+10%
+1%
+10%
+1%
+10%
+SimCLR [Chen et al., 2020a]
+48.3
+65.6
+58.5
+71.7
+63.0
+74.4
+-
+-
+BYOL [Grill et al., 2020]
+53.2
+68.8
+62.2
+73.5
+69.1
+75.7
+71.2
+77.7
+RELICv2 [Tomasev et al., 2022]
+58.1
+72.4
+64.7
+73.7
+69.5
+74.6
+72.1
+76.4
+SimCLRv2 [Chen et al., 2020b]
+57.9
+68.4
+66.3
+73.9
+-
+-
+-
+-
+CoMatch [Li et al., 2021a]
+66.0
+73.7
+-
+-
+-
+-
+-
+-
+PAWS [Assran et al., 2021]
+66.5
+75.5
+69.6
+77.8
+69.9
+79.0
+-
+-
+SimMatch [Zheng et al., 2022]
+67.2
+74.4
+-
+-
+-
+-
+-
+-
+SemPPL (ours)
+68.5
+76.0
+71.9
+78.6
+72.5
+79.3
+74.8
+80.4
+SimCLRv2 + SK [Chen et al., 2020b]
+64.5
+72.1
+70.6
+77.0
+-
+-
+-
+-
+SemPPL + SK (ours)
+72.3
+78.3
+74.5
+79.8
+-
+-
+76.0
+80.5
+3.2
+ROBUSTNESS AND OOD GENERALISATION
+We evaluate the robustness and generalisation abilities of SEMPPL on ImageNetV2 [Recht et al.,
+2019], ImageNet-C [Hendrycks & Dietterich, 2019], ImageNet-R [Hendrycks et al., 2021] and Ob-
+jectNet [Barbu et al., 2019] which have all been purposefully constructed to test different robustness
+and generalisation aspects. We evaluate all three variants on ImageNetV2: matched frequency (MF),
+Threshold 0.7 (T-0.7) and Top Images (TI). When evaluating PAWS, we used the publicly available
+checkpoints.
+5
+
+Preprint.
+Table 2 shows good robustness and generalisation ability of the representations learned with
+SEMPPL. SEMPPL sets the new state-of-the-art performance (outperforming even the supervised
+baseline) on 4 out of 5 datasets, while outperforming PAWS across all datasets. SEMPPL also out-
+performs SimMatch on 4 out of 5 datasets. For more details on the evaluation protocols and results
+for ImageNet-C see the Appendix A.2.
+Table 2: Top-1 accuracy (in %) for ImageNetV2, ImageNet-R and ObjectNet.
+Robustness
+OOD generalization
+Method
+MF
+T-0.7
+Ti
+ImageNet-R
+ObjectNet
+Supervised (100% labels) [Lim et al., 2019]
+65.1
+73.9
+78.4
+24.0
+26.6
+Semi-supervised (10% labels)
+PAWS [Assran et al., 2021]
+64.5
+73.7
+78.9
+23.5
+23.8
+SimMatch [Zheng et al., 2022]
+63.8
+73.2
+78.3
+25.0
+24.5
+SemPPL (ours)
+65.4
+74.1
+79.6
+24.4
+25.3
+3.3
+TRANSFER LEARNING
+We evaluate the generality of SEMPPL representations by testing whether the features learned on
+ImageNet are useful across different datasets and tasks. Specifically, we evaluate the transfer per-
+formance of SEMPPL on a set of 11 image classification datasets commonly used in the contrastive
+literature under the linear protocol [Grill et al., 2020; Chen et al., 2020a; Dwibedi et al., 2021; Mitro-
+vic et al., 2021; Tomasev et al., 2022]. For the linear protocol, the pretrained encoder is frozen and a
+randomly initialized linear classifier is trained on top using the training data from the target dataset.
+We report standard metrics for each dataset as well as performance on a held-out test set. For more
+details on the evaluation protocol see the Appendix A.3. Table 3 compares the transfer performance
+of representations pretrained using the supervised baseline [Chen et al., 2020a], PAWS [Assran et al.,
+2021], SimMatch [Zheng et al., 2022] and our method SEMPPL. SEMPPL outperforms the super-
+vised baseline on 8 out of 11 datasets, PAWS on 9 out of 11 datasets, while showing competitive
+performance to SimMatch, outperforming it on 4 out of 7 datasets.
+3.4
+FULL LABELLED DATASET
+Figure 2: Top-1 accuracy for ResNet50 with 100% of the
+labels across augmentations, initializations and networks.
+Method
+Params
+Top-1
+Supervised (ResNet-50)
++ AutoAugment [Cubuk et al., 2019]
+27M
+77.6
++ MaxUp [Gong et al., 2021]
+27M
+78.9
+Representation Learning (ResNet-50)
+SEMPPL (SimCLR base)
+27M
+76.0
+SEMPPL (BYOL base)
+27M
+77.7
+SEMPPL (ReLICv2 base; ours)
+27M
+79.7
+Other Architectures
+Swin-T [Liu et al., 2021b]
+29M
+81.3
+ConvNeXt [Liu et al., 2022]
+29M
+82.1
+SEMPPL + SK (ours)
+29M
+82.0
+We also assess how SEMPPL be-
+haves in a fully supervised setting.
+For this purpose, we select semantic
+positives based on the ground-truth
+labels and fine-tune the learned rep-
+resentations with the full ImageNet
+dataset. We compare against strong
+supervised baselines on ResNets as
+well as against recent performant net-
+work architectures that are extensions
+of the ResNet, e.g. [Liu et al., 2021b;
+2022].
+Our method reaches 79.7% top-1 ac-
+curacy on a ResNet 50 outperforming
+a number of strong supervised base-
+lines. When we add selective kernels
+to a ResNet 50, we achieve 82% top-1
+accuracy outperforming recent trans-
+formers architecture [Liu et al., 2021b], and matching highly tuned ConvNext [Liu et al., 2022].
+Therefore, SEMPPL may also be considered as a promising pretraining method in the supervised
+learning setting.
+6
+
+Preprint.
+Table 3: Top-1 accuracy (in %) on the full suite of transfer tasks.
+Method
+Food101 CIFAR10
+CIFAR100
+Birdsnap
+SUN397
+Cars
+Aircraft DTD
+Pets
+Caltech101
+Flowers
+Supervised-IN [Chen et al., 2020a]
+72.3
+93.6
+78.3
+53.7
+61.9
+66.7
+61.0
+74.9
+91.5
+94.5
+94.7
+Semi-supervised methods with 10% labels:
+PAWS [Assran et al., 2021]
+79.1
+92.3
+76.3
+62.0
+66.1
+75.7
+61.4
+77.0
+92.2
+91.9
+96.5
+SimMatch [Zheng et al., 2022]
+71.7
+93.6
+78.4
+–
+–
+69.7
+–
+75.1
+92.8
+–
+93.2
+SEMPPL (ours)
+80.2
+92.5
+77.6
+64.2
+66.3
+75.5
+63.9
+77.8
+92.5
+93.0
+96.3
+4
+ANALYSIS
+We analyse the impact of different design choices in SEMPPL on downstream performance. In
+this section, we focus the behaviour and impact of pseudo-labels and semantic positives on learning
+representations. For further analyses and experimental details, please see Appendix D.
+Semantic positives across self-supervised learning objectives
+With SEMPPL we extend the set
+of positives to include semantic positives based on predicted pseudo-labels; we can combine these
+ideas with other self-supervised methods. In Table 4.1, we additionally evaluate SEMPPL on the
+non-contrastive self-supervised method BYOL [Grill et al., 2020]. BYOL replaces the contrastive
+loss in Equation 4 with an l2 loss. Importantly, we follow the training pipeline (e.g. augmentation,
+hyperparameters etc.) from [Grill et al., 2020] to fairly highlight the impact of SEMPPL. We ob-
+serve a drastic improvement when adding semantic positives. With 1% labels on ImageNet BYOL
+improves by absolute +3.9% and by absolute +3.6% when using 10% labels. For completeness we
+have also highlighted the contribution of SEMPPL when using RELICv2 as the base self-supervised
+objective which is our default implementation. For 1% labeled data, we see an absolute improve-
+ment of +10.4% in top-1 accuracy, while for 10% labels we see a gain of absolute +3.6% in top-1
+accuracy. In summary, we see that SEMPPL can be easily combined with other self-supervised
+objectives to yield significant improvements and can be used a standard plug-and-play module in
+within semi-supervised learning.
+The contribution of pseudo-labels and semantic positives
+We examine the impact of omitting
+pseudo-label prediction and semantic positives from learning representations. Specifically, we ab-
+late the use of pseudo-labels when selecting semantic positives for unlabelled datapoints, i.e. we
+only use labelled images when retrieving semantic positives. In Table 4.2 (middle row), removing
+pseudo-label prediction significantly decreases performance both in the 1% and 10% label settings.
+In addition, the low-label regime (1% labels) suffers a stronger performance decrease −6.6% than
+the 10% labels regime, −4.9%. This underscores the importance of pseudo-label estimation and
+subsequent selection of semantic positives for unlabelled data especially in the low-data regime.
+Going a step further, we remove semantic positives even for the labelled data, falling back to vanilla
+RELICv2. In Table 4.2 (bottom row), we see again a significant drop in performance for both the
+1% and 10% label settings with a sharper drop for the low-label regime. Together these highlights
+the importance of both including semantic positives for labelled data as well as using pseudo-label
+prediction for selecting semantic positives for unlabelled data in order to learn informative represen-
+tations in a semi-supervised setting in a label-efficient way.
+Table 4: Top-1 test accuracy (in %) with a ResNet50 pretrained on ImageNet with 1% and 10%
+labels: (1) when trained on different self-supervised objective, (2) when removing pseudo-labelling,
+and semantic positives in SEMPPL.
+Top-1
+1%
+10%
+BYOL [Grill et al., 2020]
+53.2
+68.8
+SEMPPL with BYOL
+57.1
+72.4
+ReLICv2 [Tomasev et al., 2022]
+58.1
+72.4
+SEMPPL with RELICv2 (ours)
+68.5
+76.0
+1% labels
+10% labels
+SEMPPL
+68.5
+76.0
+- Pseudo-labels
+61.9
+71.1
+- Semantic Positives
+58.1
+72.4
+7
+
+Preprint.
+Table 5: Top-1 test accuracy (in %) with a ResNet50 pretrained on ImageNet 10% labels for 100
+epoches when using ground truth labels instead of pseudo-labels while retrieving semantic positives
+(oracle); PL accuracy helds for the accuracy of the pseudo-label.
+10% labels
+Top-1
+PL accuracy
+SEMPPL
+69.9
+69.7
+SEMPPL (+oracle)
+71.6
+76.9
+Precision and Recall of pseudo-labels.
+In Figure 3, we analyse the behaviour of pseudo-labels
+by looking at the precision and recall as training progresses. We train a ResNet-50 for 100 epochs
+using 10% labels with SEMPPL on ImageNet. As we have 4 large views there will be in total 16
+votes cast and then the pseudo-label will be estimated using majority voting. We want to measure
+how often these 16 votes agree or disagree; we denote as voting threshold the number k where at
+least k votes have been cast for one class. We see that as training progresses the precision across all
+thresholds increases as expected. This means that the pseudo-label prediction is bootstrapping itself
+to become more accurate, which enables us to select better semantic positives and thus learn more
+informative representations as training progresses, i.e. we have a virtuous cycle of representation
+learning and pseudo-label prediction. Furthermore, precision is an increasing function of the voting
+threshold throughout training and is highest for the biggest voting threshold. This indicates how
+confident we can be in the accuracy of pseudo-label prediction, and thus how confident we can be
+that an appropriate semantic positive has been selected. Yet, we see that the recall for individual
+thresholds is also increasing as training progresses but that the recall decreases as we increase the
+voting threshold. This is expected as there is always a trade-off between precision and recall.
+5000
+10000
+15000
+20000
+25000
+30000
+Steps
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Precision
+5000
+10000
+15000
+20000
+25000
+30000
+Steps
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Recall
+Voting threshold
+0
+2
+5
+7
+10
+12
+15
+Figure 3: Precision and recall for pseudo-labels computed based on k-nearest neighbours when
+trained on ImageNet with 10% labels over 100 epoches.
+Noise in pseudo-label prediction
+In Figure 3, we observe that the proportion of correctly pre-
+dicted pseudo-labels at the end of training is reasonably high (60% accuracy of voting threshold
+0). Yet, it also means that 40% of pseudo labels are still incorrectly predicted. As the incorrect
+prediction results in suboptimal semantic positives selection, i.e., SEMPPL does not select semantic
+positives from the same class as the datapoint, this behavior may ultimately worsen the quality of ex-
+tracted features. To quantify this phenomenon, we train the representation with SEMPPL where an
+oracle replaces the pseudo-label prediction with ground-truth labels and those are used for selecting
+semantic positives. In Table 5, we train the representations for 100 epochs on ImageNet with 10%
+labels. There, the oracle increases the top-1 performance of 1.7% in the test set with 10%. Besides,
+the pseudo-label accuracy also gets 6.2% higher. It thus confirms that incorrect pseudo-label predic-
+tions, and incorrect semantic positives retrieval, hurts learning informative representations and the
+downstream performance. Yet, the oracle performance remains close to the actual performance of
+SEMPPL, illustrating the method’s robustness.
+Further ablations on other design choices, such as the number of semantic positives, the use of view
+voting, the choice of k in k-NN, queue length and training duration can be found in Appendix D.2.
+8
+
+Preprint.
+5
+RELATED WORK
+Semi-supervised learning
+In the semi-supervised regime [Cheplygina et al., 2019; Van Engelen
+& Hoos, 2020; Yang et al., 2021; Alizadehsani et al., 2021], one can either pre-train a model on
+unlabelled data and subsequently fine-tune it on labelled data, or train both jointly. Joint training
+on labelled and unlabelled data often involves combining the two losses [Grandvalet & Bengio,
+2004; Miyato et al., 2018; Zhai et al., 2019; Verma et al., 2019; Berman et al., 2019; Xie et al.,
+2020a]. Pseudo-label self-training approaches [Zoph et al., 2020] present an important alternative,
+first inferring approximate pseudo-labels for the unlabelled examples, and then incorporating them
+in supervised losses. Pseudo-labels can either be generated prior to a subsequent supervised learning
+phase [Yarowsky, 1995; Riloff, 1996; Lee et al., 2013] or jointly in an online fashion [Berthelot
+et al., 2019; 2020; Sohn et al., 2020]. These methods may benefit from pseudo-label confidence
+measures [Sohn et al., 2020; Rizve et al., 2021; Zhang et al., 2021] as well as thresholding [Xu et al.,
+2021], temporal ensembling [Laine & Aila, 2017], or stronger regularization to mitigate bias in early
+model training [Sajjadi et al., 2016; Arazo et al., 2020]. The use of pseudo-labels with rebalancing
+has shown improvements, both in class-imbalanced problems [Wei et al., 2021] and in a general
+context [Wang et al., 2022]. Teacher-student network configurations for generating and utilising
+pseudo-labels have also shown promise [Tarvainen & Valpola, 2017; Luo et al., 2018; Ke et al., 2019;
+Xie et al., 2020b; Cai et al., 2021; Pham et al., 2021]. Co-training uses different feature extractors for
+different data views and alternates between pseudo-labelling and training phases [Blum & Mitchell,
+1998; Qiao et al., 2018]. Good performance has been reached by using consistency losses between
+pseudo-labels of different inputs [Verma et al., 2019; Hu et al., 2021].
+Predicting view assignments with support samples [Assran et al., 2021] (PAWS) has resulted in
+substantial performance improvements, with the idea that the assigned pseudo-labels ought to be
+similar across multiple views of the same image.
+Recent work has shown that incorporating label information in positive selection in contrastive meth-
+ods is highly promising, compared to the cross-entropy loss in the fully supervised case [Khosla
+et al., 2020]. Our method demonstrates a similar utility of pseudo-labels for semi-supervised prob-
+lems, and differs from competing ones in the following ways. Unlike DebiasPL [Wang et al., 2022]
+that uses an adaptive margin loss, SemPPL does not seek to directly address or re-shape the distri-
+bution of pseudo-labels. Unlike SimCLRv2 [Chen et al., 2020b], we do not rely on self-distillation
+procedures. In contrast with PAWS [Assran et al., 2021], we fully leverage the contrastive approach
+for semi-supervised learning; not using positives only for training means SEMPPL does not require
+specific care like pseudo-labels sharpening to stabilize learning and avoid representational collapse.
+SEMPPL is more closely related to CoMatch [Li et al., 2021a] that also uses bootstrapping to im-
+prove pseudo-labels representational quality, but is conceptually much simpler, avoiding phases of
+distributional alignment and of performing graph-based contrastive learning. In a similar vein, Sim-
+Match [Zheng et al., 2022] also uses a memory buffer to propagate pseudo-labels, but has a more
+complex objective than SEMPPL and equally requires additional phases of pseudo-labels unfolding
+and aggregation to function.
+Self-supervised learning
+Major advances in learning useful representations from unlabelled
+data [Liu et al., 2021a; Goyal et al., 2021] can be seen as a paradigm shift, since these methods
+have recently been competitive with supervised training baselines [Tomasev et al., 2022]. A num-
+ber of self-supervised learning methods involve contrasting multiple views of the data [Oord et al.,
+2018; Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Grill et al., 2020; Dwibedi et al.,
+2021].
+Similar performance were also achieved by bootstrapping-based multi-view learning [Grill et al.,
+2020; Richemond et al., 2020; Chen & He, 2021; Zbontar et al., 2021; Wang et al., 2021], or involv-
+ing explicit clustering steps [Caron et al., 2020; Asano et al., 2020; Li et al., 2021b]. An explicit
+causally-motivated invariance loss, when used in conjunction with the contrastive objective, has
+been shown to lead to more compact representations, and desirable generalisation properties [Mitro-
+vic et al., 2021; Tomasev et al., 2022].
+Contrastive approaches are not always used in self-supervised methods [He et al., 2021; Ermolov
+et al., 2021; Chen et al., 2022]. Transformer-specific methods have been devised [Caron et al., 2021;
+Chen et al., 2021; Zhai et al., 2022].
+9
+
+Preprint.
+6
+CONCLUSION
+In this work, we propose SEMPPL, a novel semi-supervised learning method to incorporate semantic
+positives in self-supervised objectives by taking advantage of pseudo-labels. Through extensive
+empirical evaluation, we demonstrated that our approach achieves state-of-the-art semi-supervised
+performance on ImageNet across several ResNet architectures as well as on the robustness, out-of-
+distribution generalization and transfer tasks. We also show that SEMPPL can be easily combined
+with other existing self-supervised methods and is a promising direction to pre-train networks also in
+a fully supervised learning regime. Our analyses suggest that the role of pseudo-labels in selecting
+positives for semi-supervised contrastive methods might be underappreciated. Despite widespread
+use in semi-supervised applications, pseudo-labels are less understood and have been explored far
+less in the context of self-supervised methods. We hope this study, which shows empirically that
+prior work has under-utilized pseudo-labels, may help bridge that gap.
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+A
+ADDITIONAL RESULTS AND IMPLEMENTATION DETAILS
+A.1
+SEMI-SUPERVISED DETAILS AND RESULTS
+Implementation details.
+In this work, we follow the protocol of Chen et al. [2020b]; Assran et al.
+[2021] for fine-tuning from the first layer of the projector and initialize both the encoder and the first
+layer of the projector with the parameters of the pretrained model. We add the randomly initialized
+classifier on top of the first layer of the projector (after the non-linearity). We train all the weights
+(pretrained and classifier weights) using either 1% or 10% of the ImageNet-1k training data, and we
+use the splits introduced in Chen et al. [2020a] and used in all the methods to compare to Grill et al.
+[2020]; Caron et al. [2020]; Dwibedi et al. [2021]; Lee et al. [2021]; Mitrovic et al. [2021]; Tomasev
+et al. [2022]; Assran et al. [2021].
+At training time we randomly crop the image, resize it to 224 × 224, and then randomly apply
+a horizontal flip. At test time we resize images to 256 pixels along the shorter side with bicubic
+resampling and apply a 224 × 224 center crop to it. Both at training and testing times we subtract
+from the color channels the average channel value and divide it by the standard deviation of the
+channel value (as computed on ImageNet-1k).
+We use a cross entropy loss and stochastic gradient descent with Nesterov momentum of 0.9
+to fine-tune the model. For both 1% and 10% settings, we train for 30 epochs and decay the
+initial learning rate by a factor 0.2 at 18 and 24 epochs.
+Following the approach of Caron
+et al. [2020], we pick different learning rates for the encoder (and the first projector layer)
+and for the classifier weights.
+We do not use any weight decay or other regularization tech-
+niques. We sweep over batch sizes values in {512, 1024, 2048}, encoder base learning rate val-
+ues in {0.005, 0.0035, 0.003, 0.0025, 0.002, 0.001}, and linear layer base learning rate values in
+{0.5, 0.3, 0.2, 0.1, 0.05, 0.025}.
+Table 6: Top-1 and Top-5 accuracies (in %), after semi-supervised fine-tuning with a fraction of
+ImageNet labels, for a ResNet-50 encoder across a number of representation learning methods.
+Method
+Top-1
+Top-5
+1%
+10%
+1%
+10%
+Supervised [Zhai et al., 2019]
+25.4
+56.4
+48.4
+80.4
+Pseudo labels in classification:
+MPL [Pham et al., 2021]
+-
+73.9
+-
+-
+Representation learning methods:
+SimCLRv2 [Chen et al., 2020b]
+57.9
+68.4
+-
+-
+SimCLRv2 + self distillation [Chen et al., 2020b]
+60.0
+70.5
+-
+-
+CoMatch [Li et al., 2021a]
+66.0
+73.7
+86.4
+91.6
+PAWS [Assran et al., 2021]
+66.5
+75.5
+-
+-
+DebiasPL [Wang et al., 2022]
+67.1
+-
+85.8
+-
+SimMatch [Zheng et al., 2022]
+67.2
+74.4
+87.1
+91.6
+SEMPPL (ours)
+68.5
+76.0
+88.2
+92.7
+SimCLRv2 + Selective Kernels [Li et al., 2019b]
+64.5
+72.1
+86.7
+91.4
+SEMPPL (ours) + Selective Kernels
+72.3
+78.2
+90.6
+93.9
+Additional results and larger networks.
+When the architecture of the ResNet-50 is modified to
+include selective kernels [Li et al., 2019b], we see significant gains in performance at the expense
+of additional weights. Our implementation of selective kernels is standard and follows rigorously Li
+et al. [2019b] for a total of 27.5 million weights instead of of 25.5 million for a regular ResNet-50.
+Specifically, we use 2 channels, two convolution kernels of (3, 3) and (5, 5) with the latter imple-
+mented as a (3, 3) dilated convolution with rate 2, and 32 grouped convolutions. Unlike SimCLRv2
+[Chen et al., 2020b], we implement our group convolutions explicitly, and do not use the additional
+ResNet-D architectural modification from He et al. [2019]. When using selective kernels our per-
+formance after finetuning with 1% of labels is the same as that of SimCLRv2 after finetuning with
+10% of labels.
+16
+
+Preprint.
+Additionally, in order to investigate the robustness and scalability of these results, we further test
+the generality of SEMPPL by learning representations on larger (both deeper and wider) ResNet en-
+coders. Table 1 testifies to SEMPPL outperforming the competing representation learning methods
+across all the architectures, both in the 1% and the 10% labelled settings. Also, as our flagship result
+we reach 80.4% top-1 accuracy on ResNet-200 2× with 10% of ImageNet-1k labels. Just as in the
+ResNet-50 1× case this figure is comparable with the fully supervised accuracy attained by historical
+methods. 80.1% top-1 is defined as in Grill et al. [2020] with standard RandAugment [Cubuk et al.,
+2020] data augmentation. However it’s certainly a few percentage accuracy points away from results
+obtained with optimal current training protocols [Bello et al., 2021]. We also note that SEMPPL is
+pre-trained for 300 epochs in all cases. This, rather than the 1000 epochs used as standard by most
+other representation learning methods, again compares with a typical figure of 200 epochs used in
+supervised learning. Overall this hints at SEMPPL having achieved close to an order of magnitude
+gain in label efficiency (compared to supervised learning) at a similar epochs budget.
+Our final networks were optimized using tranches of between 128 (for a ResNet-50) and 512 (for
+the largest ResNets) Cloud TPUv3s all during 300 epochs each irrespective of size. This required
+around a day of computation time per run and tranche for a ResNet-50 on 128 devices, time which
+scaled approximately linearly with the number of parameters on larger networks, depending on the
+actual network.
+A.2
+ROBUSTNESS AND OOD GENERALIZATION
+We test the robustness and out-of-distribution (OOD) generalization abilities of representations
+learned via SEMPPL on several detasets. We use ImageNetV2 [Recht et al., 2019] and ImageNet-
+C [Hendrycks & Dietterich, 2019] datasets to evaluate robustness and the datasets ObjectNet [Barbu
+et al., 2019] and ImageNet-R [Hendrycks et al., 2021] to evaluate the OOD generalization.
+The ImageNetV2 dataset [Recht et al., 2019] has three sets of 10000 images (matched frequency
+(MF), Threshold 0.7 (T-0.7) and Top Images (TI)) that were collected to have a similar distribution
+to the ImageNet test set. The ImageNet-C dataset [Hendrycks & Dietterich, 2019] consists of 15
+synthetically generated corruptions of 5 different severities (e.g. blur, noise) that are applied to
+the ImageNet validation set. The ImageNet-R dataset [Hendrycks et al., 2021] consists of 30000
+different renditions (e.g. paintings, cartoons) of 200 ImageNet classes; the aim of this dataset is to
+test the generalization ability to different textures and other naturally occurring style changes that
+are out-of-distribution to the ImageNet training data. The ObjectNet dataset [Barbu et al., 2019] has
+18574 images from differing viewpoints and backgrounds compared to the ImageNet training set.
+On all datasets we evaluate the representations learned on a standard ResNet50 encoder under a
+linear evaluation protocol. We freeze the pretrained representations (no gradient updates) and train
+a linear classifier on top of the output of the ResNet-50 encoder using the full labelled ImageNet
+training set. We perform the test evaluation zero-shot, i.e the above datasets are not seen during the
+training of the representation or classifier.
+We provide a detailed breakdown across the different ImageNet-C corruptions in Table 7. Our
+proposed approach SEMPPL outperforms both the supervised baseline, on 12 out of 15 corruptions,
+as well as the competing semi-supervised representation learning model PAWS, on 12 out of 15
+corruptions (notably, over all Blur, Weather and Digital corruptions).
+Table 7: Top-1 accuracies (in %) for OOD generalisation on Gauss, Shot, Impulse, Blur, Weather,
+and Digital corruption types of ImageNet-C.
+Blur
+Weather
+Digital
+Method
+Gauss
+Shot
+Impulse
+Defocus
+Glass
+Motion
+Zoom
+Snow
+Frost
+Fog
+Bright
+Contrast
+Elastic
+Pixel
+JPEG
+Supervised [Lim et al., 2019]
+37.1
+35.1
+30.8
+36.8
+25.9
+34.9
+38.1
+34.5
+40.7
+56.9
+68.1
+40.6
+45.6
+32.6
+56.0
+Semi-supervised representations:
+PAWS [Assran et al., 2021]
+43.5
+40.6
+33.5
+38.7
+19.7
+34.1
+32.8
+40.3
+44.7
+64.0
+70.5
+59.7
+42.4
+38.5
+55.1
+SEMPPL(ours)
+41.3
+39.1
+30.0
+41.9
+23.2
+37.5
+34.0
+40.5
+45.5
+64.4
+71.9
+60.6
+44.2
+45.1
+57.7
+A.3
+TRANSFER
+To further evaluate the usefulness of the learned representations, we evaluate how well they transfer
+across datasets. For this, we follow the standard evaluation protocol outlined in Grill et al. [2020];
+17
+
+Preprint.
+Chen et al. [2020a]. We evaluate SEMPPL across the linear evaluation protocol which consists of
+freezing the encoder and only training a randomly initialized linear classifier on top of the encoder.
+In line with prior work [Chen et al., 2020a; Grill et al., 2020; Dwibedi et al., 2021], we test SEMPPL
+representations on the following datasets: Food101 [Bossard et al., 2014], CIFAR10 [Krizhevsky
+et al., 2009], CIFAR100 [Krizhevsky et al., 2009], Birdsnap [Berg et al., 2014], SUN397 (split
+1) [Xiao et al., 2010], DTD (split 1) [Cimpoi et al., 2014], Cars [Krause et al., 2013] Aircraft [Maji
+et al., 2013], Pets [Parkhi et al., 2012], Caltech101 [Fei-Fei et al., 2004], and Flowers [Nilsback
+& Zisserman, 2008], where we compare the downstream performance of SEMPPL to that of other
+reported semi-supervised methods on 10% of labels. Across these datasets there are differences
+in terms of metrics used for selecting the best hyper-parameters as well the reporting of the final
+results. In line with prior work [Chen et al., 2020a; Grill et al., 2020; Dwibedi et al., 2021], for
+Food101 [Bossard et al., 2014], CIFAR10 [Krizhevsky et al., 2009], CIFAR100 [Krizhevsky et al.,
+2009], Birdsnap [Berg et al., 2014], SUN397 (split 1) [Xiao et al., 2010], DTD (split 1) [Cimpoi
+et al., 2014], and Cars [Krause et al., 2013] we report the Top-1 accuracy on the test set, and for
+Aircraft [Maji et al., 2013], Pets [Parkhi et al., 2012], Caltech101 [Fei-Fei et al., 2004], and Flow-
+ers [Nilsback & Zisserman, 2008] we report the mean per-class accuracy. For DTD and SUN397 we
+only use the first split of the 10 provided splits in the dataset as per Chen et al. [2020a]; Grill et al.
+[2020]; Dwibedi et al. [2021].
+In these experiments, models are initially trained on the training sets of the individual datasets, and
+the validation sets are used to select the best hyperparameters from the executed hyperparameter
+sweeps. Once the best hyperparameters have been selected, the final models are trained on a merged
+dataset containing both the training and the validation split and evaluated on the held-out test split.
+The final results of the transfer experiments are reported in Table 3. The performed hyperparameter
+sweeps involved sweeping over the learning rates {.001, .01, 0.1, 0.2, 0.25, 0.3, 0.35, 0.4, 1.,
+2.}, batch sizes {128, 256, 512, 1024, 2048}, weight decay {1e−6, 1e−5, 1e−4, 1e−3, 0.01, 0.1},
+warmup epochs {0, 10}, momentum {0.9, 0.99}, Nesterov {True, False}, and the number of training
+epochs. For the linear transfer protocol we considered setting epochs among {20, 30, 60, 80, 100}.
+Models were trained by stochastic gradient descent with momentum.
+18
+
+Preprint.
+B
+INVARIANCE REGULARIZATION
+We define the short-hands
+p(ˆza1
+m ; ˜za2,+
+m,t ) =
+ϕ(ˆza1
+m , ˜za2,+
+m,t )
+ϕ(ˆza1
+m , ˜za2,+
+m,t ) + �
+xn∈N (xm) ϕ(ˆza1
+m , ˜za2
+n,t)
+(5)
+and
+p(ˆza1
+m ; ˜za2
+m,t) =
+ϕ(ˆza1
+m , ˜za2
+m,t)
+ϕ(ˆza1
+m , ˜za2
+m,t) + �
+xn∈N (xm) ϕ(ˆza1
+m , ˜za2
+n,t)
+(6)
+where N(xk) is the set of negatives, randomly uniformly sampled from the current batch, ˜za2
+n,t =
+gt(ft(xn)) is the target network projection of the negative sample; ϕ(x1, x2) = τ · exp(⟨x1, x2⟩/τ)
+is the scoring function, τ > 0 is a scalar temperature, and ⟨·, ·⟩ denotes the standard Euclidean dot
+product.
+We can now rewrite the components of the overall loss
+LSEMPPL = LAUGM + αLSEMPOS
+(7)
+as
+LSEMPOS = −
+B
+�
+m=1
+log p(ˆza1
+m ; ˜za2,+
+m,t )
+(8)
+and
+LAUGM = −
+B
+�
+m=1
+log p(ˆza1
+m ; ˜za2
+m,t).
+(9)
+As discussed in the main text we add the invariance penalty introduced in Mitrovic et al. [2021] to
+further increase the similarity between the anchor and positives and regularize the learning process.
+We add this invariance penalty both for augmentation positives and semantic positives. In particular,
+we compute
+Iaugm =DKL(p(ˆza1
+m ; ˜za2
+m,t) ∥ p(ˆza2
+m ; ˜za1
+m,t))
+=sg[Ep(ˆza1
+m ;˜za2
+m,t) log p(ˆza1
+m ; ˜za2
+m,t)] − Ep(ˆza1
+m ;˜za2
+m,t) log p(ˆza2
+m ; ˜za1
+m,t)
+and
+Isempos =DKL(p(ˆza1
+m ; ˜za2,+
+m,t ) ∥ p(ˆza2,+
+m
+; ˜za1
+m,t))
+=sg[Ep(ˆza1
+m ;˜za2,+
+m,t ) log p(ˆza1
+m ; ˜za2,+
+m,t )] − Ep(ˆza1
+m ;˜za2,+
+m,t ) log p(ˆza2,+
+m
+; ˜za1
+m,t)
+where sg denotes the stop-gradient operation. Taking all this together, this gives the final form of
+the loss as
+LSEMPPL = c(LAUGM + αLSEMPOS) + λ(Iaugm + Isempos)
+(10)
+with λ the invariance scale and c is the contrastive scale. We use λ = 5 and c = 0.3 in all our
+experiments irrespective of encoder size or training time as our method is robust to the choice of
+these hyperparameters.
+19
+
+Preprint.
+C
+PSEUDO-CODE OF SEMPPL
+Listing 1 provides PyTorch-like pseudo-code for SEMPPL detailing how we compute pseudo-labels
+and use them to select the additional semantic positives, which are then used in the contrastive loss,
+along the augmentation positives.
+1
+’ ’ ’
+2 k :
+The number
+of
+neighbors
+in kNN when computing
+p s e u d o l a b e l s .
+3
+f o :
+o n l i n e
+network :
+Encoder + comparison net .
+4
+g t :
+t a r g e t
+network :
+Encoder + comparison net .
+5 gamma :
+Target EMA c o e f f i c i e n t .
+6
+n e :
+Number of
+n e g a t i v e s .
+7 p m : Mask apply
+p r o b a b i l i t y
+.
+8
+’ ’ ’
+9
+f o r
+i
+in
+range ( num large views ) :
+10
+queue i = queue . i n i t ( queue size ,
+FIFO )
+11
+12
+f o r x ,
+y in
+batch :
+# Load
+batch
+of B samples
+each
+with
+data x and
+( maybe )
+l a b e l
+y .
+13
+x m = mask background ( x )
+14
+f o r
+i
+in
+range ( num large views ) :
+15
+#
+S t o c h a s t i c a l l y
+apply
+background
+removal .
+16
+x = B e r n o u l l i ( p m )
+? x m
+:
+x
+17
+# Create
+an augmented
+l a r g e
+view .
+18
+x l i
+= augment ( c r o p l a r g e ( x ) )
+19
+o l i
+= f o ( x l i )
+20
+t l
+i
+= g t ( x l i )
+21
+# Enqueue
+the
+l a b e l e d
+images
+in
+the
+batch .
+22
+i f
+y
+i s
+not None :
+23
+queue i . enqueue ( ( t l i ,
+y i ) )
+24
+25
+f o r
+i
+in
+range ( num small views ) :
+26
+x s i = augment ( c r o p s m a l l ( x ) )
+27
+# Small
+views
+only go
+through
+the
+o n l i n e
+network
+28
+o s i = f o ( x s i )
+29
+30
+# Pseudo − l a b e l
+computation
+f o r
+u n l a b e l l e d
+examples .
+31
+i f
+y
+i s
+None :
+# Missing
+l a b e l .
+32
+votes = [ knn ( k ,
+queue i ,
+o l j )
+f o r
+i ,
+j
+in
+a l l p a i r s ( num large views ) ]
+33
+y = mode ( votes )
+34
+35
+l o s s = 0
+36
+# Compute
+the
+l o s s
+between
+a l l
+the
+p a i r s
+of
+l a r g e
+views .
+37
+f o r
+i
+in
+range ( num large views ) :
+38
+f o r
+j
+in
+range ( num large views ) :
+39
+l o s s +=
+c o n t r a s t i v e l o s s ( o l i ,
+t l j ,
+n e )
+# Augmentation
+p o s i t i v e s .
+40
+f o r
+in
+range ( n u m s e m a n t i c p o s i t i v e s ) :
+41
+# Sample
+semantic
+p o s i t i v e s
+from
+the
+queue ,
+and add
+to
+the
+l o s s .
+42
+z = sample ( queue j . f i l t e r ( y ) )
+43
+l o s s +=
+c o n t r a s t i v e l o s s ( o l i ,
+z ,
+n e )
+44
+45
+# Compute
+the
+l o s s
+between
+the
+small
+and
+l a r g e
+views .
+46
+f o r
+i
+in
+range ( num small views ) :
+47
+f o r
+j
+in
+range ( num large views ) :
+48
+l o s s +=
+c o n t r a s t i v e l o s s ( os i ,
+t l j ,
+n e )
+# Augmentation
+p o s i t i v e s .
+49
+f o r
+in
+range ( n u m s e m a n t i c p o s i t i v e s ) :
+50
+# Sample
+semantic
+p o s i t i v e s
+from
+the
+queue ,
+and add
+to
+the
+l o s s .
+51
+z = sample ( queue j . f i l t e r ( y ) )
+52
+l o s s +=
+c o n t r a s t i v e l o s s ( o l i ,
+z ,
+n e )
+53
+54
+l o s s
+/=
+( ( num large views + num small views )
+55
+* num large views * (1 + n u m s e m a n t i c p o s i t i v e s ) )
+56
+57
+# Compute
+the
+g r a d i e n t s ,
+and
+update
+the
+o n l i n e
+and
+t a r g e t
+network .
+58
+l o s s . backward ( )
+59
+update ( f o )
+60
+g t = gamma *
+g t + (1 − gamma) *
+f o
+Listing 1 Pseudo-code for SEMPPL.
+20
+
+Preprint.
+D
+ANALYSIS
+D.1
+IMPLEMENTATION DETAILS
+We perform all the ablation experiments using 10% of labelled data and train a standard ResNet-
+50 encoder with SEMPPL for 100 epochs (except in the training duration ablation). We report the
+top-1 accuracies on the ImageNet test set after fine-tuning from the first layer of the projector. As
+in Grill et al. [2020] and for the main results in this paper, we use multi-layer perceptrons for the
+projector and predictor with 2 linear layers—the first one followed by batch normalization [Ioffe
+& Szegedy, 2015] and rectified linear activation with output sizes 4096 and 256 for the two layers
+respectively. We use the same augmentations as for the experiments in the main paper—the stan-
+dard SIMCLR augmentations [Chen et al., 2020a] and the RELICV2 multi-crop and saliency-based
+masking [Tomasev et al., 2022]. Following the hyperparameter settings of the main results, we use
+• batch size: B = 4096
+• queue capacity C = 20B (unless specifically ablated)
+• number of nearest neighbours k = 1 (unless specifically ablated)
+• view voting is used (unless specifically ablated)
+• weight decay: 1e − 6 (exclude biases and batch normalization parameters)
+• optimizer: LARS (exclude biases and batch normalization parameters from adaptation)
+• base learning rate: 0.3 (scaled linearly with batch size [Goyal et al., 2017])
+• warm-up: 10 epochs
+• cosine decay schedule for learning rate
+• exponential moving average parameter: 0.996
+• views: 4 large views of size 224 × 224 and 2 small views of size 96 × 96
+• temperature: τ = 0.2
+• number of semantic positives: 3 (unless specifically ablated)
+• 10 randomly subsampled negatives per anchor
+• α = 1/5 (unless specifically ablated), λ = 5 and c = 0.3.
+21
+
+Preprint.
+D.2
+ADDITIONAL ANALYSES
+Number of semantic positives
+We study the effect of varying the number of semantic positives in
+SEMPPL. Table 8a shows that increasing this number from 1 to 3 only has an effect on the amount
+of correctly predicted pseudo-labels, but no effect on downstream performance. On the other hand,
+using 5 or 10 semantic positives significantly improves performance and also yields much more
+accurate pseudo-labels prediction.
+Training duration
+Next, we vary the length of training representations and examine downstream
+performance. As can be seen from Table 8b, both the proportion of correctly predicted pseudo-
+labels and downstream performance improve with longer training up to 300 epochs but decrease if
+we continue training up to 500 epochs. This indicates with training longer than 300 epochs SEMPPL
+is starting to overfit, an observation consistent with phenomena reported elsewhere in the literature
+involving noisy labels [Li et al., 2019a; Kim et al., 2019; Liu et al., 2020; Han et al., 2018; Zhang &
+Sabuncu, 2018; Wang et al., 2019b].
+Table 8: Top-1 accuracy (in %) on the ImageNet-1k test set, and accuracy (in %) of correctly pre-
+dicted pseudo-labels at the end of training for semantic positives and training length experiments.
+Num. positives
+Top-1
+Pseudo-label acc.
+1
+69.9
+68.6
+2
+69.9
+70.9
+3
+69.9
+69
+5
+71.0
+72.8
+10
+70.7
+72.9
+(a) Varying the number of semantic positives.
+Training time (epochs)
+Top-1
+Pseudo-label acc.
+100
+69.9
+69.2
+200
+72.4
+76.8
+300
+72.7
+77.9
+500
+72.3
+75.6
+(b) Varying the length of training.
+View voting
+SEMPPL generates multiple views from a single instance image in order to learn rep-
+resentations. Those different views can be leveraged towards better pseudo-labels prediction. Rather
+than only picking one randomly selected data view to compute a single pseudo-label, we perform
+majority voting over (noisy) pseudo-labels computed from all available image views. Specifically,
+we compare the online predictor embedding of one view with the queue of the target projector em-
+beddings of the same data view from previous batches in the first setting; in the second setting we
+compare the online predictor embedding of each view with the queue of the target projector embed-
+dings of each other data view from previous batches.
+Since SEMPPL relies on 4 large views, this yields up to 16 different pairs of views to compare
+and compute pseudo-labels from, i.e. we get 16 pseudo-label predictions; this setting we call view
+voting. Table 9 shows that using all available views to compute pseudo-labels significantly increases
+pseudo-labels accuracy which in turn significantly improves downstream performance.
+Table 9: Top-1 accuracy (in %) on the ImageNet-1k test set, and accuracy (in %) of correctly pre-
+dicted pseudo-labels at the end of training for view voting experiments (using all views to compute
+pseudo-labels vs using just a single view).
+View voting
+Top-1
+Pseudo-label acc.
+On
+69.9
+68.6
+Off
+69.0
+62.5
+Number of nearest neighbours
+In order to compute pseudo-labels we use k-nearest neighbour
+lookup on the queue. While in the main results Section 3 we consistently assume k = 1 here we
+ablate the effect of varying k on downstream performance. As can be seen Table 10a, increasing
+k actually leads to a small decrease in performance. This is attributable to the decrease in the
+proportion of correctly predicted pseudo-labels as k increases.
+22
+
+Preprint.
+Queue length
+How long should the queue of target projector embeddings for computing pseudo-
+labels be? As the queue grows in size, it contains increasingly stale embeddings and threatens to
+hamper the accuracy of predicted pseudo-labels. On the other hand, increasing queue size increases
+the amount and diversity of labelled embeddings available which we would expect to be beneficial.
+Those two opposing forces—staleness and increasing coverage—govern the accuracy with which we
+can correctly predict pseudo-labels in turn directly affecting the selection of semantic positives and
+their quality. We resolve this ambiguity empirically. As seen in Table 10b, increasing the coverage
+and diversity of labelled embeddings has a strong positive effect on representation learning and
+downstream performance. Staleness of embeddings is far less of a problem at practical (i.e., not
+very large) queue sizes, showing diversity and coverage to be the dominant factor.
+Table 10: Top-1 accuracy (in %) on the ImageNet-1k test set, and accuracy (in %) of correctly
+predicted pseudo-labels at the end of training for k-NN and queue length experiments.
+k-nn
+Top-1
+Pseudo-label acc.
+1
+70.0
+70.5
+2
+69.9
+69.5
+3
+69.8
+70.0
+5
+69.8
+70.0
+10
+69.1
+68.0
+(a) Varying the number of nearest neighbours.
+Queue size (C)
+Top-1
+Pseudo-label acc.
+4000
+67.9
+53.1
+8000
+68.6
+60.4
+12000
+68.8
+62.0
+20000
+69.2
+64.3
+40000
+69.6
+66.9
+80000
+69.9
+68.8
+200000
+69.8
+69.5
+(b) Varying queue size.
+The effect of the loss weight α
+We use α in Equation 4 to weight the contribution of the loss
+coming from semantic positives against the loss coming from augmentation positives. In SEMPPL
+we use multiple semantic positives for learning and thus we need to adjust α to appropriately weigh
+the contribution of the individual semantic positives. In our main experiments, we use 3 semantic
+positives and for this reason α is not equal to 1.
+In Table 11 we vary α from 0.0 to 1.0, with α = 0.0 effectively recovering RELICV2. The results
+indicate that the SEMPPL loss notably improves over RELICV2, but that the exact choice of α
+should be treated as a hyperparameter. Note that the value 0.2 is very close to the 1
+3 which is exactly
+1/number of semantic positives. Thus, the optimal choice of α is very close to that fraction.
+Table 11: Top-1 accuracy (in %) on the ImageNet-1k test set for varied values of the loss weight α
+Loss weight (α)
+Top-1
+0.0 (RELICV2)
+72.4
+0.2
+76.0
+1.0
+74.6
+23
+
+Preprint.
+E
+AUGMENTATIONS
+In this work, we follow the established data augmentations protocols and pipelines of Chen et al.
+[2020a]; Grill et al. [2020]; Caron et al. [2020]; Mitrovic et al. [2021]; Chen et al. [2020b]. Specifi-
+cally, SEMPPL uses a set of augmentations to generate different views of the original image which
+has three channels, red r, green g and blue b with r, g, b ∈ [0, 1].
+The augmentations we use are generated by applying the following sequence of operations in the
+following order
+1. Crop the image: Randomly select a patch of the image, between a minimum and maximum
+crop area of the image, with aspect ratio sampled log-uniformly in [3/4, 4/3]. Upscale the
+patch, via bicubic interpolation, to a square image of size s × s.
+2. Flip the image horizontally.
+3. Colour jitter: randomly adjust brightness, contrast, saturation and hue of the image, in a
+random order, uniformly by a value in [−a, a] where a is the maximum adjustment (speci-
+fied below).
+4. Grayscale the image, such that the channels are combined into one channel with value
+0.2989r + 0.5870g + 0.1140b.
+5. Randomly blur. Apply a 23 × 23 Gaussian kernel with standard deviation sampled uni-
+formly in [0.1, 2.0].
+6. Randomly solarize: threshold each channel value such that all values less than 0.5 are
+replaced by 0 and all values above or equal to 0.5 are replaced with 1.
+Apart from the initial step of image cropping, each subsequent step is applied with a certain proba-
+bility to generate the augmented view of the original image. These probabilities and other augmen-
+tation parameters are given in Table 12. SEMPPL uses 4 large views of size 224 × 224 pixels and 2
+small views of 96 × 96 pixels; to get the first and third large views and the first small view we use
+the parameters listed below for odd views, while for the second and fourth large view and the second
+small view we use the parameters for even views. Note that these are the same augmentations used
+also in Chen et al. [2020a]; Grill et al. [2020]; Caron et al. [2020]; Mitrovic et al. [2021]; Chen et al.
+[2020b].
+In addition to these augmentations, we also randomly apply the saliency masking augmentation
+proposed in Tomasev et al. [2022] which enables us to remove a large part of the background. We
+follow the protocol described in Tomasev et al. [2022] for computing the saliency masking for an
+image and we apply this augmentation with probability 0.1 to the 4 large views. In keeping with
+Tomasev et al. [2022], we fill out the removed background of the image with homogeneous grayscale
+noise with the grayscale level randomly sampled for each view. We only apply the saliency masking
+when the remaining foreground covers at least 20% of the total image.
+24
+
+Preprint.
+Parameter
+Even views
+Odd views
+Probability of randomly cropping
+50%
+50%
+Probability of horizontal flip
+50%
+50%
+Probability of colour jittering
+80%
+80%
+Probability of grayscaling
+20%
+20%
+Probability of blurring
+100%
+10%
+Probability of solarization
+0%
+20%
+Maximum adjustment a of brightness
+0.4
+0.4
+Maximum adjustment a of contrast
+0.4
+0.4
+Maximum adjustment a of saturation
+0.2
+0.2
+Maximum adjustment a of hue
+0.1
+0.1
+Crop size s
+224
+96 (small), 224 (large)
+Crop minimum area
+8%
+5% (small), 14% (large)
+Crop maximum area
+100%
+14% (small), 100% (large)
+Table 12: Parameters of data augmentation scheme. Small/large indicates small or large crop.
+25
+
diff --git a/RdE4T4oBgHgl3EQflQ1W/content/tmp_files/load_file.txt b/RdE4T4oBgHgl3EQflQ1W/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..aacf322f8f68d20b1f020e43c727fae2b2ff8344
--- /dev/null
+++ b/RdE4T4oBgHgl3EQflQ1W/content/tmp_files/load_file.txt
@@ -0,0 +1,1587 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf,len=1586
+page_content='Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  SEMPPL: PREDICTING PSEUDO-LABELS FOR BETTER  CONTRASTIVE REPRESENTATIONS  Matko Boˇsnjak, Pierre H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Richemond, Nenad Tomasev, Florian Strub, Jacob C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Walker,  Felix Hill, Lars Holger Buesing, Razvan Pascanu, Charles Blundell, Jovana Mitrovic  DeepMind  {matko, richemond, mitrovic}@deepmind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='com  ABSTRACT  Learning from large amounts of unsupervised data and a small amount of super-  vision is an important open problem in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We propose a new semi-  supervised learning method, Semantic Positives via Pseudo-Labels (SEMPPL),  that combines labelled and unlabelled data to learn informative representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Our method extends self-supervised contrastive learning—where representations  are shaped by distinguishing whether two samples represent the same underlying  datum (positives) or not (negatives)—with a novel approach to selecting posi-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' To enrich the set of positives, we leverage the few existing ground-truth  labels to predict the missing ones through a k-nearest neighbours classifier by  using the learned embeddings of the labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We thus extend the set of  positives with datapoints having the same pseudo-label and call these semantic  positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We jointly learn the representation and predict bootstrapped pseudo-  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This creates a reinforcing cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Strong initial representations enable better  pseudo-label predictions which then improve the selection of semantic positives  and lead to even better representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' SEMPPL outperforms competing semi-  supervised methods setting new state-of-the-art performance of 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5% and 76%  top-1 accuracy when using a ResNet-50 and training on 1% and 10% of labels on  ImageNet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Furthermore, when using selective kernels, SEMPPL sig-  nificantly outperforms previous state-of-the-art achieving 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3% and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3% top-1  accuracy on ImageNet with 1% and 10% labels, respectively, which improves  absolute +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8% and +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2% over previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' SEMPPL also exhibits state-  of-the-art performance over larger ResNet models as well as strong robustness,  out-of-distribution and transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  1  INTRODUCTION  In recent years, self-supervised learning has made significant strides in learning useful visual fea-  tures from large unlabelled datasets [Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Mitrovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Moreover, self-supervised representations have matched the  performance of historical supervised baselines on the ImageNet-1k benchmark [Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2015] in like-for-like comparisons as well as outperformed supervised learning in many transfer  settings [Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' While such results show exciting progress in the field, in many  real-wold applications often there exists a small amount of ground-truth labelled datapoints making  the problem of representation learning semi-supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In this work we propose a novel approach to semi-supervised learning called Semantic Positives  via Pseudo-Labels (SEMPPL) which incorporates supervised information during the representation  learning stage within a self-supervised loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Unlike previous work which uses the available super-  vision as targets within a cross-entropy objective, we propose to use the supervised information to  help inform which points should have similar representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We propose to learn representations  using a contrastive approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we learn the representation of a datapoint (anchor) by maximizing  the similarity of the embedding of that datapoint with a set of similar points (positives), while simul-  taneously minimizing the similarity of that embedding with a set of dissimilar points (negatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='05158v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='CV]  12 Jan 2023  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  As such, the appropriate construction of these sets of positives and negatives is crucial to the success  of contrastive learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' While strategies for sampling negatives have been extensively  studied in the literature [Schroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Harwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020c], the sampling of positives has received far less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We propose a novel approach to selecting positives which leverages supervised information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Specif-  ically, we propose using the small amount of available ground-truth labels in order to non-  parametrically predict the missing labels (pseudo-labels) for the unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Note that many  previous semi-supervised approaches use pseudo-labels as targets within a cross-entropy-based ob-  jective [Van Engelen & Hoos, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In SEMPPL we use pseudo-labels in a very  different way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we use them to select positives based on whether two datapoints (we call these  semantic positives) share the same (pseudo-)label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' By maximizing the similarity of a datapoint with  its semantic positives we expect to learn representations that are more semantically aligned and as a  consequence encode more abstract, higher-level features which should generalise better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' To predict  informative pseudo-labels, we compare the representations of the unlabelled data with those of the  labelled subset and use a k-nearest neighbours (k-NN) classifier to impute the missing labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We simultaneously learn the representation, predict pseudo-labels and select semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  This creates a virtuous cycle: better representations enable better pseudo-label prediction which in  turn enables better selection of semantic positives and thus helps us learn better representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Im-  portantly, as the prediction of pseudo-labels and selection of semantic positives does not depend on  the exact form of the contrastive objective employed, SEMPPL is compatible with and complements  all contrastive losses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Mitrovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2021] and may even be extended to non-contrastive losses [Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Chen & He, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We evaluate the representations learned with SEMPPL across a varied set of tasks and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In  particular, SEMPPL sets new state-of-the-art in semi-supervised learning on ImageNet with 1% and  10% of labels on the standard ResNet-50 (1×) architecture with respectively 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5% and 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0% top-  1 performance and across larger architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' When combined with Selective Kernels [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2019b], we achieve 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3% and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3% top-1 performance with 1% and 10% labels, respectively,  significantly outperforming previous state-of-the-art by absolute +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8% and +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2% in top-1 perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We also outperform previous state-of-the-art on robustness and out-of-distribution (OOD)  generalisation benchmarks while retaining competitive performance in transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Our main contributions are:  We extend contrastive learning to the semi-supervised setting by introducing the idea of es-  timating pseudo-labels for selecting semantic positives as a key component especially in the  low-label regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We propose a novel semi-supervised method SEMPPL that jointly estimates pseudo-labels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  selects semantic positives and learns representations which creates a virtuous cycle and enables  us to learn more informative representations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We extensively evaluate SEMPPL and achieve a new state-of-the-art in semi-supervised learn-  ing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' robustness and out-of-distribution generalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' and competitive performance in transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  2  SEMANTIC POSITIVES VIA PSEUDO-LABELS  The selection of appropriate positive and negative examples are the cornerstone of contrastive learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Though the research community has mainly focused on the selection of negatives, positives  are equally important as they play a vital role in learning semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We thus leverage  labelled information as it encodes semantic information to improve the selection of informative pos-  itives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Specifically, we expand a self-supervised model to use this labelled data to non-parametrically  predict pseudo-labels for the remaining unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Using both ground-truth labels and the pre-  dicted pseudo-labels, we expand the set of positives with semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Notations Let D = Dl ∪Du be a dataset consisting of labelled training data Dl = {(xi, yi)}N  i=1 and  unlabelled training data Du = {(xj)}M  j=N+1 with M ≫ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Let B be a batch of data of size B with  B = {(xi, yi)}b  i=1 ∪ {xj}B  j=b+1 where (xi, yi) ∈ Dl and xj ∈ Du, where the indices i, j and m to  denote labelled, unlabelled, and all datapoints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Following established self-supervised  learning practices [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Mitrovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Dwibedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Embedding  Projection  Semantic  positive  Loss  Prediction  Projection  Projection  Projection  Projection  K-NN  Query  Pseudo  labels  Embedding  Projection  Target Network  Negatives  Loss  Target Network  Online Network  Online Network  Embedding  Projection  Prediction  Online Network  Prediction  Queue  Figure 1: Sketch of SEMPPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' (Left) Standard contrastive pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' (Middle) Unlabelled data are  tagged with pseudo-labels by using a k-NN over projected labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' (Right) Semantic positives  are queried from the queue and processed to compute an additional contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022], we create different views of the data by applying pairs of randomly  sampled augmentations a1, a2 ∼ A from the augmentation distribution A proposed in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  [2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For every datapoint xm ∈ D we denote the corresponding augmentations as xa1  m , xa2  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Augmentation positives  We embed one data view xa1  m via an online encoder network f and embed  the other data view xa2  m with a target encoder network ft, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we get latent representations za1  m =  f(xa1  m ) and za2  m,t = ft(xa2  m ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Note that the weights of ft are an exponential moving average of the  weights of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Next, we pass these latent representations through projection and prediction multi-  layer perceptrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Specifically, we use an online projector g and target projector gt, as well as an  online predictor h, to further transform za1  m and za2  m,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' again, the weights of gt are an exponential  moving average of the weights of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We then get ˆza1  m = h(g(za1  m )) and ˜za2  m,t = gt(za2  m,t) and l2-  normalise these;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we use ˆza1  m , ˜za2  m,t onward as the normalised latent embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In order to learn the representation of ˆza1  m , we contrast it against the augmentation-based positive  ˜za2  m,t as well as against negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For this, we use the contrastive loss:  LAUGM = −  B  �  m=1  log  ϕ(ˆza1  m , ˜za2  m,t)  ϕ(ˆza1  m , ˜za2  m,t) + �  xn∈N (xm) ϕ(ˆza1  m , ˜za2  n,t)  (1)  where N(xk) is the set of negatives, randomly uniformly sampled from the current batch, ˜za2  n,t =  gt(ft(xn)) the target network projection of the negative sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ϕ(x1, x2) = τ · exp(⟨x1, x2⟩/τ)  is the scoring function, τ > 0 is a scalar temperature, and ⟨·, ·⟩ denotes the Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Since the representations we contrast are l2-normalised, the dot product effectively turns into cosine  similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Pseudo-label prediction and semantic positives  Since we have access to a small labelled dataset,  we can use the label information to select more informative positives beyond just augmentations of  the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Specifically, we can associate images with the same label as positives and we call  these semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We want to select semantic positives for all the data, not just the labelled  subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For this purpose, we propose to compute pseudo-labels for the unlabelled data and use this  to select semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' To compute pseudo-labels we compare the current latent embeddings  of the unlabelled data to those of the labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Specifically, we propose to use a first-in-first-out  3  020202Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  queue Q with capacity C for storing labelled embeddings which we use for computing the pseudo-  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' At the start of training, we simply initialise the queue with random vectors, and use the queue  from the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For each batch B, we add the target projection of only the labelled data to the  queue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Q ← (˜za2  i,t, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' To predict a pseudo-label for an unlabelled datapoint xj, we first compute  the online predictor output ˆza1  j , before retrieving its k-nearest neighbours {(˜za2  s,t, ys)}k  s=1 in cosine  similarity from the queue Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1 Finally, we compute the pseudo-label ¯yj of xj as:  ¯yj = mode  ys  {(˜za2  s,t, ys)}k  s=1  (2)  where mode is the mode of the set, tasked with obtaining the most frequent class in the k-nearest  neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We use the ground-truth labels (for the labelled data) or the computed pseudo-labels  (for the unlabelled data) to select semantic positives for every datapoint in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For each xm ∈ B,  we uniformly sample over all the embeddings in Q that share the same (pseudo-) label as xm to  get a semantic positive ˜za2,+  m,t  ∼ U({(˜za2  l,t, yl) ∈ Q | yl = pl(xm)}), where pl(xm) = ym if xm is  labelled and pl(xm) = ¯ym if xm is unlabelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Next, we include these semantic positives within our  representation learning process through the contrastive objective  LSEMPOS = −  B  �  m=1  log  ϕ(ˆza1  m , ˜za2,+  m,t )  ϕ(ˆza1  m , ˜za2,+  m,t ) + �  xn∈N (xm) ϕ(ˆza1  m , ˜za2  n,t)  (3)  Taking these two losses (1) and (3) together, we propose to learn representations in our method  SemPPL by minimising the following total loss  LSEMPPL = LAUGM + αLSEMPOS  (4)  where α controls the ratio between these sub-losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  IMPLEMENTATION DETAILS  Architecture We use Residual Networks [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2016] (v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' pre-activation as customary in the  literature) for f and ft and use either 50 or 200 layers deep networks and with a width multiplier  ranging from 1× to 4×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' As in [Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022], we use multi-layer per-  ceptrons with 2 layers of size 4096 and 256, with batch normalisation [Ioffe & Szegedy, 2015] and  rectified linear activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Self-supervised learning method We use RELICv2 [Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022] as our default self-  supervised training objective due to its competitive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Therefore, we add an invariance  penalty on top of Equation 4 to further enforce the similarity constraints and regularize the learning  process as detailed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We also explore other self-supervised learning objectives in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Algorithm parameters We use a queue of capacity C = 20B, with batch size B = 4096, and  temperature τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2 while randomly sampling negatives from the current batch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we take |N(x)| =  10 negatives in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For augmentations, we use the standard SIMCLR augmentations [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2020a] and the RELICV2 multi-crop and saliency-based masking [Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we use 4  large views and 2 small views for augmentation positives and 3 semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' The semantic  positives are computed with a k-NN with k = 1 (see the analysis section in Appendix D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we build  a single k-NN instance per augmentation a queried with all the augmentations where |a| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This  produces |a|2 = 16 k-NN induced pseudo-labels in total for each unlabelled image among which  we then perform majority voting to compute the final pseudo-label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Optimisation Our networks are optimized with LARS [You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Our base learning rate  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3 and we train our models for 300 epochs with a learning rate warm-up period of 10 epochs  and cosine decay schedule thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We use a weight decay of 10−6 and batch size B = 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We exclude the biases and batch normalisation parameters both from LARS adaptation and weight  decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' The exponential moving average parameter for target networks is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Our pseudo-code  is described in the appendix along with precise architectural and implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Pretrained  model checkpoints and code will be made available on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  1We use the cosine similarity as the embeddings are normalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  4  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  3  EXPERIMENTAL RESULTS  To evaluate SEMPPL, we pre-train representations using 1% and 10% labelled data from the Ima-  geNet dataset [Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2015] based on the splits from Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020a]  We then test SEMPPL in semi-supervised classification, robustness and out-of-distribution general-  isation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Lastly, we probe the transfer capabilities of the representations to other image classi-  fication datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For a complete set of results and experimental details, please see the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  SEMI-SUPERVISED LEARNING  In Table 1, we report top-1 accuracy on the ImageNet test set when either 1% or 10% of the data is  labelled for the ResNet-50 architecture as well as deeper and wider ResNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  SEMPPL achieves top-1 accuracy of 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5% with 1% of labels, significantly outperforming the previ-  ous state-of-the-art SimMatch [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022] by an absolute +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3% in ImageNet test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  With 10% of label data, our top-1 accuracy on ResNet-50 reaches 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0%, outperforming the previ-  ous state-of-the-art PAWS [Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021] in semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' SEMPPL outperforms  competing representation learning methods across the board, achieving state-of-the-art performance  on all ResNet-50 2×, ResNet-50 4× and , in both the 1% and 10% labelled settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' SEMPPL does  not use, and therefore excludes from comparison, distillation from larger networks as in [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Similar to [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020b], we also tested SEMPPL on ResNets with Selective Kernels (SK) [Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  This increases the encoder parameter count to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We thus achieve a new absolute state-of-the-art of 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3% and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3% top-1 accuracies, respectively,  when using 1% and 10% of labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Finally, SEMPPL reaches a new state-of-the-art using 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  and 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5 on 1% and 10% of labels without self-distillation with a ResNet-200 2× + SK architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  For implementation details of the semi-supervised results and additional results, see the Ap-  pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 1: Top-1 accuracy (in %) for ResNet encoders with different depth and width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  ResNet-50 1×  ResNet-50 2×  ResNet-50 4×  ResNet-200 2×  Method  Top-1  Top-1  Top-1  Top-1  1%  10%  1%  10%  1%  10%  1%  10%  SimCLR [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a]  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4 BYOL [Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020]  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  RELICv2 [Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022]  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  SimCLRv2 [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020b]  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9 CoMatch [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021a]  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7 PAWS [Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021]  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0 SimMatch [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022]  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4 SemPPL (ours)  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  SimCLRv2 + SK [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020b]  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0 SemPPL + SK (ours)  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  ROBUSTNESS AND OOD GENERALISATION  We evaluate the robustness and generalisation abilities of SEMPPL on ImageNetV2 [Recht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2019], ImageNet-C [Hendrycks & Dietterich, 2019], ImageNet-R [Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021] and Ob-  jectNet [Barbu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019] which have all been purposefully constructed to test different robustness  and generalisation aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We evaluate all three variants on ImageNetV2: matched frequency (MF),  Threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7 (T-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7) and Top Images (TI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' When evaluating PAWS, we used the publicly available  checkpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  5  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 2 shows good robustness and generalisation ability of the representations learned with  SEMPPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' SEMPPL sets the new state-of-the-art performance (outperforming even the supervised  baseline) on 4 out of 5 datasets, while outperforming PAWS across all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' SEMPPL also out-  performs SimMatch on 4 out of 5 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For more details on the evaluation protocols and results  for ImageNet-C see the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 2: Top-1 accuracy (in %) for ImageNetV2, ImageNet-R and ObjectNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Robustness  OOD generalization  Method  MF  T-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  Ti  ImageNet-R  ObjectNet  Supervised (100% labels) [Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019]  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  Semi-supervised (10% labels)  PAWS [Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021]  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  SimMatch [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022]  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  SemPPL (ours)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  TRANSFER LEARNING  We evaluate the generality of SEMPPL representations by testing whether the features learned on  ImageNet are useful across different datasets and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Specifically, we evaluate the transfer per-  formance of SEMPPL on a set of 11 image classification datasets commonly used in the contrastive  literature under the linear protocol [Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Dwibedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Mitro-  vic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For the linear protocol, the pretrained encoder is frozen and a  randomly initialized linear classifier is trained on top using the training data from the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We report standard metrics for each dataset as well as performance on a held-out test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For more  details on the evaluation protocol see the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Table 3 compares the transfer performance  of representations pretrained using the supervised baseline [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a], PAWS [Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2021], SimMatch [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022] and our method SEMPPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' SEMPPL outperforms the super-  vised baseline on 8 out of 11 datasets, PAWS on 9 out of 11 datasets, while showing competitive  performance to SimMatch, outperforming it on 4 out of 7 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  FULL LABELLED DATASET  Figure 2: Top-1 accuracy for ResNet50 with 100% of the  labels across augmentations, initializations and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Method  Params  Top-1  Supervised (ResNet-50)  + AutoAugment [Cubuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019]  27M  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  + MaxUp [Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021]  27M  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  Representation Learning (ResNet-50)  SEMPPL (SimCLR base)  27M  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  SEMPPL (BYOL base)  27M  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  SEMPPL (ReLICv2 base;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ours)  27M  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  Other Architectures  Swin-T [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021b]  29M  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  ConvNeXt [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022]  29M  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  SEMPPL + SK (ours)  29M  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  We also assess how SEMPPL be-  haves in a fully supervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  For this purpose, we select semantic  positives based on the ground-truth  labels and fine-tune the learned rep-  resentations with the full ImageNet  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We compare against strong  supervised baselines on ResNets as  well as against recent performant net-  work architectures that are extensions  of the ResNet, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Our method reaches 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7% top-1 ac-  curacy on a ResNet 50 outperforming  a number of strong supervised base-  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' When we add selective kernels  to a ResNet 50, we achieve 82% top-1  accuracy outperforming recent trans-  formers architecture [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021b], and matching highly tuned ConvNext [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Therefore, SEMPPL may also be considered as a promising pretraining method in the supervised  learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  6  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 3: Top-1 accuracy (in %) on the full suite of transfer tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Method  Food101 CIFAR10  CIFAR100  Birdsnap  SUN397  Cars  Aircraft DTD  Pets  Caltech101  Flowers  Supervised-IN [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a]  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  Semi-supervised methods with 10% labels:  PAWS [Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021]  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  SimMatch [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022]  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  –  –  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  –  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  –  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  SEMPPL (ours)  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  4  ANALYSIS  We analyse the impact of different design choices in SEMPPL on downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In  this section, we focus the behaviour and impact of pseudo-labels and semantic positives on learning  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For further analyses and experimental details, please see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Semantic positives across self-supervised learning objectives  With SEMPPL we extend the set  of positives to include semantic positives based on predicted pseudo-labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we can combine these  ideas with other self-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1, we additionally evaluate SEMPPL on the  non-contrastive self-supervised method BYOL [Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' BYOL replaces the contrastive  loss in Equation 4 with an l2 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Importantly, we follow the training pipeline (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' augmentation,  hyperparameters etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=') from [Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020] to fairly highlight the impact of SEMPPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We ob-  serve a drastic improvement when adding semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' With 1% labels on ImageNet BYOL  improves by absolute +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9% and by absolute +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6% when using 10% labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For completeness we  have also highlighted the contribution of SEMPPL when using RELICv2 as the base self-supervised  objective which is our default implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For 1% labeled data, we see an absolute improve-  ment of +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4% in top-1 accuracy, while for 10% labels we see a gain of absolute +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6% in top-1  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In summary, we see that SEMPPL can be easily combined with other self-supervised  objectives to yield significant improvements and can be used a standard plug-and-play module in  within semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  The contribution of pseudo-labels and semantic positives  We examine the impact of omitting  pseudo-label prediction and semantic positives from learning representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Specifically, we ab-  late the use of pseudo-labels when selecting semantic positives for unlabelled datapoints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we  only use labelled images when retrieving semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2 (middle row), removing  pseudo-label prediction significantly decreases performance both in the 1% and 10% label settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In addition, the low-label regime (1% labels) suffers a stronger performance decrease −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6% than  the 10% labels regime, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This underscores the importance of pseudo-label estimation and  subsequent selection of semantic positives for unlabelled data especially in the low-data regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Going a step further, we remove semantic positives even for the labelled data, falling back to vanilla  RELICv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2 (bottom row), we see again a significant drop in performance for both the  1% and 10% label settings with a sharper drop for the low-label regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Together these highlights  the importance of both including semantic positives for labelled data as well as using pseudo-label  prediction for selecting semantic positives for unlabelled data in order to learn informative represen-  tations in a semi-supervised setting in a label-efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 4: Top-1 test accuracy (in %) with a ResNet50 pretrained on ImageNet with 1% and 10%  labels: (1) when trained on different self-supervised objective, (2) when removing pseudo-labelling,  and semantic positives in SEMPPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Top-1  1%  10%  BYOL [Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020]  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  SEMPPL with BYOL  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  ReLICv2 [Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022]  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  SEMPPL with RELICv2 (ours)  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  1% labels  10% labels  SEMPPL  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  Pseudo-labels  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  Semantic Positives  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  7  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 5: Top-1 test accuracy (in %) with a ResNet50 pretrained on ImageNet 10% labels for 100  epoches when using ground truth labels instead of pseudo-labels while retrieving semantic positives  (oracle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' PL accuracy helds for the accuracy of the pseudo-label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  10% labels  Top-1  PL accuracy  SEMPPL  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  SEMPPL (+oracle)  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  Precision and Recall of pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In Figure 3, we analyse the behaviour of pseudo-labels  by looking at the precision and recall as training progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We train a ResNet-50 for 100 epochs  using 10% labels with SEMPPL on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' As we have 4 large views there will be in total 16  votes cast and then the pseudo-label will be estimated using majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We want to measure  how often these 16 votes agree or disagree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we denote as voting threshold the number k where at  least k votes have been cast for one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We see that as training progresses the precision across all  thresholds increases as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This means that the pseudo-label prediction is bootstrapping itself  to become more accurate, which enables us to select better semantic positives and thus learn more  informative representations as training progresses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we have a virtuous cycle of representation  learning and pseudo-label prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Furthermore, precision is an increasing function of the voting  threshold throughout training and is highest for the biggest voting threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This indicates how  confident we can be in the accuracy of pseudo-label prediction, and thus how confident we can be  that an appropriate semantic positive has been selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Yet, we see that the recall for individual  thresholds is also increasing as training progresses but that the recall decreases as we increase the  voting threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This is expected as there is always a trade-off between precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  5000  10000  15000  20000  25000  30000  Steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  Precision  5000  10000  15000  20000  25000  30000  Steps  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  Recall  Voting threshold  0  2  5  7  10  12  15  Figure 3: Precision and recall for pseudo-labels computed based on k-nearest neighbours when  trained on ImageNet with 10% labels over 100 epoches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Noise in pseudo-label prediction  In Figure 3, we observe that the proportion of correctly pre-  dicted pseudo-labels at the end of training is reasonably high (60% accuracy of voting threshold  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Yet, it also means that 40% of pseudo labels are still incorrectly predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' As the incorrect  prediction results in suboptimal semantic positives selection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', SEMPPL does not select semantic  positives from the same class as the datapoint, this behavior may ultimately worsen the quality of ex-  tracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' To quantify this phenomenon, we train the representation with SEMPPL where an  oracle replaces the pseudo-label prediction with ground-truth labels and those are used for selecting  semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Table 5, we train the representations for 100 epochs on ImageNet with 10%  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' There, the oracle increases the top-1 performance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7% in the test set with 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Besides,  the pseudo-label accuracy also gets 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2% higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' It thus confirms that incorrect pseudo-label predic-  tions, and incorrect semantic positives retrieval, hurts learning informative representations and the  downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Yet, the oracle performance remains close to the actual performance of  SEMPPL, illustrating the method’s robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Further ablations on other design choices, such as the number of semantic positives, the use of view  voting, the choice of k in k-NN, queue length and training duration can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  8  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  5  RELATED WORK  Semi-supervised learning  In the semi-supervised regime [Cheplygina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Van Engelen  & Hoos, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Alizadehsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021], one can either pre-train a model on  unlabelled data and subsequently fine-tune it on labelled data, or train both jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Joint training  on labelled and unlabelled data often involves combining the two losses [Grandvalet & Bengio,  2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Miyato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Verma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Berman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Pseudo-label self-training approaches [Zoph et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020] present an important alternative,  first inferring approximate pseudo-labels for the unlabelled examples, and then incorporating them  in supervised losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Pseudo-labels can either be generated prior to a subsequent supervised learning  phase [Yarowsky, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Riloff, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2013] or jointly in an online fashion [Berthelot  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' These methods may benefit from pseudo-label confidence  measures [Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Rizve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021] as well as thresholding [Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2021], temporal ensembling [Laine & Aila, 2017], or stronger regularization to mitigate bias in early  model training [Sajjadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Arazo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' The use of pseudo-labels with rebalancing  has shown improvements, both in class-imbalanced problems [Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021] and in a general  context [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Teacher-student network configurations for generating and utilising  pseudo-labels have also shown promise [Tarvainen & Valpola, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Co-training uses different feature extractors for  different data views and alternates between pseudo-labelling and training phases [Blum & Mitchell,  1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Good performance has been reached by using consistency losses between  pseudo-labels of different inputs [Verma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Predicting view assignments with support samples [Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021] (PAWS) has resulted in  substantial performance improvements, with the idea that the assigned pseudo-labels ought to be  similar across multiple views of the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Recent work has shown that incorporating label information in positive selection in contrastive meth-  ods is highly promising, compared to the cross-entropy loss in the fully supervised case [Khosla  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Our method demonstrates a similar utility of pseudo-labels for semi-supervised prob-  lems, and differs from competing ones in the following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Unlike DebiasPL [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022]  that uses an adaptive margin loss, SemPPL does not seek to directly address or re-shape the distri-  bution of pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Unlike SimCLRv2 [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020b], we do not rely on self-distillation  procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In contrast with PAWS [Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021], we fully leverage the contrastive approach  for semi-supervised learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' not using positives only for training means SEMPPL does not require  specific care like pseudo-labels sharpening to stabilize learning and avoid representational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  SEMPPL is more closely related to CoMatch [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021a] that also uses bootstrapping to im-  prove pseudo-labels representational quality, but is conceptually much simpler, avoiding phases of  distributional alignment and of performing graph-based contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In a similar vein, Sim-  Match [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022] also uses a memory buffer to propagate pseudo-labels, but has a more  complex objective than SEMPPL and equally requires additional phases of pseudo-labels unfolding  and aggregation to function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Self-supervised learning  Major advances in learning useful representations from unlabelled  data [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021] can be seen as a paradigm shift, since these methods  have recently been competitive with supervised training baselines [Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' A num-  ber of self-supervised learning methods involve contrasting multiple views of the data [Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Bachman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Dwibedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Similar performance were also achieved by bootstrapping-based multi-view learning [Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Richemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Chen & He, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Zbontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021], or involv-  ing explicit clustering steps [Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' An explicit  causally-motivated invariance loss, when used in conjunction with the contrastive objective, has  been shown to lead to more compact representations, and desirable generalisation properties [Mitro-  vic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Contrastive approaches are not always used in self-supervised methods [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Ermolov  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Transformer-specific methods have been devised [Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  9  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  6  CONCLUSION  In this work, we propose SEMPPL, a novel semi-supervised learning method to incorporate semantic  positives in self-supervised objectives by taking advantage of pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Through extensive  empirical evaluation, we demonstrated that our approach achieves state-of-the-art semi-supervised  performance on ImageNet across several ResNet architectures as well as on the robustness, out-of-  distribution generalization and transfer tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We also show that SEMPPL can be easily combined  with other existing self-supervised methods and is a promising direction to pre-train networks also in  a fully supervised learning regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Our analyses suggest that the role of pseudo-labels in selecting  positives for semi-supervised contrastive methods might be underappreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Despite widespread  use in semi-supervised applications, pseudo-labels are less understood and have been explored far  less in the context of self-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We hope this study, which shows empirically that  prior work has under-utilized pseudo-labels, may help bridge that gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  REFERENCES  Roohallah Alizadehsani, Danial Sharifrazi, Navid Hoseini Izadi, Javad Hassannataj Joloudari, Afshin Shoeibi,  Juan M Gorriz, Sadiq Hussain, Juan E Arco, Zahra Alizadeh Sani, Fahime Khozeimeh, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
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+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
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+page_content=' Asano, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Rupprecht, and Andrea Vedaldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Self-labelling via simultaneous clustering and representa-  tion learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of International Conference on Learning Representations (ICLR), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Mahmoud Assran, Mathilde Caron, Ishan Misra, Piotr Bojanowski, Armand Joulin, Nicolas Ballas, and  Michael Rabbat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Semi-supervised learning of visual features by non-parametrically predicting view assign-  ments with support samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Conference on Computer Vision and Pattern Recognition (CVPR),  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Philip Bachman, R Devon Hjelm, and William Buchwalter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Learning representations by maximizing mutual  information across views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Advances in Neural Information Processing Systems (NeurIPS), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Andrei Barbu, David Mayo, Julian Alverio, William Luo, Christopher Wang, Dan Gutfreund, Josh Tenenbaum,  and Boris Katz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Objectnet: A large-scale bias-controlled dataset for pushing the limits of object recognition  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Advances in Neural Information Processing Systems (NeurIPS), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Irwan Bello, William Fedus, Xianzhi Du, Ekin Dogus Cubuk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Srinivas, Tsung-Yi Lin, Jonathon Shlens, and  Barret Zoph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Revisiting resnets: Improved training and scaling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Advances in Neural  Information Processing Systems (NeurIPS), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Thomas Berg, Jiongxin Liu, Seung Woo Lee, Michelle L Alexander, David W Jacobs, and Peter N Belhumeur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Birdsnap: Large-scale fine-grained visual categorization of birds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Conference on Computer Vision  and Pattern Recognition (CVPR), 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Maxim Berman, Herv´e J´egou, Andrea Vedaldi, Iasonas Kokkinos, and Matthijs Douze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Multigrain: a unified  image embedding for classes and instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' arxiv preprint arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
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+page_content='  David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin A Raffel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Mix-  match: A holistic approach to semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Advances in Neural Information Pro-  cessing Systems (NeurIPS), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  David Berthelot, Nicholas Carlini, Ekin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Cubuk, Alex Kurakin, Kihyuk Sohn, Han Zhang, and Colin Raffel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
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+page_content=' of  International Conference on Learning Representations (ICLR), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Avrim Blum and Tom Mitchell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Combining labeled and unlabeled data with co-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of the Annual  Conference on Computational Learning Theory, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Lukas Bossard, Matthieu Guillaumin, and Luc Van Gool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Food-101–mining discriminative components with  random forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of the European conference on computer vision (ECCV), 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  10  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Zhaowei Cai, Avinash Ravichandran, Subhransu Maji, Charless Fowlkes, Zhuowen Tu, and Stefano Soatto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Exponential moving average normalization for self-supervised and semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of  Conference on Computer Vision and Pattern Recognition (CVPR), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Unsupervised  learning of visual features by contrasting cluster assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Advances in Neural Information  Processing Systems (NeurIPS), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Mathilde Caron, Hugo Touvron, Ishan Misra, Herv´e J´egou, Julien Mairal, Piotr Bojanowski, and Armand  Joulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Emerging properties in self-supervised vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Conference on Computer  Vision and Pattern Recognition (CVPR), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' A simple framework for contrastive  learning of visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of International Conference on Machine Learning (ICML),  2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Ting Chen, Simon Kornblith, Kevin Swersky, Mohammad Norouzi, and Geoffrey E Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Big self-supervised  models are strong semi-supervised learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Advances in Neural Information Processing Systems  (NeurIPS), 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Xiaokang Chen, Mingyu Ding, Xiaodi Wang, Ying Xin, Shentong Mo, Yunhao Wang, Shumin Han, Ping Luo,  Gang Zeng, and Jingdong Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Context autoencoder for self-supervised representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' arxiv  preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
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+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of International Conference on Machine Learning (ICML)g  (ICML), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
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+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of International Conference on Machine Learning (ICML), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Shuangfei Zhai, Navdeep Jaitly, Jason Ramapuram, Dan Busbridge, Tatiana Likhomanenko, Joseph Yitan  Cheng, Walter Talbott, Chen Huang, Hanlin Goh, and Joshua Susskind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Position prediction as an effec-  tive pretraining strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of International Conference on Machine Learning (ICML), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Xiaohua Zhai, Avital Oliver, Alexander Kolesnikov, and Lucas Beyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' S4l: Self-supervised semi-supervised  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of the International Conference on Computer Vision (ICCV), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Bowen Zhang, Yidong Wang, Wenxin Hou, Hao Wu, Jindong Wang, Manabu Okumura, and Takahiro Shi-  nozaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Flexmatch: Boosting semi-supervised learning with curriculum pseudo labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Ad-  vances in Neural Information Processing Systems (NeurIPS), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Zhilu Zhang and Mert Rory Sabuncu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Generalized cross entropy loss for training deep neural networks with  noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Advances in Neural Information Processing Systems (NeurIPS), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Mingkai Zheng, Shan You, Lang Huang, Fei Wang, Chen Qian, and Chang Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Simmatch: Semi-supervised  learning with similarity matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Conference on Computer Vision and Pattern Recognition  (CVPR), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Barret Zoph, Golnaz Ghiasi, Tsung-Yi Lin, Yin Cui, Hanxiao Liu, Ekin Dogus Cubuk, and Quoc Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Rethinking  pre-training and self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' of Advances in Neural Information Processing Systems (NeurIPS),  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  15  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  A  ADDITIONAL RESULTS AND IMPLEMENTATION DETAILS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  SEMI-SUPERVISED DETAILS AND RESULTS  Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In this work, we follow the protocol of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  [2021] for fine-tuning from the first layer of the projector and initialize both the encoder and the first  layer of the projector with the parameters of the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We add the randomly initialized  classifier on top of the first layer of the projector (after the non-linearity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We train all the weights  (pretrained and classifier weights) using either 1% or 10% of the ImageNet-1k training data, and we  use the splits introduced in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020a] and used in all the methods to compare to Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  [2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Dwibedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Mitrovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Tomasev  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2022];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  At training time we randomly crop the image, resize it to 224 × 224, and then randomly apply  a horizontal flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' At test time we resize images to 256 pixels along the shorter side with bicubic  resampling and apply a 224 × 224 center crop to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Both at training and testing times we subtract  from the color channels the average channel value and divide it by the standard deviation of the  channel value (as computed on ImageNet-1k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We use a cross entropy loss and stochastic gradient descent with Nesterov momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  to fine-tune the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For both 1% and 10% settings, we train for 30 epochs and decay the  initial learning rate by a factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2 at 18 and 24 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Following the approach of Caron  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020], we pick different learning rates for the encoder (and the first projector layer)  and for the classifier weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We do not use any weight decay or other regularization tech-  niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We sweep over batch sizes values in {512, 1024, 2048}, encoder base learning rate val-  ues in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0035, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='001}, and linear layer base learning rate values in  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='025}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 6: Top-1 and Top-5 accuracies (in %), after semi-supervised fine-tuning with a fraction of  ImageNet labels, for a ResNet-50 encoder across a number of representation learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Method  Top-1  Top-5  1%  10%  1%  10%  Supervised [Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019]  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  Pseudo labels in classification:  MPL [Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9 Representation learning methods:  SimCLRv2 [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020b]  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4 SimCLRv2 + self distillation [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020b]  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5 CoMatch [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021a]  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  PAWS [Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021]  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5 DebiasPL [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022]  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8 SimMatch [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022]  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  SEMPPL (ours)  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  SimCLRv2 + Selective Kernels [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019b]  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  SEMPPL (ours) + Selective Kernels  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  Additional results and larger networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  When the architecture of the ResNet-50 is modified to  include selective kernels [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019b], we see significant gains in performance at the expense  of additional weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Our implementation of selective kernels is standard and follows rigorously Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2019b] for a total of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5 million weights instead of of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5 million for a regular ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Specifically, we use 2 channels, two convolution kernels of (3, 3) and (5, 5) with the latter imple-  mented as a (3, 3) dilated convolution with rate 2, and 32 grouped convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Unlike SimCLRv2  [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020b], we implement our group convolutions explicitly, and do not use the additional  ResNet-D architectural modification from He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' When using selective kernels our per-  formance after finetuning with 1% of labels is the same as that of SimCLRv2 after finetuning with  10% of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  16  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Additionally, in order to investigate the robustness and scalability of these results, we further test  the generality of SEMPPL by learning representations on larger (both deeper and wider) ResNet en-  coders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Table 1 testifies to SEMPPL outperforming the competing representation learning methods  across all the architectures, both in the 1% and the 10% labelled settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Also, as our flagship result  we reach 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4% top-1 accuracy on ResNet-200 2× with 10% of ImageNet-1k labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Just as in the  ResNet-50 1× case this figure is comparable with the fully supervised accuracy attained by historical  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1% top-1 is defined as in Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020] with standard RandAugment [Cubuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2020] data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' However it’s certainly a few percentage accuracy points away from results  obtained with optimal current training protocols [Bello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We also note that SEMPPL is  pre-trained for 300 epochs in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This, rather than the 1000 epochs used as standard by most  other representation learning methods, again compares with a typical figure of 200 epochs used in  supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Overall this hints at SEMPPL having achieved close to an order of magnitude  gain in label efficiency (compared to supervised learning) at a similar epochs budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Our final networks were optimized using tranches of between 128 (for a ResNet-50) and 512 (for  the largest ResNets) Cloud TPUv3s all during 300 epochs each irrespective of size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This required  around a day of computation time per run and tranche for a ResNet-50 on 128 devices, time which  scaled approximately linearly with the number of parameters on larger networks, depending on the  actual network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  ROBUSTNESS AND OOD GENERALIZATION  We test the robustness and out-of-distribution (OOD) generalization abilities of representations  learned via SEMPPL on several detasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We use ImageNetV2 [Recht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019] and ImageNet-  C [Hendrycks & Dietterich, 2019] datasets to evaluate robustness and the datasets ObjectNet [Barbu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019] and ImageNet-R [Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021] to evaluate the OOD generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  The ImageNetV2 dataset [Recht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019] has three sets of 10000 images (matched frequency  (MF), Threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7 (T-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7) and Top Images (TI)) that were collected to have a similar distribution  to the ImageNet test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' The ImageNet-C dataset [Hendrycks & Dietterich, 2019] consists of 15  synthetically generated corruptions of 5 different severities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' blur, noise) that are applied to  the ImageNet validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' The ImageNet-R dataset [Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021] consists of 30000  different renditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' paintings, cartoons) of 200 ImageNet classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' the aim of this dataset is to  test the generalization ability to different textures and other naturally occurring style changes that  are out-of-distribution to the ImageNet training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' The ObjectNet dataset [Barbu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019] has  18574 images from differing viewpoints and backgrounds compared to the ImageNet training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  On all datasets we evaluate the representations learned on a standard ResNet50 encoder under a  linear evaluation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We freeze the pretrained representations (no gradient updates) and train  a linear classifier on top of the output of the ResNet-50 encoder using the full labelled ImageNet  training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We perform the test evaluation zero-shot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e the above datasets are not seen during the  training of the representation or classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We provide a detailed breakdown across the different ImageNet-C corruptions in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Our  proposed approach SEMPPL outperforms both the supervised baseline, on 12 out of 15 corruptions,  as well as the competing semi-supervised representation learning model PAWS, on 12 out of 15  corruptions (notably, over all Blur, Weather and Digital corruptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 7: Top-1 accuracies (in %) for OOD generalisation on Gauss, Shot, Impulse, Blur, Weather,  and Digital corruption types of ImageNet-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Blur  Weather  Digital  Method  Gauss  Shot  Impulse  Defocus  Glass  Motion  Zoom  Snow  Frost  Fog  Bright  Contrast  Elastic  Pixel  JPEG  Supervised [Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019]  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  Semi-supervised representations:  PAWS [Assran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021]  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  SEMPPL(ours)  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
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+page_content='5  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  TRANSFER  To further evaluate the usefulness of the learned representations, we evaluate how well they transfer  across datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For this, we follow the standard evaluation protocol outlined in Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  17  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We evaluate SEMPPL across the linear evaluation protocol which consists of  freezing the encoder and only training a randomly initialized linear classifier on top of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In line with prior work [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Dwibedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021], we test SEMPPL  representations on the following datasets: Food101 [Bossard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2014], CIFAR10 [Krizhevsky  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2009], CIFAR100 [Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2009], Birdsnap [Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2014], SUN397 (split  1) [Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2010], DTD (split 1) [Cimpoi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2014], Cars [Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2013] Aircraft [Maji  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2013], Pets [Parkhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2012], Caltech101 [Fei-Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2004], and Flowers [Nilsback  & Zisserman, 2008], where we compare the downstream performance of SEMPPL to that of other  reported semi-supervised methods on 10% of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Across these datasets there are differences  in terms of metrics used for selecting the best hyper-parameters as well the reporting of the final  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In line with prior work [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Dwibedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2021], for  Food101 [Bossard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2014], CIFAR10 [Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2009], CIFAR100 [Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2009], Birdsnap [Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2014], SUN397 (split 1) [Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2010], DTD (split 1) [Cimpoi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2014], and Cars [Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2013] we report the Top-1 accuracy on the test set, and for  Aircraft [Maji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2013], Pets [Parkhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2012], Caltech101 [Fei-Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2004], and Flow-  ers [Nilsback & Zisserman, 2008] we report the mean per-class accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For DTD and SUN397 we  only use the first split of the 10 provided splits in the dataset as per Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  [2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Dwibedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In these experiments, models are initially trained on the training sets of the individual datasets, and  the validation sets are used to select the best hyperparameters from the executed hyperparameter  sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Once the best hyperparameters have been selected, the final models are trained on a merged  dataset containing both the training and the validation split and evaluated on the held-out test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  The final results of the transfer experiments are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' The performed hyperparameter  sweeps involved sweeping over the learning rates {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='001, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=',  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' }, batch sizes {128, 256, 512, 1024, 2048}, weight decay {1e−6, 1e−5, 1e−4, 1e−3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1},  warmup epochs {0, 10}, momentum {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='99}, Nesterov {True, False}, and the number of training  epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' For the linear transfer protocol we considered setting epochs among {20, 30, 60, 80, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Models were trained by stochastic gradient descent with momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  18  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  B  INVARIANCE REGULARIZATION  We define the short-hands  p(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za2,+  m,t ) =  ϕ(ˆza1  m , ˜za2,+  m,t )  ϕ(ˆza1  m , ˜za2,+  m,t ) + �  xn∈N (xm) ϕ(ˆza1  m , ˜za2  n,t)  (5)  and  p(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za2  m,t) =  ϕ(ˆza1  m , ˜za2  m,t)  ϕ(ˆza1  m , ˜za2  m,t) + �  xn∈N (xm) ϕ(ˆza1  m , ˜za2  n,t)  (6)  where N(xk) is the set of negatives, randomly uniformly sampled from the current batch, ˜za2  n,t =  gt(ft(xn)) is the target network projection of the negative sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ϕ(x1, x2) = τ · exp(⟨x1, x2⟩/τ)  is the scoring function, τ > 0 is a scalar temperature, and ⟨·, ·⟩ denotes the standard Euclidean dot  product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We can now rewrite the components of the overall loss  LSEMPPL = LAUGM + αLSEMPOS  (7)  as  LSEMPOS = −  B  �  m=1  log p(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za2,+  m,t )  (8)  and  LAUGM = −  B  �  m=1  log p(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za2  m,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  (9)  As discussed in the main text we add the invariance penalty introduced in Mitrovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2021] to  further increase the similarity between the anchor and positives and regularize the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  We add this invariance penalty both for augmentation positives and semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In particular,  we compute  Iaugm =DKL(p(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za2  m,t) ∥ p(ˆza2  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za1  m,t))  =sg[Ep(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='˜za2  m,t) log p(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za2  m,t)] − Ep(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='˜za2  m,t) log p(ˆza2  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za1  m,t)  and  Isempos =DKL(p(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za2,+  m,t ) ∥ p(ˆza2,+  m  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za1  m,t))  =sg[Ep(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='˜za2,+  m,t ) log p(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za2,+  m,t )] − Ep(ˆza1  m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='˜za2,+  m,t ) log p(ˆza2,+  m  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' ˜za1  m,t)  where sg denotes the stop-gradient operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Taking all this together, this gives the final form of  the loss as  LSEMPPL = c(LAUGM + αLSEMPOS) + λ(Iaugm + Isempos)  (10)  with λ the invariance scale and c is the contrastive scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We use λ = 5 and c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3 in all our  experiments irrespective of encoder size or training time as our method is robust to the choice of  these hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  19  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  C  PSEUDO-CODE OF SEMPPL  Listing 1 provides PyTorch-like pseudo-code for SEMPPL detailing how we compute pseudo-labels  and use them to select the additional semantic positives, which are then used in the contrastive loss,  along the augmentation positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  1  ’ ’ ’  2 k :  The number  of  neighbors  in kNN when computing  p s e u d o l a b e l s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  3  f o :  o n l i n e  network :  Encoder + comparison net .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  4  g t :  t a r g e t  network :  Encoder + comparison net .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  5 gamma :  Target EMA c o e f f i c i e n t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  6  n e :  Number of  n e g a t i v e s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  7 p m : Mask apply  p r o b a b i l i t y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  8  ’ ’ ’  9  f o r  i  in  range ( num large views ) :  10  queue i = queue .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' i n i t ( queue size ,  FIFO )  11  12  f o r x ,  y in  batch :  # Load  batch  of B samples  each  with  data x and  ( maybe )  l a b e l  y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  13  x m = mask background ( x )  14  f o r  i  in  range ( num large views ) :  15  #  S t o c h a s t i c a l l y  apply  background  removal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  16  x = B e r n o u l l i ( p m )  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' x m  :  x  17  # Create  an augmented  l a r g e  view .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  18  x l i  = augment ( c r o p l a r g e ( x ) )  19  o l i  = f o ( x l i )  20  t l  i  = g t ( x l i )  21  # Enqueue  the  l a b e l e d  images  in  the  batch .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  22  i f  y  i s  not None :  23  queue i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' enqueue ( ( t l i ,  y i ) )  24  25  f o r  i  in  range ( num small views ) :  26  x s i = augment ( c r o p s m a l l ( x ) )  27  # Small  views  only go  through  the  o n l i n e  network  28  o s i = f o ( x s i )  29  30  # Pseudo − l a b e l  computation  f o r  u n l a b e l l e d  examples .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  31  i f  y  i s  None :  # Missing  l a b e l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  32  votes = [ knn ( k ,  queue i ,  o l j )  f o r  i ,  j  in  a l l p a i r s ( num large views ) ]  33  y = mode ( votes )  34  35  l o s s = 0  36  # Compute  the  l o s s  between  a l l  the  p a i r s  of  l a r g e  views .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  37  f o r  i  in  range ( num large views ) :  38  f o r  j  in  range ( num large views ) :  39  l o s s +=  c o n t r a s t i v e l o s s ( o l i ,  t l j ,  n e )  # Augmentation  p o s i t i v e s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  40  f o r  in  range ( n u m s e m a n t i c p o s i t i v e s ) :  41  # Sample  semantic  p o s i t i v e s  from  the  queue ,  and add  to  the  l o s s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  42  z = sample ( queue j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' f i l t e r ( y ) )  43  l o s s +=  c o n t r a s t i v e l o s s ( o l i ,  z ,  n e )  44  45  # Compute  the  l o s s  between  the  small  and  l a r g e  views .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  46  f o r  i  in  range ( num small views ) :  47  f o r  j  in  range ( num large views ) :  48  l o s s +=  c o n t r a s t i v e l o s s ( os i ,  t l j ,  n e )  # Augmentation  p o s i t i v e s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  49  f o r  in  range ( n u m s e m a n t i c p o s i t i v e s ) :  50  # Sample  semantic  p o s i t i v e s  from  the  queue ,  and add  to  the  l o s s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  51  z = sample ( queue j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' f i l t e r ( y ) )  52  l o s s +=  c o n t r a s t i v e l o s s ( o l i ,  z ,  n e )  53  54  l o s s  /=  ( ( num large views + num small views )  55  num large views * (1 + n u m s e m a n t i c p o s i t i v e s ) )  56  57  # Compute  the  g r a d i e n t s ,  and  update  the  o n l i n e  and  t a r g e t  network .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  58  l o s s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' backward ( )  59  update ( f o )  60  g t = gamma *  g t + (1 − gamma) *  f o  Listing 1 Pseudo-code for SEMPPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  20  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  D  ANALYSIS  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  IMPLEMENTATION DETAILS  We perform all the ablation experiments using 10% of labelled data and train a standard ResNet-  50 encoder with SEMPPL for 100 epochs (except in the training duration ablation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We report the  top-1 accuracies on the ImageNet test set after fine-tuning from the first layer of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' As  in Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020] and for the main results in this paper, we use multi-layer perceptrons for the  projector and predictor with 2 linear layers—the first one followed by batch normalization [Ioffe  & Szegedy, 2015] and rectified linear activation with output sizes 4096 and 256 for the two layers  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We use the same augmentations as for the experiments in the main paper—the stan-  dard SIMCLR augmentations [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020a] and the RELICV2 multi-crop and saliency-based  masking [Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Following the hyperparameter settings of the main results, we use  batch size: B = 4096  queue capacity C = 20B (unless specifically ablated)  number of nearest neighbours k = 1 (unless specifically ablated)  view voting is used (unless specifically ablated)  weight decay: 1e − 6 (exclude biases and batch normalization parameters)  optimizer: LARS (exclude biases and batch normalization parameters from adaptation)  base learning rate: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3 (scaled linearly with batch size [Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2017])  warm-up: 10 epochs  cosine decay schedule for learning rate  exponential moving average parameter: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='996  views: 4 large views of size 224 × 224 and 2 small views of size 96 × 96  temperature: τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  number of semantic positives: 3 (unless specifically ablated)  10 randomly subsampled negatives per anchor  α = 1/5 (unless specifically ablated), λ = 5 and c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  21  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  ADDITIONAL ANALYSES  Number of semantic positives  We study the effect of varying the number of semantic positives in  SEMPPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Table 8a shows that increasing this number from 1 to 3 only has an effect on the amount  of correctly predicted pseudo-labels, but no effect on downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' On the other hand,  using 5 or 10 semantic positives significantly improves performance and also yields much more  accurate pseudo-labels prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Training duration  Next, we vary the length of training representations and examine downstream  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' As can be seen from Table 8b, both the proportion of correctly predicted pseudo-  labels and downstream performance improve with longer training up to 300 epochs but decrease if  we continue training up to 500 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This indicates with training longer than 300 epochs SEMPPL  is starting to overfit, an observation consistent with phenomena reported elsewhere in the literature  involving noisy labels [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Zhang &  Sabuncu, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', 2019b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 8: Top-1 accuracy (in %) on the ImageNet-1k test set, and accuracy (in %) of correctly pre-  dicted pseudo-labels at the end of training for semantic positives and training length experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' positives  Top-1  Pseudo-label acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  1  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  2  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  3  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  69  5  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  10  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  (a) Varying the number of semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Training time (epochs)  Top-1  Pseudo-label acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  100  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  200  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  300  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='7  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  500  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  (b) Varying the length of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  View voting  SEMPPL generates multiple views from a single instance image in order to learn rep-  resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Those different views can be leveraged towards better pseudo-labels prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Rather  than only picking one randomly selected data view to compute a single pseudo-label, we perform  majority voting over (noisy) pseudo-labels computed from all available image views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Specifically,  we compare the online predictor embedding of one view with the queue of the target projector em-  beddings of the same data view from previous batches in the first setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' in the second setting we  compare the online predictor embedding of each view with the queue of the target projector embed-  dings of each other data view from previous batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Since SEMPPL relies on 4 large views, this yields up to 16 different pairs of views to compare  and compute pseudo-labels from, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' we get 16 pseudo-label predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' this setting we call view  voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Table 9 shows that using all available views to compute pseudo-labels significantly increases  pseudo-labels accuracy which in turn significantly improves downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 9: Top-1 accuracy (in %) on the ImageNet-1k test set, and accuracy (in %) of correctly pre-  dicted pseudo-labels at the end of training for view voting experiments (using all views to compute  pseudo-labels vs using just a single view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  View voting  Top-1  Pseudo-label acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  On  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  Off  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  Number of nearest neighbours  In order to compute pseudo-labels we use k-nearest neighbour  lookup on the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' While in the main results Section 3 we consistently assume k = 1 here we  ablate the effect of varying k on downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' As can be seen Table 10a, increasing  k actually leads to a small decrease in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' This is attributable to the decrease in the  proportion of correctly predicted pseudo-labels as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  22  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Queue length  How long should the queue of target projector embeddings for computing pseudo-  labels be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' As the queue grows in size, it contains increasingly stale embeddings and threatens to  hamper the accuracy of predicted pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' On the other hand, increasing queue size increases  the amount and diversity of labelled embeddings available which we would expect to be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Those two opposing forces—staleness and increasing coverage—govern the accuracy with which we  can correctly predict pseudo-labels in turn directly affecting the selection of semantic positives and  their quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We resolve this ambiguity empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' As seen in Table 10b, increasing the coverage  and diversity of labelled embeddings has a strong positive effect on representation learning and  downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Staleness of embeddings is far less of a problem at practical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=', not  very large) queue sizes, showing diversity and coverage to be the dominant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 10: Top-1 accuracy (in %) on the ImageNet-1k test set, and accuracy (in %) of correctly  predicted pseudo-labels at the end of training for k-NN and queue length experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  k-nn  Top-1  Pseudo-label acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  1  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  2  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  3  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  5  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  10  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  (a) Varying the number of nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Queue size (C)  Top-1  Pseudo-label acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  4000  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  8000  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  12000  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  20000  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='3  40000  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  80000  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='9  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  200000  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='8  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5  (b) Varying queue size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  The effect of the loss weight α  We use α in Equation 4 to weight the contribution of the loss  coming from semantic positives against the loss coming from augmentation positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In SEMPPL  we use multiple semantic positives for learning and thus we need to adjust α to appropriately weigh  the contribution of the individual semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In our main experiments, we use 3 semantic  positives and for this reason α is not equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In Table 11 we vary α from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0, with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0 effectively recovering RELICV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' The results  indicate that the SEMPPL loss notably improves over RELICV2, but that the exact choice of α  should be treated as a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Note that the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2 is very close to the 1  3 which is exactly  1/number of semantic positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Thus, the optimal choice of α is very close to that fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Table 11: Top-1 accuracy (in %) on the ImageNet-1k test set for varied values of the loss weight α  Loss weight (α)  Top-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0 (RELICV2)  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='6  23  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  E  AUGMENTATIONS  In this work, we follow the established data augmentations protocols and pipelines of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  [2020a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Mitrovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Specifi-  cally, SEMPPL uses a set of augmentations to generate different views of the original image which  has three channels, red r, green g and blue b with r, g, b ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  The augmentations we use are generated by applying the following sequence of operations in the  following order  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Crop the image: Randomly select a patch of the image, between a minimum and maximum  crop area of the image, with aspect ratio sampled log-uniformly in [3/4, 4/3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Upscale the  patch, via bicubic interpolation, to a square image of size s × s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Flip the image horizontally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Colour jitter: randomly adjust brightness, contrast, saturation and hue of the image, in a  random order, uniformly by a value in [−a, a] where a is the maximum adjustment (speci-  fied below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Grayscale the image, such that the channels are combined into one channel with value  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2989r + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5870g + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1140b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Randomly blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Apply a 23 × 23 Gaussian kernel with standard deviation sampled uni-  formly in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Randomly solarize: threshold each channel value such that all values less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5 are  replaced by 0 and all values above or equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='5 are replaced with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Apart from the initial step of image cropping, each subsequent step is applied with a certain proba-  bility to generate the augmented view of the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' These probabilities and other augmen-  tation parameters are given in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' SEMPPL uses 4 large views of size 224 × 224 pixels and 2  small views of 96 × 96 pixels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' to get the first and third large views and the first small view we use  the parameters listed below for odd views, while for the second and fourth large view and the second  small view we use the parameters for even views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Note that these are the same augmentations used  also in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Mitrovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  [2020b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  In addition to these augmentations, we also randomly apply the saliency masking augmentation  proposed in Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2022] which enables us to remove a large part of the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We  follow the protocol described in Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2022] for computing the saliency masking for an  image and we apply this augmentation with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1 to the 4 large views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' In keeping with  Tomasev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' [2022], we fill out the removed background of the image with homogeneous grayscale  noise with the grayscale level randomly sampled for each view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' We only apply the saliency masking  when the remaining foreground covers at least 20% of the total image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  24  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  Parameter  Even views  Odd views  Probability of randomly cropping  50%  50%  Probability of horizontal flip  50%  50%  Probability of colour jittering  80%  80%  Probability of grayscaling  20%  20%  Probability of blurring  100%  10%  Probability of solarization  0%  20%  Maximum adjustment a of brightness  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  Maximum adjustment a of contrast  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='4  Maximum adjustment a of saturation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='2  Maximum adjustment a of hue  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='1  Crop size s  224  96 (small), 224 (large)  Crop minimum area  8%  5% (small), 14% (large)  Crop maximum area  100%  14% (small), 100% (large)  Table 12: Parameters of data augmentation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content=' Small/large indicates small or large crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
+page_content='  25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQflQ1W/content/2301.05158v1.pdf'}
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+Manipulation of nonequilibrium spin dynamics of an ultracold gas in a moving optical
+lattice
+Z. N. Hardesty-Shaw,1 Q. Guan,2, 3, 4 J. O. Austin,1 D. Blume,2, 3 R. J. Lewis-Swan,2, 3, ∗ and Y. Liu1, †
+1Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
+2Homer L. Dodge Department of Physics and Astronomy,
+The University of Oklahoma, Norman, Oklahoma 73019, USA
+3Center for Quantum Research and Technology, The University of Oklahoma, Norman, Oklahoma 73019, USA
+4Department of Physics and Astronomy, Washington State University, Pullman, WA 99164, USA
+(Dated: January 10, 2023)
+The isolation and control of disparate degrees of freedom underpin quantum simulators.
+We
+advance the programmability of cold atom quantum simulators with a first realization of the dynamic
+interplay of spatial and spin degrees of freedom. We experimentally demonstrate that violent spatial
+evolutions tune long-lived coherent spin dynamics and develop a model of quantum spin-mixing
+incorporating the spatial evolution via time-dependent spin-spin interactions. Our results open new
+paths towards the simulation of quantum spin models with tunable interactions via tailored spatial
+dynamics.
+Introduction – Ultracold quantum gases that feature spa-
+tial and spin degrees of freedom offer a powerful plat-
+form for simulating quantum magnetism in controlled,
+isolated settings [1–5]. When combined with optical lat-
+tices, these simulation capabilities are exemplified by ex-
+perimental studies featuring tunable dimensionality and
+filling factors [6–9].
+Possessing long coherence times,
+these systems also provide an ideal platform for study-
+ing out-of-equilibrium phenomena such as spin-mixing
+[10–13], transport [14, 15], dynamical phases of matter
+[16], and critical dynamics across quantum phase tran-
+sitions [7, 8, 17]. Simultaneously, advances in spin- and
+spatially-resolved probes [7, 9, 18, 19] and the control of
+time- and spin-dependent lattice potentials are opening
+up new opportunities, including the study of multi-state
+tunneling physics [20] and driven-dissipative phases [21],
+in the presence of the spin degree of freedom.
+Typically, the energy scales of the spin and spatial de-
+grees of freedom are disparate. This has been exploited
+to obtain a reduced description of the spin dynamics that
+depends only on a spatial profile that remains frozen due
+to, e.g., a strong confining potential [11, 16, 22–28]. In
+the context of spinor Bose-Einstein condensates (BECs),
+this decoupled regime has received significant attention
+[29–31] and, amongst other applications, has been uti-
+lized to generate entangled states in the highly control-
+lable spin degree of freedom [32–37], which can also be
+mapped to the motional degrees of freedom [38–40].
+In contrast, the interplay of spatial and spin degrees of
+freedom remains largely unexplored, and, although typi-
+cally weak, can provide a powerful avenue for controlling
+the spin dynamics through tailored dynamical manipula-
+tion of the spatial properties of the gas [41–45]. We inves-
+tigate this interplay, providing a first example of how spa-
+tial degrees of freedom can be utilized to manipulate the
+spin dynamics. We experimentally observe that a one-
+dimensional (1D) moving lattice, combined with a skew
+optical dipole trap (ODT), induces violent transient spa-
+tial motion which is nevertheless accompanied by long-
+lived spin-mixing dynamics. We develop a theoretical un-
+derstanding of these observations based on a dynamical
+single spatial-mode approximation (dSMA), which leads
+to an effective spin model with a time-dependent spin-
+spin interaction coefficient that depends on the tempo-
+ral evolution of the BEC density profile. Experimental
+observations — including a robust critical regime featur-
+ing divergent timescales for the spin dynamics, which is
+tuned by the applied moving optical lattice and associ-
+ated spatial motion — are qualitatively described by our
+model.
+Our results open the way for the exploitation of classi-
+cal spatial dynamics for simulating many-body quantum
+spin dynamics with highly tunable, time-dependent in-
+teractions [46, 47], thereby enhancing the class of quan-
+tum spin models accessible in spinor BECs. In addition,
+our findings imply spatial dynamics can provide new con-
+trol knobs for the nonequilibrium generation of entangled
+spin states for, e.g., quantum-enhanced sensing [38].
+Experimental setup – Each experimental cycle begins
+with a sodium spin-1 BEC at quadratic Zeeman energy
+q in an ODT (see Supplemental Materials [48]). A key
+feature of spinor BECs is their spin degree of freedom
+characterized by the spin-dependent interaction coeffi-
+cient c2. Spin-mixing and other nonequilibrium phenom-
+ena driven by a static c2 have been studied in various
+contexts [7, 8, 11, 23, 24, 28, 31]. Here, in contrast, we
+demonstrate that c2 can be tuned dynamically by uti-
+lizing a moving lattice to change the BEC’s spatial den-
+sity profile. We construct a 1D moving lattice with two
+nearly orthogonal optical beams whose frequency differ-
+ence ∆f determines the moving lattice speed [48]. Our
+initial BEC has a fractional population ρ0 = 0.5 of atoms
+in the |S = 1, m = 0⟩ state and zero magnetization (equal
+populations in the |S = 1, m = ±1⟩ states). The BEC
+is then adiabatically loaded into the lattice, which is sta-
+tionary at time t = 0 and quenched to the desired speed
+arXiv:2301.02707v1  [cond-mat.quant-gas]  6 Jan 2023
+
+2
+0.6
+0.4
+0.2
+ρ0
+(a)
+(b)
+(c)
+40
+35
+30
+25
+20
+15
+10
+5
+Spin oscillation period T (ms)
+q/h (Hz)
+Δf = 4.6 ER/h
+ Markers: Experiment
+ Line: Theory
+70
+60
+50
+40
+30
+20
+10
+(g)
+c2(t) / c2(0)
+(h)
+0
+BEC
+tF
+t1
+0
+Time
+(i)
+tF (ms)
+tF (ms)
+tF (ms)
+tF (ms)
+30
+20
+10
+Δf = 0
+Δf = 4.6 ER/h
+1.2
+0.8
+0.4
+0.0
+Δf
+0
+uL
+ρ0
+30
+20
+10
+30
+20
+10
+30
+20
+10
+0.6
+0.4
+0.2
+(d)
+(e)
+(f)
+q < q*
+q ≈ q*
+q > q*
+FIG. 1. (a)-(f) Exemplary time traces of ρ0 for spinor BECs. Panels (a)-(c) show experimental results for ρ0 (red markers)
+at q/h = 15 Hz (a), 25 Hz (b), and 65 Hz (c) as well as sSMA predictions (dotted lines) with c2,fit/h = 23.7(1)Hz extracted
+from Fig. 1(g). Panel (a) compares static (black) and moving (red) lattice results. Solid lines are sinusoidal fits to guide the
+eye. Panels (d)-(f) compare predicted ρ0 for q/h = 15 Hz (d), 22 Hz (e), and 65 Hz (f) from 1D GP simulations (solid lines) to
+sSMA (dotted lines) with c2 obtained through fits to the GP data [48], and dSMA (dashed lines) with c2(t) obtained from GP
+data. The chosen q values exemplify the interaction dominated (q < q∗) and Zeeman (q > q∗) regimes separated by the critical
+region q ≈ q∗. (g) Observed T (markers) versus q fit by analytical sSMA expressions (solid line) with the fitting parameter
+c2,fit/h = 23.7(1)Hz [48]. (h) Red (black) lines are evolution of c2(t)/c2(0) for q/h = 15 Hz obtained from 1D GP simulations
+of the moving (static) lattice. All moving lattice data use a lattice depth uL = 2.3ER, t1 = 1.43ms, and fixed ∆f = 4.6ER/h.
+(i) Timeline of the lattice depth (lower panel) and moving lattice speed (upper panel).
+for t > 0 (see Fig. 1(i)). We study the ensuing non-trivial
+spin (Fig. 1) and spatial (Fig. 2) dynamics of the atoms
+by holding them in the moving lattice for a time tF before
+releasing them for ballistic expansion and imaging [9, 48].
+Spin dynamics – We first study the nonequilibrium spin
+dynamics generated by experimental sequences (Fig. 1(i)
+∆f = 4.6ER/h) which near-resonantly couple the initial
+stationary BEC with the p = 2ℏkL momentum state.
+Here, ER is the recoil energy, h (ℏ) is the (reduced)
+Planck constant, and kL is the lattice vector [6, 48].
+Spin-mixing oscillations, arising from coherent intercon-
+versions among two m = 0 atoms and a pair of atoms
+in the m = ±1 Zeeman states [3], constitute a useful
+tool in understanding the spin dynamics. The periods
+T of these oscillations are determined by the competi-
+tion between c2 and q, illustrated by typical examples of
+the interaction dominated region (Fig. 1(a)) and Zeeman
+dominated region (Fig. 1(c)).
+We also see convincing
+experimental signatures of a critical separatrix regime
+near q = q∗ where T diverges (see Fig. 1(b)).
+These
+observations are qualitatively consistent with expecta-
+tions based on established theory formulations referred
+to as a static single spatial-mode approximation (sSMA)
+in this paper, which assumes c2 is time-independent
+[3, 7, 8, 11, 23, 24, 28, 31].
+Fig. 1(g) shows the ob-
+served T can be used to estimate the effective static spin-
+spin interaction c2,fit/h = 23.7(1) Hz as sSMA predicts
+q∗ ≈ c2,fit for our initial state [6, 11]. However, direct
+comparisons to the sSMA predicted time traces (dotted
+lines in Figs. 1(a)-(c)) demonstrate that the model fails
+to capture experimentally observed features such as the
+damping of the oscillation amplitude and the drift of the
+oscillations. Another notable observation that cannot be
+explained by sSMA is the shift of the separatrix location
+induced by the moving lattice, as shown by a compari-
+son between static (∆f = 0) and moving lattice results
+in Fig. 1(a).1 These experimental observations suggest
+sSMA provides an incomplete description of our system.
+Theoretical model – To explain the sSMA’s shortcomings,
+we develop the following dSMA model which assumes c2
+varies with time and describes our system with the spin
+Hamiltonian [6, 22, 39, 48],
+ˆHeff(t) = c2(t)
+2N
+ˆS · ˆS + q(ˆn1 + ˆn−1).
+(1)
+Here, ˆS = �N
+i=1 ˆsi where ˆsi denotes the spin-1 operator
+for the ith of the total N atoms and ˆnm is the number
+operator for the Zeeman state m. The time-dependent
+c2(t) arises from the temporal evolution of the BEC’s
+spatial density profile and, in turn, modulation of the
+effective interaction strength of the spin model, driven
+by the moving lattice. Formally, c2(t) emerges from the
+time-dependence of the Gross-Pitaevskii (GP) orbitals
+ψm(r, t) that describe the spatial dynamics of the mth
+Zeeman component.
+By assuming the spatial density
+1 Fig. 1(a) indicates that for static lattices at an identical q in the
+interaction dominated regime T is smaller and thus, using the
+same sSMA interpretation, would lie on a curve shifted to higher
+q (indicating a larger characteristic c2) relative to the moving
+lattice data shown in Fig. 1(g).
+
+3
+-4
+-2
+0
+2
+-4
+-2
+0
+2
+-4
+-2
+0
+2
+-4
+-2
+0
+2
+tF = 1.22 ms
+tF = 3.72 ms
+L
+R1
+R2
+px/ħkL
+pz/ħkL
+pz/ħkL
+(a)
+-4
+-2
+0
+2
+-4
+-2
+0
+2
+tF = 2.22 ms
+L
+R1
+pz/ħkL
+(b)
+(c)
+Experimental
+Theoretical
+-4
+-2
+0
+2
+-4
+-2
+0
+2
+tF = 1.22 ms
+-4
+-2
+0
+2
+-4
+-2
+0
+2
+tF = 2.22 ms
+L
+R1
+-4
+-2
+0
+2
+-4
+-2
+0
+2
+px/ħkL
+tF = 3.72 ms
+L
+R1
+R2
+(d)
+(e)
+(f)
+0.6
+0.4
+0.2
+0.0
+0.6
+0.4
+0.2
+0.0
+0.8
+0.4
+0.0
+Optical density
+Optical density
+Optical density
+FIG. 2. Time-of-flight snapshots of 2D integrated momentum
+distribution with ∆f = 4.6ER/h, uL = 1.2ER, t1 = 0.72 ms,
+and tF = 1.22 ms (a), 2.22 ms (b), and 3.72 ms (c) in a shal-
+low ODT with LODT,z = 20 µm [48] at q/h = 42Hz. (d)-(f)
+Analogous theoretical results of in-situ momentum distribu-
+tions based on 2D GP simulations [48]. The colorbar scale
+indicates the optical density of the images for each row.
+profiles |ψm(r, t)|2 for the different m states are the same
+but time-dependent (as we find in theory calculations dis-
+cussed in more detail later) it is implicit that while the
+spatial degree of freedom may contribute to the spin dy-
+namics the converse is not true, i.e., the spin degree of
+freedom does not feed back onto the evolution of the spa-
+tial profile. Setting |ψm(r, t)|2 ∝ |φ(r, t)|2 and integrat-
+ing out the spatial degrees of freedom leads to Eq. (1)
+with c2(t) ∝ (N − 1)
+�
+d3r |φ(r, t)|4 [48]. A key result
+of the presented experiment-theory work is that dSMA
+enables a transparent understanding of the non-trivial
+spin dynamics triggered by violent spatial evolution of
+the BEC that occurs on faster characteristic time scales
+than the spin dynamics.
+Interplay of spin and spatial dynamics – To illustrate
+the typical spatial dynamics driving the spin-mixing ob-
+served in Fig. 1, we show experimental BEC momentum
+distributions in Figs. 2(a)-(c), which capture the emer-
+gence of violent spatial motion due to the interplay of
+momentum kicks generated by the moving lattice and
+the shallow ODT harmonic confinement on a timescale
+significantly shorter than the observed spin dynamics.
+The rapid appearance of many of discrete momentum
+peaks and associated spatial dynamics shown in Fig. 2
+simultaneously suggests that the deviations from sSMA
+predictions in Figs. 1(a)-(c) are to be expected but also
+entices us to reconcile elements of the good qualitative
+agreement between the experimental data and sSMA cal-
+culations in Fig. 1(g). We note that the creation of the
+discrete momentum peaks is a coherent process and does
+not conflict with the assumption of the single spatial-
+mode approximation. In fact, the Supplemental Materi-
+als [48] show that Figs. 1(a)-(c) are replicated if different
+momentum components are used to construct ρ0, thereby
+providing experimental support for dSMA.
+To gain further insight, we use numerical GP calcu-
+lations [48], which provides a mean-field description of
+the full spinor BEC dynamics including both spatial and
+spin degrees of freedom. The complexity of the experi-
+mental system, in particular the disparate timescales of
+spin and spatial dynamics, precludes a full quantitative
+3D GP treatment. Instead, we use a reduced dimension-
+ality 1D spinor GP calculation with parameters tuned to
+capture essential aspects of the experimental 3D system.
+This simplified treatment enables us to develop a qual-
+itative understanding of the experimental results [48].
+The GP simulations (solid lines in Figs. 1(d)-(f)) quali-
+tatively replicate the coherent spin dynamics, including
+a diverging oscillation period for q ≈ q∗ (Fig. 1(e)) and
+robust harmonic oscillations for q < q∗ (Fig. 1(d)) and
+q > q∗ (Fig. 1(f)) with damped amplitude and drift-
+ing mean value, respectively.
+We note that the up-
+ward drifting mean value in the experimental data for
+q > q∗ (Fig. 1(c)), notably not captured by the 1D GP
+theory, may be induced by a subtle resonance mecha-
+nism between the spin and spatial dynamics [31] that
+depends sensitively on the dimensionality of the system.
+Higher dimensionality calculations will be presented else-
+where [49].
+We use GP calculations to make a more fine-grained
+theoretical investigation of the relationship between the
+spatial and spin dynamics and, in particular, certify that
+while the BEC undergoes violent motion on fast time-
+scales: i) all Zeeman components are described by a com-
+mon spatial density profile |φ(r, t)|2, and ii) the sophis-
+ticated interplay of the moving lattice and ODT drive
+complex dynamics of c2(t). These observations lead us
+to self-consistently compare the 1D GP results to dSMA
+predictions, i.e., mean-field dynamics based on Eq. (1)
+with c2(t) computed via the GP density |φ(r, t)|2 [48].
+The GP and dSMA time traces in Figs. 1(d)-(f) show
+excellent agreement with each other.
+This implicitly
+demonstrates the spin dynamics do not feed back into the
+spatial evolution, in agreement with expectations based
+on the disparity of energy scales. The dSMA results also
+show significantly improved qualitative agreements with
+experimental data than sSMA results, providing further
+support for our dSMA model.
+GP calculations of the spatial dynamics in the pres-
+ence of a moving lattice lead to appreciable variation of
+
+4
+c2(t) (red line in Fig. 1(h)) at t ≲ 3 ms. Over longer
+timescales c2(t) features an overall decrease, which we un-
+derstand as being driven by the relaxation of the spatial
+density profile as the BEC fractures into many momen-
+tum components. This behavior is in stark contrast with
+predictions for a static lattice (black line in Fig. 1(h))
+that indicate c2(t) instead fluctuates around a well de-
+fined time-averaged value with small oscillations due to
+excitations created during the loading phase. After the
+initial transient behavior in the moving lattice, our re-
+sults (Fig. 1(h)) indicate that the decay of c2(t) is slow
+relative to the characteristic time of the spin dynamics,
+and therefore observables such as the spin oscillation pe-
+riod are captured by sSMA [48]. Our calculations show
+that the precise details of the spin dynamics (e.g., quali-
+tative features including damping of the spin oscillations
+in Fig. 1(a)) can depend greatly on the temporal vari-
+ation of c2(t) and hence a more rigorous description is
+provided by the GP and dSMA models.
+Single-particle resonances – The precise evolution of
+the spatial density profile, seen in Figs. 2(a)-(c) as a
+menagerie of seemingly irregularly distributed wavepack-
+ets in momentum space, can be understood with the aid
+of GP simulations (Figs. 2(d)-(f)).
+To more precisely
+capture the impact of gravity and the finite trap depth,
+the simulations in Fig. 2 are performed using an axially
+symmetric 2D setup. This numerical treatment is feasi-
+ble due to the relatively short time scales over which the
+spatial dynamics are studied in detail. Using 2D simula-
+tions also enables us to capture key details of the momen-
+tum kicks that 1D simulations, such as those employed
+in Fig. 1, miss. At short times, the lattice kicks atoms
+from the initial BEC with momentum p = 0 to the near-
+resonant state with momentum p = 2ℏkL (Figs. 2(a)
+and 2(d)). Subsequently, an additional momentum com-
+ponent, referred to as the lead peak (L-peak), splits from
+the p = 2ℏkL peak and decelerates as it travels away
+from the minima of the relatively shallow ODT poten-
+tial. As the L-peak slows, its momentum evolves until
+it sweeps through the approximate resonance region cen-
+tered on the line pz ≈ 0.86px + 0.20ℏkL (see the gray
+shaded region in Fig. 3(b) and Supplemental Materi-
+als [48]), where the lattice couples two nearly resonant
+momentum states, corresponding to the L-peak and a
+new peak labelled R1, that are separated by another
+2ℏkL momentum kick (Figs. 2(b) and 2(e)). Similarly,
+the R1-peak also decelerates until it crosses the resonance
+region and the lattice generates a new peak labelled R2
+(Figs. 2(c) and 2(f)).
+This pattern continues and the
+BEC fractures into a multitude of momentum states.
+Figure 3(a) confirms our prior analysis, which suggests
+a dependence on both the confining ODT potential and
+moving lattice, by tracking the position of the L- and
+R1-peaks in time and momentum space in a compressed
+ODT. The position of the L-peak initially follows a tra-
+jectory consistent with a Lissajous curve derived from a
+-1
+0
+1
+2
+1
+0
+-1
+pz/ħkL
+px/ħkL
+(b)
+Red: R1-Peak
+Blue: L-Peak
+-1
+0
+1
+pz/ħkL
+px/ħkL
+-1
+0
+1
+1
+2
+3
+4
+tF (ms)
+5
+(a)
+FIG. 3. (a) Time evolution trajectories of the mean position
+of the L-peak (circles) and R1-peak (squares) in the pz-px
+plane taken in a compressed ODT with LODT,z = 33 µm [48],
+extracted from experimental images similar to those shown in
+Figs. 2(a)-(c) at q/h = 42Hz. With increasing time, markers
+appear larger due to movement in the px axis.
+Blue (red)
+solid lines are the L-peak (R1-peak) trajectories correspond-
+ing to Lissajous curves (see text) [48].
+(b) Trajectories of
+panel (a) projected into the pz-px plane. The gray shaded
+region encompasses an approximate resonance region where
+decelerated atoms are kicked by the lattice (to create, e.g.,
+the R1-peak) [48].
+simplified classical treatment of a single particle initially
+moving with momentum 2ℏkL in the ODT (see Supple-
+mental Materials [48]). A sudden momentum kick im-
+parted by the lattice couples the L- and R1-peaks as the
+former crosses the approximate resonance region, shown
+by the gray region in Fig. 3(b) [48]. As time increases
+(i.e., deeper into the trajectory of each peak), the agree-
+ment between the experimentally observed and the the-
+oretically predicted trajectories for the L- and R1-peaks
+deteriorates, as shown near the end of the time axis in
+Fig. 3(a). The deterioration is most clearly seen when
+comparing an ideal (effectively LODT,i = ∞) harmonic
+trap with our typical shallow ODT depth that features a
+comparatively small curvature (see Supplemental Mate-
+rials [48]). In Fig. 3, we use compressed ODTs instead of
+shallow ODTs, trading reduced visibility and condensate
+
+2
+3
+55
+fraction, to better illustrate the bending of the trajectory
+in the px − pz plane [48].
+Conclusion – Our results open a new direction for the
+exploitation of spatial dynamics as a control knob for
+quantum simulations of many-body quantum spin models
+with tunable, time-dependent interactions [46, 47]. Tai-
+lored modulation of the spatial profile could be used to
+control the precise time dependence of the spin-spin inter-
+actions and realize Floquet-driven spin dynamics [59, 60].
+This can have immediate applications for the dynam-
+ical generation of entangled spin states for quantum-
+enhanced sensing [36, 39, 61–63].
+The observed short
+time dynamics also raise intriguing questions about equi-
+libration of spinor BECs.
+For example, future studies
+might utilize the time dependence of c2(t) to force these
+systems along different equilibration trajectories.
+Acknowledgements – D. B. acknowledges support by the
+National Science Foundation (NSF) through grant No.
+PHY-2110158. R. J. L-S. acknowledges support by NSF
+through Grant No. PHY-2110052 and the Dodge Family
+College of Arts and Sciences at the University of Okla-
+homa (OU). Z. N. H-S., J. O. A., and Y. L. acknowledge
+support by the Noble Foundation and the NSF through
+Grant Nos. PHY-1912575 and PHY-2207777. This work
+used the OU Supercomputing Center for Education and
+Research (OSCER).
+∗ lewisswan@ou.edu
+† yingmei.liu@okstate.edu
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+Supplemental Material: Manipulation of nonequilibrium spin dynamics of an ultracold
+gas in a moving optical lattice
+Z. N. Hardesty-Shaw,1 Q. Guan,2, 3, 4 J. O. Austin,1 D. Blume,2, 3 R. J. Lewis-Swan,2, 3 and Y. Liu1
+1Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
+2Homer L. Dodge Department of Physics and Astronomy,
+The University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
+3Center for Quantum Research and Technology, The University of Oklahoma,
+440 W. Brooks Street, Norman, Oklahoma 73019, USA
+4Department of Physics and Astronomy, Washington State University, Pullman, WA 99164, USA
+(Dated: January 10, 2023)
+arXiv:2301.02707v1  [cond-mat.quant-gas]  6 Jan 2023
+
+S2
+I.
+EXPERIMENTAL DETAILS
+A.
+Experimental sequence
+Our experimental sequences start by creating a S = 1 spinor BEC of up to 105 sodium (23Na) atoms in a crossed,
+anisotropic harmonic optical dipole trap (ODT) at a particular quadratic Zeeman shift q tuned by external magnetic
+fields, similar to our previous work [1–3].
+We apply a resonant radio-frequency (RF) pulse to prepare an initial
+state with fractional population ρ0 = ⟨ˆn0⟩/N = 0.5 in the |S = 1, m = 0⟩ state and zero magnetization, M =
+⟨ˆn1 − ˆn−1⟩/N = 0. We then adiabatically load the initial state into a one-dimensional moving optical lattice. The
+lattice is constructed from two nearly orthogonal lattice beams originating from a single-mode laser with wavelength
+1064 nm and characterized by the potential Vlat(r, t) = uL cos2[kL ·r−2π∆f(t)t/4], with lattice vector kL oriented at
+approximately 40◦ from the z-axis defined by gravity. The resulting standing wave potential has a lattice spacing of
+λL/2 = 0.81µm. The time-dependent frequency difference ∆f(t) = |fH −fV |, where fH and fV are the corresponding
+lattice beam frequencies, determines the velocity v of the moving lattice, v = λL(fH − fV ).
+The velocity v is
+manipulated via a linear ramping rate α = h(∆f(t2) − ∆f(t1))
+t2 − t1
+such that when v < 0 (v > 0) the atoms move in
+the p = 2ℏkL (p = −2ℏkL) direction. In the main text, the data in Fig. 1 were taken with positive velocities, while
+the data in Figs. 2 and 3 were taken with negative velocities. The value of ∆f is initially set to zero and, after an
+adiabatic ramp of the lattice depth uL to its final value at t = t1, is quenched to its final value. The total time the
+atoms spend in the lattice is denoted by tF (for details see Fig. 1(i) of the main text). At the conclusion of each
+sequence, the trapping potentials are turned off so that the atoms can ballistically expand and be captured using a
+two-step microwave imaging after a given time of flight (TOF) [1]. Each data point in this paper is an average of at
+least 8 repeated measurements and all error bars reported are estimated one standard deviations.
+B.
+Optical dipole trap
+An essential element of our experimental setup is a harmonic confinement potential skew to the moving lattice
+potential. The interplay of these two potentials triggers the nontrivial spatial dynamics that are key to our findings.
+The harmonic confinement is provided by a crossed optical dipole trap (ODT) constructed by two orthogonal beams
+with wavelength λ = 1064 nm. One ODT beam (ODT1) is orthogonal to gravity while the other (ODT2) is at a
+76 degree angle relative to gravity (see Fig. S1). ODT1 is along the xODT axis, while the projection of ODT2 into
+the plane normal to gravity falls along the yODT axis (see Fig. S1). The moving lattice lies 72 degrees horizontally
+from ODT1 and is tilted at a 40 degree angle relative to gravity. Due to experimental considerations, our theoretical
+calculations therefore occur in three distinct coordinate systems that share a common z axis defined by gravity: the
+coordinate systems defined by the ODT potential, the moving lattice, and the imaging plane, as illustrated in Fig. S1.
+xODT
+px
+yODT
+py
+kL,x
+kL,y
+72°
+30°
+pz  & kL,z 
+zODT
+xODT
+yODT
+pz  & kL,z zODT
+kL,x
+kL,y
+px
+py
+14°
+76°
+g
+a)
+b)
+FIG. S1.
+a) The three coordinate systems used in our theoretical calculations projected into the plane spanned by px and py.
+Blue (red) [black] axes refer to the ODT (moving lattice, with associated lattice vector kL) [imaging plane, with associated
+momentum vector p] coordinate system. b) Similar to a) but projected into the plane spanned by yODT and zODT. The green
+vector labeled by g indicates the direction of gravity. The projections in this figure are to scale.
+
+S3
+The potential generated by our crossed ODT can be parameterized to a good approximation by
+V3D(xODT, yODT, z) = −V0
+�
+�
+�
+1
+1 + x2
+ODT
+z2
+0
+exp
+�
+�−2(y2
+ODT + z2)
+w2
+0
+�
+1 + x2
+ODT
+z2
+0
+�
+�
+� +
+1
+1 + y2
+ODT
+z2
+0
+exp
+�
+�−2(x2
+ODT + z2)
+w2
+0
+�
+1 + y2
+ODT
+z2
+0
+�
+�
+�
+�
+�
+� ,
+(S1)
+where V0 =
+P0hαfsλ2λNa
+2
+2π3mec2w2
+0(λ2−λNa2), w0 = 33 µm is the ODT beam waist, z0 = πw2
+0
+λ
+is the associated Rayleigh length,
+P0 is the ODT power, λNa is the D2 line of sodium atoms, λ is the wavelength of the ODT beam, me is the mass
+of an electron, h (ℏ) is the (reduced) Planck constant, αfs ≈
+1
+137 is the fine structure constant, g is the gravitational
+acceleration, and c is the speed of light. The ODT power, P0, can be varied to change the effective trap depth and
+size.
+Shallow ODT
+Compressed ODT
+LODT,z
+LODT,z
+40
+30
+20
+10
+0
+-10
+-60
+-40
+-20
+0
+20
+ODT Potential (kHz)
+zODT (μm)
+FIG. S2.
+Cross-sectional cut of the ODT potential after accounting for gravity in the zODT-direction at a local minimum in
+the xODT and yODT directions. The effective length of the ODT along the zODT-direction is LODT,z = 20µm for the shallow
+ODT (red solid line) and LODT,z = 33µm for the compressed ODT (blue dashed line). Similar plots can be made for the ODT
+potential in the yODT-direction, giving an effective length of LODT,y = 31µm (LODT,y = 40µm) for the shallow (compressed)
+ODT.
+In Fig. S2 we utilize Eq. (S1) and take into account the effects of gravity to generate cross-sectional cuts for the
+compressed ODT trap (P0 ≈ 35 mW), which was utilized for the experiments discussed in Fig. 3 of the main text
+and Fig. S3 of the Supplementary Information, and the shallow ODT trap (P0 ≈ 17 mW), which was utilized for all
+other experimental figures and discussions. The effective trap length LODT,i is defined as the difference between the
+values for which V3D takes on a local maximum and local minimum (see Fig. S2) in the i-coordinate axis. Thus, the
+shallow trap (red solid line) is characterized by a smaller effective trap length LODT,i than the compressed trap (blue
+dashed line), i.e., atoms with high momentum exit the shallow trap more easily than the compressed trap. Since the
+trapping extends over a larger spatial region for the larger LODT,i, the compressed trap extends the time scales over
+which the complex spatial dynamics can be observed. However, the extended trapping times come at the cost of an
+increased average atom temperature, which in turn tends to reduce the coherence and condensate fractions. This is
+demonstrated in Fig. S3, which exhibits less coherent peaks than those shown in the analogous time-of-flight (TOF)
+image in Fig. 2(c) of the main text.
+C.
+Spin-mixing oscillations
+All spin oscillation data presented in the main text are extracted from considering just the zero momentum, p = 0,
+peak. For completeness, Fig. S4 presents spin oscillation data derived from the same data sets as Figs. 1(a)-(c) of the
+main text but obtained by counting all trapped atoms. Using this data, we obtain similar results to Figs. 1(a)-(c);
+specifically, the respective fitted periods agree within the margins of the errors. The agreement between the distinct
+counting methods provides further support for the assumption of a dynamical single spatial-mode approximation
+(dSMA), as it indicates that coherence is retained between different momentum components as they evolve under the
+dynamics driven by the lattice potential.
+In Fig. 1(g) of the main text we see good agreement between the experimental data and the sSMA predictions in
+the Zeeman-dominated region. This is because we expect the dynamics of c2(t) to be less important in this region.
+To support this, Fig. S5 shows experimental data for time traces of ρ0, the fractional population of the |S = 1, m = 0⟩
+state, obtained for a large range of ∆f at q/h = 42 Hz, in the Zeeman-dominated region. We observe no significant
+
+S4
+-4
+-2
+0
+2
+4
+-4
+-2
+0
+2
+L
+R1
+tF = 3.72 ms
+px/ħkL
+pz/ħkL
+Optical Density
+0.0
+0.1
+0.2
+0.3
+R2
+FIG. S3.
+Typical experimental TOF images obtained with the compressed ODT (LODT,z = 33µm, LODT,y = 40µm) displayed
+in Fig. S2. Data are taken following the same experimental sequence as in Fig. 2(c) of the main text, which instead used the
+shallow ODT (LODT,z = 20µm, LODT,y = 31µm). The colorbar scale on the right indicates the optical density of the image.
+ρ0
+tF (ms)
+tF (ms)
+q < q*
+q ≈ q*
+q > q*
+tF (ms)
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+35
+30
+25
+20
+15
+10
+5
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+35
+30
+25
+20
+15
+10
+5
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+35
+30
+25
+20
+15
+10
+5
+(a)
+(b)
+(c)
+FIG. S4.
+Example time traces of ρ0, the fractional population of the |S = 1, m = 0⟩ state, obtained by counting all trapped
+atoms (see text) for: (a) q/h = 15 Hz, (b) q/h = 25 Hz, and (c) q/h = 65 Hz. Here, q is the quadratic Zeeman energy. All
+data is obtained from an experimental sequence illustrated in Fig. 1(i) of the main text with final lattice depth uL = 2.3ER
+and fixed ∆f = 4.6ER/h. Sinusoidal fits (lines), which are used to extract the spin oscillation period T, are shown to guide
+the eye.
+dependence of the period or amplitude of the oscillations on the actual value of ∆f (i.e., the speed of the moving
+lattice), and thus c2(t) associated with the resulting spatial dynamics.
+II.
+DETAILS ON THEORETICAL TREATMENT OF SPIN-1 GASES
+A.
+Gross-Pitaevskii treatment of spin-1 condensates
+We consider a spin-1 condensate of N sodium atoms of mass MNa under external confinement in the presence of a
+moving lattice. The two-body interactions are characterized by the spin-independent and spin-dependent interaction
+coefficients g0 and g2,
+g0 = 4πℏ2(aS=0 + 2aS=2)
+3MNa
+(S2)
+and
+g2 = 4πℏ2(aS=2 − aS=0)
+3MNa
+,
+(S3)
+where aS=0 = 48.9a0 and aS=2 = 54.5a0 are, respectively, the s-wave scattering lengths for the S = 0 and S = 2
+states with a0 the Bohr radius [1, 4]. Our treatment includes the quadratic Zeeman shift term (proportional to q) but
+not the linear Zeeman shift, which does not play a role for the dynamics since it is conserved.
+At the mean-field level, the coupled dynamics of the spin and spatial degrees of freedom is described by the time-
+dependent spinor Gross-Pitaevskii (GP) equation,
+
+S5
+0.8
+0.6
+0.4
+0.2
+25
+20
+15
+10
+5
+tF (ms)
+ρ0
+ Δf = 4.6 ER/h
+ Δf = 7 ER/h
+ Δf = 9.3 ER/h
+q/h = 42 Hz 
+FIG. S5.
+Time traces of ρ0 at fixed q/h = 42Hz and a range of lattice speeds, set by ∆f = 4.6ER/h (blue), 7ER/h (green),
+and 9.3ER/h (black). All data is for a final lattice depth uL = 2.3ER. Sinusoidal fits (lines), which are used to extract the
+spin oscillation period T, are shown to guide the eye. The ∆f = 7ER/h (∆f = 9.3ER/h) curves are offset by 0.2 (0.4) in the
+y direction for visual clarity.
+iℏ ∂
+∂t
+�
+�
+ψ−1
+ψ0
+ψ1
+�
+� =
+�
+− ℏ2∇2
+2MNa
++ V (r, t) + g0(N − 1)
+�
+|ψ−1|2 + |ψ0|2 + |ψ1|2�� �
+�
+ψ−1
+ψ0
+ψ1
+�
+� +
+�
+�
+q 0 0
+0 0 0
+0 0 q
+�
+�
+�
+�
+ψ−1
+ψ0
+ψ1
+�
+�
+(S4)
++ g2(N − 1)
+�
+�
+|ψ−1|2 + |ψ0|2 − |ψ1|2
+ψ∗
+1ψ0
+0
+ψ1ψ∗
+0
+|ψ−|2 + |ψ−1|2
+ψ−1ψ∗
+0
+0
+ψ∗
+−1ψ0
+|ψ1|2 + |ψ0|2 − |ψ−1|2
+�
+�
+�
+�
+ψ−1
+ψ0
+ψ1
+�
+� ,
+where V (r, t) includes contributions from the moving optical lattice Vlat(r, t), the confining potential (see Sec. I A),
+and gravity. Within the GP formalism, each Zeeman component of the BEC is described by a mean-field wavefunc-
+tion ψm(r, t), such that both spin and spatial degrees of freedom, and their interplay, are simultaneously captured.
+The model ignores quantum fluctuations, which are expected to contribute minimally for the regimes in which the
+experiment operates.
+For the scenarios reported in the main text, solution of the GP equation for the full 3D system by direct numerical
+integration is not feasible. This is due to both the disparate timescales for the spin and spatial degrees of freedom
+as well as the large spatial region occupied by the BEC when it is kicked by the lattice. Specifically, the repeated
+momentum kicks imparted by the moving optical lattice on the fractured BEC lead to intricate spatial structure
+as well as a rapidly expanding cloud that must be tracked over comparatively long time scales (36 ms in Fig. 1 of
+the main text), requiring a large simulation box with good spatial resolution. Thus, the GP result presented in the
+main text are from numerical simulations of reduced dimensionality 1D (Fig. 1) or 2D (Fig. 2) models, wherein the
+interaction coefficients g0 and g2, mean spatial density of the BEC and trapping potential V (r, t) are adjusted to
+capture the features of the experimental setup. The GP results presented are generated by numerically integrating
+the coupled GP equations using the XMDS2 software package [5].
+The dynamics shown in Figs. 2-3 of the main text are restricted to a few ms, i.e., much shorter times than those on
+which the spin oscillation dynamics occurs (see Fig. 1 of the main text). These dynamics out to 3 ms are amenable
+to 2D GP equations. To this end, we define a “2D simulation plane” by the direction of the px − pz imaging plane.
+The reduced dimensionality trap V2D is obtained by projecting the 3D trap V3D onto the 2D simulation plane. Our
+simulations capture the role of the ODT trap (including, e.g., non-harmonic corrections to the confining potential)
+and its interplay with gravity. Approximating the azimuthal and polar angles introduced in Sec. I B by 30◦ and 76◦,
+respectively, the 2D trap is given by
+V2D(y, z) = V3D [cos(76◦) cos(30◦)y, − sin(30◦)y, sin(76◦)z] .
+(S5)
+The projection of the moving lattice in the px − py plane is described by kL,x (see Fig. S1). Neglecting the 12◦ angle
+between the kL,x axis and the negative px axis, the moving lattice is in the px − pz plane. Given the fact that the
+
+S6
+lattice has a 40◦ angle relative to the pz axis (see the discussion above Fig. S1), the lattice vector is described by
+kL = kL[− sin(40◦), 0, cos(40◦)]. The reduction from 3D to 2D does change the mean density and, correspondingly,
+the chemical potential. This is accounted for by introducing the effective 2D interaction strength geff
+2D; similar to the
+1D case, g0(N − 1) and g2(N − 1) are replaced by geff
+2D and geff
+2D/28.06, respectively. The coupling constant geff
+2D is set
+by enforcing that the chemical potential of the 2D system, obtained by solving the scalar GP equation, is equal to
+that of the 3D system at t = 0.
+The procedure for mapping the full 3D system to a reduced dimensionality model is not unique. Figure S6(c)
+illustrates this exemplary for the 1D case. Since the chemical potential µ is the characteristic energy scale of the
+time-independent scalar GP equation and c2 the characteristic energy scale of the effective spin Hamiltonian, it is
+natural to demand that the reduced dimensionality scalar GP model reproduces these two energy scales. The solid
+and dashed horizontal lines in Fig. S6(c) show the chemical potential µ and the spin-spin interaction strength c2 of
+the 3D system in the presence of a static optical lattice with uL = 2.3 ER (note that the initial state preparation
+within the full 3D framework is significantly simpler than tracking the time evolution); this is the final lattice depth
+used in the experiment. Fixing the parameters of the scalar 1D GP equation requires setting the values of ℏωz/ER
+and geff
+1D/(ERk−1
+L ). The 1D scalar GP equation is given by Eq. (S7) with the r-vector replaced by z and g0(N − 1)
+replaced by geff
+1D, where geff
+1D has units of “energy times length”. The 1D spinor GP equation is obtained analogously,
+i.e., g0(N − 1) and g2(N − 1) are replaced by geff
+1D and geff
+1D/28.06, respectively. The black and red dashed lines in
+Fig. S6(c) show c2 and µ as a function of geff
+1D/(ERk−1
+L ) for fixed ℏωz/ER (plugging in uL = 2.3 ER, this corresponds
+to ωz = 197 Hz). It can be seen that the dashed lines cross the solid lines at geff
+1D/(ERk−1
+L ) ≈ 7 and 12.5, respectively,
+i.e., for the ℏωz/ER value chosen there exists no unique geff
+1D/(ERk−1
+L ) at which the values of c2 and µ calculated within
+the 1D framework agree with the respective values calculated within the 3D framework. Since we do not find a unique
+geff
+1D/(ERk−1
+L ) for other values of ℏωz/ER either, there exists—unless additional conditions are added—an arbitrariness
+in the chosen 1D simulation parameters. With this in mind, we select parameters that place the divergence of T at
+about the same q value as observed experimentally (see Fig. S6(a)). Specifically, the simulations shown in Figs. 1(d)-
+1(f) of the main text (and in Fig. S6(a)) use the same final lattice depth as the experiment (namely, uL = 2.3 ER),
+ωz = 197 Hz, and geff
+1D/(ERk−1
+L ) = 12.0.
+(c)
+FIG. S6.
+(a) Spin oscillation period T as a function of Zeeman shift q. The black markers show T extracted from 1D spinor
+GP calculations that account for the spin-independent and spin-dependent interactions. The spin oscillation period diverges
+around q∗/h = 22 Hz. Emulating the analysis of the experimental data shown in Fig. 1(g) of the main text, the solid line is
+obtained by fitting the 1D GP data using the analytical sSMA expressions and treating the spin-spin interaction as a fitting
+parameter (namely, c2,fit). (b) Time evolution of the overlap of the Zeeman states (see Eq. (S10)) as a function of the Zeeman
+shift q. An overlap F of one indicates that all Zeeman states share a common spatial density profile, consistent with the
+dSMA. The only significant deviation from unity is observed for relatively long times (t ≳ 20 ms) when q ≈ q∗. (c) Parameter
+calibration of reduced dimensionality 1D GP equation. The black and red solid horizontal lines show the interaction energy
+c2 (left axis) and chemical potential µ (right axis), respectively, for a 3D scalar BEC in the presence of a static lattice with
+uL = 2.3 ER for N = 80, 000 and angular trapping frequencies ωx = 125 Hz, ωy = 125 Hz, and ωz = 155 Hz. The red and
+black dashed lines show c2 and µ, obtained by solving the 1D scalar GP equation as a function of the 1D coupling constant
+geff
+1D. The 1D calculations use ωz = 197 Hz and uL = 2.3 ER.
+B.
+Single spatial-mode approximation and effective spin model
+We now motivate and introduce an approximate treatment that decouples the spin and spatial degrees of freedom.
+Typically, the energy scales associated with the spin-independent terms of the Hamiltonian (i.e., the energy scales
+of the external harmonic and lattice confinement, the interactions that are proportional to g0, and the chemical
+potential µ of the system) are of the order of kilohertz and much larger than those of the spin-dependent terms of
+
+70
+5.5
+(b)
+60
+(2H)
+5
+α (kHz)
+50
+40
+4.5
+30
+20
+4
+5
+10
+15
+20
+81Dk/ER60
+(a)
+45
+(su) L
+30
+15
+0
+0
+15
+30
+45
+60
+q/h (Hz)ur(b)S7
+the Hamiltonian (i.e., the value of the Zeeman shift q and the interactions that are proportional to g2), which are of
+the order of hertz. This scale separation motivates an approximate treatment wherein the spin and spatial degrees of
+freedom are treated independently [6], with the spatial degree of freedom controlled solely by the spin-independent
+terms of the Hamiltonian and the spin dynamics governed by the spin-dependent terms of the Hamiltonian.
+Following the literature [7], we make a single spatial-mode approximation (SMA) wherein the bosonic field operators
+are decomposed as
+ˆψm(r) = ˆamφ(r, t),
+(S6)
+where ˆam (ˆa†
+m) is a bosonic operator that destroys (creates) a particle in Zeeman state m in a spatial mode defined
+by the spatial mean-field wavefunction φ(r, t), which is normalized to one, i.e.,
+�
+d3r|φ(r, t)|2 = 1; in Eq. (S6), m
+labels the Zeeman states (m = 0 and ±1). The key assumption of the SMA is that φ(r, t) is identical for all three
+Zeeman states. The mean-field wave function φ(r, t) is the solution to the time-dependent scalar Gross-Pitaevskii
+(GP) equation
+iℏ∂φ(r, t)
+∂t
+≈
+�
+− ℏ2∇2
+2MNa
++ V (r, t) + g0(N − 1)|φ(r, t)|2
+�
+φ(r, t),
+(S7)
+where V (r, t) is the same as Eq. (S4). The spin dynamics, in turn, is governed by the effective Hamiltonian ˆHeff(t),
+ˆHeff(t) = c2(t)
+2N
+ˆS · ˆS + q(ˆa†
+1ˆa1 + ˆa†
+−1ˆa−1),
+(S8)
+where the quantity ˆS denotes a collective spin operator. The time-dependent interaction strength c2(t),
+c2(t) = (N − 1)g2
+�
+d3r|φ(r, t)|4,
+(S9)
+is driven by the time dependence of the spatial mean-field wave function φ(r, t), i.e., the spin dynamics is governed—
+through the coefficient c2(t)—by the spatial dynamics. In our experiment, the spatial dynamics is, to a large degree,
+induced by the moving optical lattice potential. The time dependence of the interaction coefficient c2(t) is distinct
+from prior works (e.g., Refs. [2, 8, 9]), which assumed that the condensate is prepared in the ground-state of a static
+confining potential such that subsequent spatial motion is minimal and to a good approximation |φ(r, t)|2 = |φ(r, 0)|2.
+For this reason, we use the distinguishing nomenclature of dynamical SMA (dSMA; time-dependent c2) and static
+SMA (sSMA; time-independent c2) for our and prior works, respectively. The latter is recovered from Eq. (S8) by
+assuming c2(t) = c2. To zeroth-order, the dynamics of the moving lattice experiments during the first few milli-
+seconds is dominated by spatial dynamics. At later times, however, the spin degrees of freedom become increasingly
+important as evidenced by the observation of spin oscillations (see Fig. 1 of the main text).
+The dSMA is supported by our 1D spinor GP calculations. First, Fig. 1 of the main text shows good agreement
+between dSMA and GP predictions for ρ0(t). Second, we can explicitly validate the assumption that each Zeeman
+state occupies a single common spatial mode by computing the overlap,
+F =
+|
+�
+dxψ∗
+0(x)ψ1(x)|
+��
+dx|ψ0(x)|2
+��
+dx|ψ1(x)|2
+.
+(S10)
+The time evolution of this quantity over a range of Zeeman shifts is plotted in Fig. S6(b). We observe that F remains
+near unity across the interaction dominated regime (q < q∗) throughout the timescales we investigate (up to 40 ms in
+the GP calculations, which is longer than the 30 ms covered by the experiment). In the Zeeman regime, the overlap
+remains close to unity for t ≲ 20 ms before minor deviations appear. As might be expected naively, a substantial
+breakdown of the single spatial-mode approximation occurs in a narrow region around the critical regime, q ≈ q∗.
+The experimental spin oscillation data in Fig. 1 of the main text are analyzed using the sSMA, i.e., the mean-
+field equations associated with ˆHeff(t) for a time-independent spin-spin interaction coefficient. Specifically, the spin
+oscillation period T is extracted by fitting the experimentally measured fractional population ρ0(t) for various q
+with sinusoidal functions, which provide good approximations to the spin oscillation dynamics away from the critical
+regime where the period diverges. All fits include at least one full period of oscillation. To determine the spin-spin
+interaction coefficient from the extracted periods, we perform a nonlinear least squares fit of the T-versus-q data using
+the analytical solutions to the mean-field equations associated with ˆHeff [10]. The fit uses ρ0(0) = 0.5 and θ(0) = 0
+and treats c2 as a free parameter (we denote the fit result by c2,fit). While ρ0(0) is measured experimentally, θ(0)
+is not. However, based on the experimental sequence used to prepare the initial Zeeman populations we expect that
+θ(0) is equal to 0. As reported in the main text, our fitting procedure yields c2,fit = 23.7(1) Hz.
+
+S8
+To gain additional insights, we perform an analogous analysis for the spin oscillation data obtained by solving the
+1D spinor GP equations. Specifically, we obtain ρ0(t) and the associated period T by solving the spinor GP equations
+(see markers in Fig. S6(a)), and then extract c2,fit from the T-versus-q data using the sSMA (solid line in Fig. S6(a)).
+For the 1D spinor GP simulations shown in Fig. S6(a), the spin oscillation period diverges at q∗/h ≈ 22 Hz, i.e.,
+at roughly the same value of the Zeeman energy as in the experiment presented in the main text. The qualitative
+agreement between the experimental and theoretical analysis, particularly the consistency with the sSMA results, is
+encouraging and suggests that the 1D spinor GP simulations provide a qualitatively correct description of the moving
+lattice experiments.
+The GP simulations also enable us to better understand why the spin oscillation period extracted from the exper-
+imental data is consistent with the predictions of sSMA, even though the time traces show significant discrepancies.
+In the regime q ≤ q∗, the sSMA theory predicts that the spin oscillation period should be strongly correlated with
+the spin-spin interaction strength [3, 10, 11], which the GP results (see, e.g., Fig. 1(h) of the main text) predict to
+decay relatively slowly, compared to the typical timescales of the spin degree of freedom, apart from initial transient
+dynamics for t ≲ 3 ms. This suggests that the spin oscillation period obtained from the experimental data for q ≤ q∗
+should be interpreted as being reflective of the characteristic scale of c2(t) over the experimental sequence. On the
+other hand, in the regime q ≫ q∗ the oscillation period is expected to be dominated by the quadratic Zeeman shift.
+Thus, the experimentally observed spin oscillation periods are fit well by the sSMA predictions as the time dependence
+of c2(t) is less relevant for larger q.
+III.
+CLASSICAL INTERPRETATION OF SPATIAL DYNAMICS: LISSAJOUS CURVES
+A.
+Model
+Figure 3 of the main text presents an analysis of the experimentally observed spatial dynamics of the condensate.
+Specifically, the plot tracks the positions of the L- and R1-peaks in momentum space. These peaks are extracted
+by fitting each momentum peak in a TOF image with a 2D Gaussian and extracting the fitted position. Positions
+are then converted to quasimomentum. The main text compared the experimental results to the predicted Lissajous
+curves. This section introduces the classical trajectory model that generates these Lissajous curves, using a single
+confined particle and initial conditions that are determined self-consistently from the properties of the experimental
+optical lattice and confining potential.
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+pcom,x/ℏkL
+pcom,y/ℏk
+FIG. S7.
+Example trajectory of the center-of-mass momentum pcom extracted from a 2D GPE simulation via pcom =
+�
+d2p p �
+m |ψm(p, t)|2, for a lattice depth of uL = 1.2ER and ∆f = 4.6ER/h.
+Our model is underpinned by the insight that the spatial motion of the condensate atoms is predominantly controlled
+by the approximately harmonic trap and the main effect of the optical lattice is to stroboscopically impart sudden
+(instantaneous) kicks to the atoms when a resonance condition is satisfied dynamically. This is well supported by
+an inspection of the time evolution of the center-of-mass (COM) momentum of the entire atomic cloud according
+to a 2D GP calculation, as shown in Fig. S7. The phase-space trajectory of the COM momentum is dominated by
+smooth evolution – as different momentum components move freely within the confining potential V2D(x, z) – with
+the exception of a sequence of regular abrupt reversals, indicating the sudden imparting of 2ℏkL to a subset of atoms
+by the moving optical lattice. Our classical model constructs analogous Lissajous curves for an ensemble of individual
+(non-interacting) momentum components – defined with respect to their initial position and velocity within the trap
+
+S9
+– that are treated as classical particles traversing the 3D confining potential V3D(r). Note that our classical trajectory
+simulations employ the full 3D potential since these calculations are not numerically challenging. The initial position
+and momentum for each particle are determined by solving for the dynamically appearing resonances at which the
+moving lattice couples different momentum components.
+For concreteness, we introduce the specifics of our model by using it to describe the trajectory of the L-peak, which
+is assumed to be instantaneously populated at t = t1 (i.e., when the lattice detuning is suddenly quenched) due to
+a near-resonant coupling mediated by the moving lattice between the initial BEC at p = 0 and a second non-zero
+momentum state p = 2ℏkL whose kinetic energy differs by 4ER/h ≈ ∆f. Already, we note that the assumption
+that the L-peak is “born” instantaneously at t = t1 is an oversimplification, but we argue that for lattice depths uL
+on the order of ER the coupling between momentum states generated by the moving lattice is much faster than any
+subsequent spatial motion. Based on this description, the L-peak is initially located at the central minimum of the
+trap, which herein we define as the origin of our co-ordinate system, rL(t = t1) = (0, 0, 0), as it is directly outcoupled
+from the stationary BEC, and has initial momentum pL(t = t1) = 2ℏkL.
+We expect the L-peak to quickly separate from the stationary BEC and then to begin climbing the walls of the
+confining potential and decelerate. To describe this evolution, we assume that the mean position and momentum of
+the L-peak follow a trajectory defined by an equivalent classical particle subject to the confining potential V3D. Such
+a trajectory is described by solving the equations,
+MNa
+d2r(t)
+dt2
+= −∇V3D [cos(30◦)x − sin(30◦)y, cos(76◦) sin(30◦)x + cos(76◦) cos(30◦)y + sin(76◦)z, sin(76◦)z − cos(76◦)y]
+(S11)
+and
+p(t) = MNa
+dr(t)
+dt ,
+(S12)
+for the mean position and momentum of the corresponding classical particle, respectively. At this point, we highlight
+that we are assuming that the moving lattice is effectively invisible for the L-peak’s motion; this is justified since
+the L-peak’s motion rapidly changes its “status” from being on resonance with the p = 0 BEC at t = t1 to being
+off-resonant with the p = 0 BEC for t just a bit larger than t1.
+The L-peak moves freely within the confining potential until it dynamically satisfies a resonance condition where
+the optical lattice instantaneously couples the L-peak with momentum pL(t) to a new state – the R1-peak – with
+momentum pR1(t) = pL(t)+2ℏkL. The resonance condition is defined by equating the difference between the kinetic
+energies of these states (it is assumed that the potential energy is unchanged as the new momentum component is
+created at the same location within the trap) with the energy imparted by the moving lattice, i.e., 4.6ER,
+|pL(t) + 2ℏkL|2
+2MNa
+− |pL(t)|2
+2MNa
+= 4.6ER.
+(S13)
+For the lattice vector kL used in our moving-lattice experiment, this resonance condition can be written as a relation
+between different momentum components of the L-peak,
+pL(t) · kL = (0.6/4)ℏk2
+L.
+(S14)
+In the experiment, we approximately have kL = kL[− sin(40◦), 0, cos(40◦)], which leads to the resonance condition
+of pL,z = 0.86pL,x + 0.20ℏkL. The critical time when this resonance is fulfilled is estimated to be tcr = 34.6ℏ/ER =
+1.67 ms. The time tcr also defines the time at which we assume the R1-peak is “born”. The initial condition for the R1-
+peak is then straightforward to compute as rR1(tcr) = rL(tcr) = (−65.0, 0, 55.1)k−1
+L
+and pR1(tcr) = pL(tcr) + 2ℏkL =
+(−0.42, 0, −0.35)ℏkL + 2ℏkL. The computation of the Lissajous curve for the R1-peak, and subsequent creation of,
+e.g., the R2-peak, follows the same scheme as discussed above.
+We reiterate that these theoretical calculations should only be relied upon as a qualitative guide to help understand
+the mechanism of trap-induced resonances that lead to the experimentally observed dynamical fracturing of the BEC
+in momentum space. The Lissajous curves can only provide an approximate guide to the observed trajectories of
+the fractured components in momentum space for a number of reasons. Primarily, these are that the phase-space
+trajectories generated by Eqs. (S11) and (S12) ignore the role of: i) interactions and ii) time-dependent corrugation
+of the confining potential due to the moving lattice.
+Lastly, we comment on the width of the approximate resonance. As previously discussed, the optical lattice couples
+momentum states separated by 2ℏkL over a finite energy window, i.e., states that do not exactly fulfill the resonance
+criterion Eq. (S14). While quantifying the width of this energy window, or resonance region, is not straightforward,
+and further complicated by the role of interactions and motion of the BEC components in the ODT, we can provide
+a rough estimate.
+
+S10
+We use that the coupling of the initial p = 0 BEC to the p = 2ℏkL momentum state is off-resonance (the
+energy difference between these states is 4ER rather than 4.6ER) to determine an estimate of a minimum width of
+the resonance region. Specifically, the fact that we see atoms transfer from the zero momentum state to the 2ℏkL
+momentum state at t ≈ t1 for a detuning of 4.6ER, even though the true resonance for these momentum states occurs
+for a detuning of 4ER, indicates that the lattice-induced coupling of momentum states is near resonance for a window
+of at least 4.6ER ± 0.6ER. Correspondingly, the upper and lower boundaries of the resonance width can be derived
+by replacing the right hand side of Eq. (S13) with 5.2ER and 4ER, respectively. This width is what is plotted in
+Fig. 3(b) of the main text and Fig. S8 as gray shaded regions.
+B.
+Experimental comparison with Lissajous curves
+pz/ħkL
+px/ħkL
+Red (Blue): R1-Peak (L-Peak)
+Solid lines: theory
+Markers: experiment
+-2
+-1
+0
+1
+-2
+-1
+0
+1
+FIG. S8.
+Trajectories of the mean position of L-peak (blue triangles) and R1-peak (red squares), extracted similarly to Fig. 3
+of the main text. Markers represent data taken in a typical ODT (LODT,z = 20 µm). Blue (red) solid lines are L-peak (R1-peak)
+positions according to Lissajous curves based on a calculation of a classical point particle in an ideal (effectively, LODT,i = ∞)
+harmonic trap. Preceding L-peak formation, atoms occupy the p = 0 state at pz = px = 0 (prepared BEC) and the p = 2ℏkL
+state. The gray shaded region marks the approximate resonance region where decelerated atoms are kicked by the lattice (to
+create, e.g., the R1-peak).
+As illustrated in Fig. 3 of the main text, theoretical calculations of the Lissajous trajectories qualitatively reproduce
+the experimental trajectories that utilize the compressed ODT. However, as discussed in the main text, ODT com-
+pression is accompanied by lower condensate fractions and reduced visibility (see also Fig. S3). In Fig. S8 we compare
+the Lissajous curves with the ODT typically employed in our experiment (uncompressed trap with LODT,z = 20 µm).
+We find stark differences between experiment and theory curves for these parameters, including an early exiting of
+atoms out of the ODT trap due to gravity. The disagreements found in the compressed trap of Fig. 3 and Fig. S8
+illustrate that fully capturing all the aspects of the experiment is necessarily beyond the simplified classical model,
+which is designed to minimally capture the dynamical appearance of resonances between different momentum states.
+A closer quantitative comparison would be provided by, e.g., a full 3D GP simulation and analysis of momentum
+space dynamics analogous to what is carried out in Figs. 3 and S8 for the experimental data.
+[1] Chen, Z. et al. Quantum quench and nonequilibrium dynamics in lattice-confined spinor condensates. Phys. Rev. Lett.
+123, 113002 (2019). And references therein.
+[2] Zhao, L., Jiang, J., Tang, T., Webb, M. & Liu, Y. Dynamics in spinor condensates tuned by a microwave dressing field.
+Phys. Rev. A 89, 023608 (2014).
+[3] Zhao, L., Jiang, J., Tang, T., Webb, M. & Liu, Y. Antiferromagnetic spinor condensates in a two-dimensional optical
+lattice. Phys. Rev. Lett. 114, 225302 (2015).
+
+S11
+[4] Knoop, S. et al. Feshbach spectroscopy and analysis of the interaction potentials of ultracold sodium. Phys. Rev. A 83,
+042704 (2011). URL https://link.aps.org/doi/10.1103/PhysRevA.83.042704.
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+article/pii/S0010465512002822.
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+5257–5261 (1998). URL https://link.aps.org/doi/10.1103/PhysRevLett.81.5257.
+[7] Stamper-Kurn, D. M. & Ueda, M. Spinor Bose gases: Symmetries, magnetism, and quantum dynamics. Rev. Mod. Phys.
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+
diff --git a/S9E0T4oBgHgl3EQf2AKe/content/tmp_files/load_file.txt b/S9E0T4oBgHgl3EQf2AKe/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
+++ b/S9E0T4oBgHgl3EQf2AKe/content/tmp_files/load_file.txt
@@ -0,0 +1,1632 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf,len=1631
+page_content='Manipulation of nonequilibrium spin dynamics of an ultracold gas in a moving optical  lattice  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Hardesty-Shaw,1 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Guan,2, 3, 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Austin,1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Blume,2, 3 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Lewis-Swan,2, 3, ∗ and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Liu1, †  1Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA  2Homer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Dodge Department of Physics and Astronomy,  The University of Oklahoma, Norman, Oklahoma 73019, USA  3Center for Quantum Research and Technology, The University of Oklahoma, Norman, Oklahoma 73019, USA  4Department of Physics and Astronomy, Washington State University, Pullman, WA 99164, USA  (Dated: January 10, 2023)  The isolation and control of disparate degrees of freedom underpin quantum simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  We  advance the programmability of cold atom quantum simulators with a first realization of the dynamic  interplay of spatial and spin degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We experimentally demonstrate that violent spatial  evolutions tune long-lived coherent spin dynamics and develop a model of quantum spin-mixing  incorporating the spatial evolution via time-dependent spin-spin interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Our results open new  paths towards the simulation of quantum spin models with tunable interactions via tailored spatial  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Introduction – Ultracold quantum gases that feature spa-  tial and spin degrees of freedom offer a powerful plat-  form for simulating quantum magnetism in controlled,  isolated settings [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' When combined with optical lat-  tices, these simulation capabilities are exemplified by ex-  perimental studies featuring tunable dimensionality and  filling factors [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Possessing long coherence times,  these systems also provide an ideal platform for study-  ing out-of-equilibrium phenomena such as spin-mixing  [10–13], transport [14, 15], dynamical phases of matter  [16], and critical dynamics across quantum phase tran-  sitions [7, 8, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Simultaneously, advances in spin- and  spatially-resolved probes [7, 9, 18, 19] and the control of  time- and spin-dependent lattice potentials are opening  up new opportunities, including the study of multi-state  tunneling physics [20] and driven-dissipative phases [21],  in the presence of the spin degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Typically, the energy scales of the spin and spatial de-  grees of freedom are disparate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This has been exploited  to obtain a reduced description of the spin dynamics that  depends only on a spatial profile that remains frozen due  to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', a strong confining potential [11, 16, 22–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' In  the context of spinor Bose-Einstein condensates (BECs),  this decoupled regime has received significant attention  [29–31] and, amongst other applications, has been uti-  lized to generate entangled states in the highly control-  lable spin degree of freedom [32–37], which can also be  mapped to the motional degrees of freedom [38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  In contrast, the interplay of spatial and spin degrees of  freedom remains largely unexplored, and, although typi-  cally weak, can provide a powerful avenue for controlling  the spin dynamics through tailored dynamical manipula-  tion of the spatial properties of the gas [41–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We inves-  tigate this interplay, providing a first example of how spa-  tial degrees of freedom can be utilized to manipulate the  spin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We experimentally observe that a one-  dimensional (1D) moving lattice, combined with a skew  optical dipole trap (ODT), induces violent transient spa-  tial motion which is nevertheless accompanied by long-  lived spin-mixing dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We develop a theoretical un-  derstanding of these observations based on a dynamical  single spatial-mode approximation (dSMA), which leads  to an effective spin model with a time-dependent spin-  spin interaction coefficient that depends on the tempo-  ral evolution of the BEC density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Experimental  observations — including a robust critical regime featur-  ing divergent timescales for the spin dynamics, which is  tuned by the applied moving optical lattice and associ-  ated spatial motion — are qualitatively described by our  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Our results open the way for the exploitation of classi-  cal spatial dynamics for simulating many-body quantum  spin dynamics with highly tunable, time-dependent in-  teractions [46, 47], thereby enhancing the class of quan-  tum spin models accessible in spinor BECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' In addition,  our findings imply spatial dynamics can provide new con-  trol knobs for the nonequilibrium generation of entangled  spin states for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', quantum-enhanced sensing [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Experimental setup – Each experimental cycle begins  with a sodium spin-1 BEC at quadratic Zeeman energy  q in an ODT (see Supplemental Materials [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' A key  feature of spinor BECs is their spin degree of freedom  characterized by the spin-dependent interaction coeffi-  cient c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Spin-mixing and other nonequilibrium phenom-  ena driven by a static c2 have been studied in various  contexts [7, 8, 11, 23, 24, 28, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Here, in contrast, we  demonstrate that c2 can be tuned dynamically by uti-  lizing a moving lattice to change the BEC’s spatial den-  sity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We construct a 1D moving lattice with two  nearly orthogonal optical beams whose frequency differ-  ence ∆f determines the moving lattice speed [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Our  initial BEC has a fractional population ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5 of atoms  in the |S = 1, m = 0⟩ state and zero magnetization (equal  populations in the |S = 1, m = ±1⟩ states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The BEC  is then adiabatically loaded into the lattice, which is sta-  tionary at time t = 0 and quenched to the desired speed  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='02707v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='quant-gas]  6 Jan 2023  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  ρ0  (a)  (b)  (c)  40  35  30  25  20  15  10  5  Spin oscillation period T (ms)  q/h (Hz)  Δf = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6 ER/h  Markers: Experiment  Line: Theory  70  60  50  40  30  20  10  (g)  c2(t) / c2(0)  (h)  0  BEC  tF  t1  0  Time  (i)  tF (ms)  tF (ms)  tF (ms)  tF (ms)  30  20  10  Δf = 0  Δf = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6 ER/h  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  Δf  0  uL  ρ0  30  20  10  30  20  10  30  20  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  (d)  (e)  (f)  q < q*  q ≈ q*  q > q*  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (a)-(f) Exemplary time traces of ρ0 for spinor BECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Panels (a)-(c) show experimental results for ρ0 (red markers)  at q/h = 15 Hz (a), 25 Hz (b), and 65 Hz (c) as well as sSMA predictions (dotted lines) with c2,fit/h = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='7(1)Hz extracted  from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Panel (a) compares static (black) and moving (red) lattice results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Solid lines are sinusoidal fits to guide the  eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Panels (d)-(f) compare predicted ρ0 for q/h = 15 Hz (d), 22 Hz (e), and 65 Hz (f) from 1D GP simulations (solid lines) to  sSMA (dotted lines) with c2 obtained through fits to the GP data [48], and dSMA (dashed lines) with c2(t) obtained from GP  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The chosen q values exemplify the interaction dominated (q < q∗) and Zeeman (q > q∗) regimes separated by the critical  region q ≈ q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (g) Observed T (markers) versus q fit by analytical sSMA expressions (solid line) with the fitting parameter  c2,fit/h = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='7(1)Hz [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (h) Red (black) lines are evolution of c2(t)/c2(0) for q/h = 15 Hz obtained from 1D GP simulations  of the moving (static) lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' All moving lattice data use a lattice depth uL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3ER, t1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='43ms, and fixed ∆f = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (i) Timeline of the lattice depth (lower panel) and moving lattice speed (upper panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  for t > 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We study the ensuing non-trivial  spin (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1) and spatial (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2) dynamics of the atoms  by holding them in the moving lattice for a time tF before  releasing them for ballistic expansion and imaging [9, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Spin dynamics – We first study the nonequilibrium spin  dynamics generated by experimental sequences (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(i)  ∆f = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER/h) which near-resonantly couple the initial  stationary BEC with the p = 2ℏkL momentum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Here, ER is the recoil energy, h (ℏ) is the (reduced)  Planck constant, and kL is the lattice vector [6, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Spin-mixing oscillations, arising from coherent intercon-  versions among two m = 0 atoms and a pair of atoms  in the m = ±1 Zeeman states [3], constitute a useful  tool in understanding the spin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The periods  T of these oscillations are determined by the competi-  tion between c2 and q, illustrated by typical examples of  the interaction dominated region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(a)) and Zeeman  dominated region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  We also see convincing  experimental signatures of a critical separatrix regime  near q = q∗ where T diverges (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  These  observations are qualitatively consistent with expecta-  tions based on established theory formulations referred  to as a static single spatial-mode approximation (sSMA)  in this paper, which assumes c2 is time-independent  [3, 7, 8, 11, 23, 24, 28, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(g) shows the ob-  served T can be used to estimate the effective static spin-  spin interaction c2,fit/h = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='7(1) Hz as sSMA predicts  q∗ ≈ c2,fit for our initial state [6, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' However, direct  comparisons to the sSMA predicted time traces (dotted  lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(a)-(c)) demonstrate that the model fails  to capture experimentally observed features such as the  damping of the oscillation amplitude and the drift of the  oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Another notable observation that cannot be  explained by sSMA is the shift of the separatrix location  induced by the moving lattice, as shown by a compari-  son between static (∆f = 0) and moving lattice results  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1 These experimental observations suggest  sSMA provides an incomplete description of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Theoretical model – To explain the sSMA’s shortcomings,  we develop the following dSMA model which assumes c2  varies with time and describes our system with the spin  Hamiltonian [6, 22, 39, 48],  ˆHeff(t) = c2(t)  2N  ˆS · ˆS + q(ˆn1 + ˆn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (1)  Here, ˆS = �N  i=1 ˆsi where ˆsi denotes the spin-1 operator  for the ith of the total N atoms and ˆnm is the number  operator for the Zeeman state m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The time-dependent  c2(t) arises from the temporal evolution of the BEC’s  spatial density profile and, in turn, modulation of the  effective interaction strength of the spin model, driven  by the moving lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Formally, c2(t) emerges from the  time-dependence of the Gross-Pitaevskii (GP) orbitals  ψm(r, t) that describe the spatial dynamics of the mth  Zeeman component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  By assuming the spatial density  1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(a) indicates that for static lattices at an identical q in the  interaction dominated regime T is smaller and thus, using the  same sSMA interpretation, would lie on a curve shifted to higher  q (indicating a larger characteristic c2) relative to the moving  lattice data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  3  4  2  0  2  4  2  0  2  4  2  0  2  4  2  0  2  tF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='22 ms  tF = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='72 ms  L  R1  R2  px/ħkL  pz/ħkL  pz/ħkL  (a)  4  2  0  2  4  2  0  2  tF = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='22 ms  L  R1  pz/ħkL  (b)  (c)  Experimental  Theoretical  4  2  0  2  4  2  0  2  tF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='22 ms  4  2  0  2  4  2  0  2  tF = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='22 ms  L  R1  4  2  0  2  4  2  0  2  px/ħkL  tF = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='72 ms  L  R1  R2  (d)  (e)  (f)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  Optical density  Optical density  Optical density  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Time-of-flight snapshots of 2D integrated momentum  distribution with ∆f = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER/h, uL = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2ER, t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='72 ms,  and tF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='22 ms (a), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='22 ms (b), and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='72 ms (c) in a shal-  low ODT with LODT,z = 20 µm [48] at q/h = 42Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (d)-(f)  Analogous theoretical results of in-situ momentum distribu-  tions based on 2D GP simulations [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The colorbar scale  indicates the optical density of the images for each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  profiles |ψm(r, t)|2 for the different m states are the same  but time-dependent (as we find in theory calculations dis-  cussed in more detail later) it is implicit that while the  spatial degree of freedom may contribute to the spin dy-  namics the converse is not true, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', the spin degree of  freedom does not feed back onto the evolution of the spa-  tial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Setting |ψm(r, t)|2 ∝ |φ(r, t)|2 and integrat-  ing out the spatial degrees of freedom leads to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (1)  with c2(t) ∝ (N − 1)  �  d3r |φ(r, t)|4 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' A key result  of the presented experiment-theory work is that dSMA  enables a transparent understanding of the non-trivial  spin dynamics triggered by violent spatial evolution of  the BEC that occurs on faster characteristic time scales  than the spin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Interplay of spin and spatial dynamics – To illustrate  the typical spatial dynamics driving the spin-mixing ob-  served in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1, we show experimental BEC momentum  distributions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2(a)-(c), which capture the emer-  gence of violent spatial motion due to the interplay of  momentum kicks generated by the moving lattice and  the shallow ODT harmonic confinement on a timescale  significantly shorter than the observed spin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The rapid appearance of many of discrete momentum  peaks and associated spatial dynamics shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2  simultaneously suggests that the deviations from sSMA  predictions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(a)-(c) are to be expected but also  entices us to reconcile elements of the good qualitative  agreement between the experimental data and sSMA cal-  culations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We note that the creation of the  discrete momentum peaks is a coherent process and does  not conflict with the assumption of the single spatial-  mode approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' In fact, the Supplemental Materi-  als [48] show that Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(a)-(c) are replicated if different  momentum components are used to construct ρ0, thereby  providing experimental support for dSMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  To gain further insight, we use numerical GP calcu-  lations [48], which provides a mean-field description of  the full spinor BEC dynamics including both spatial and  spin degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The complexity of the experi-  mental system, in particular the disparate timescales of  spin and spatial dynamics, precludes a full quantitative  3D GP treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Instead, we use a reduced dimension-  ality 1D spinor GP calculation with parameters tuned to  capture essential aspects of the experimental 3D system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  This simplified treatment enables us to develop a qual-  itative understanding of the experimental results [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The GP simulations (solid lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(d)-(f)) quali-  tatively replicate the coherent spin dynamics, including  a diverging oscillation period for q ≈ q∗ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(e)) and  robust harmonic oscillations for q < q∗ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(d)) and  q > q∗ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(f)) with damped amplitude and drift-  ing mean value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  We note that the up-  ward drifting mean value in the experimental data for  q > q∗ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(c)), notably not captured by the 1D GP  theory, may be induced by a subtle resonance mecha-  nism between the spin and spatial dynamics [31] that  depends sensitively on the dimensionality of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Higher dimensionality calculations will be presented else-  where [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  We use GP calculations to make a more fine-grained  theoretical investigation of the relationship between the  spatial and spin dynamics and, in particular, certify that  while the BEC undergoes violent motion on fast time-  scales: i) all Zeeman components are described by a com-  mon spatial density profile |φ(r, t)|2, and ii) the sophis-  ticated interplay of the moving lattice and ODT drive  complex dynamics of c2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' These observations lead us  to self-consistently compare the 1D GP results to dSMA  predictions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', mean-field dynamics based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (1)  with c2(t) computed via the GP density |φ(r, t)|2 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The GP and dSMA time traces in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(d)-(f) show  excellent agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  This implicitly  demonstrates the spin dynamics do not feed back into the  spatial evolution, in agreement with expectations based  on the disparity of energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The dSMA results also  show significantly improved qualitative agreements with  experimental data than sSMA results, providing further  support for our dSMA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  GP calculations of the spatial dynamics in the pres-  ence of a moving lattice lead to appreciable variation of  4  c2(t) (red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(h)) at t ≲ 3 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Over longer  timescales c2(t) features an overall decrease, which we un-  derstand as being driven by the relaxation of the spatial  density profile as the BEC fractures into many momen-  tum components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This behavior is in stark contrast with  predictions for a static lattice (black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(h))  that indicate c2(t) instead fluctuates around a well de-  fined time-averaged value with small oscillations due to  excitations created during the loading phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' After the  initial transient behavior in the moving lattice, our re-  sults (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(h)) indicate that the decay of c2(t) is slow  relative to the characteristic time of the spin dynamics,  and therefore observables such as the spin oscillation pe-  riod are captured by sSMA [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Our calculations show  that the precise details of the spin dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', quali-  tative features including damping of the spin oscillations  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(a)) can depend greatly on the temporal vari-  ation of c2(t) and hence a more rigorous description is  provided by the GP and dSMA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Single-particle resonances – The precise evolution of  the spatial density profile, seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2(a)-(c) as a  menagerie of seemingly irregularly distributed wavepack-  ets in momentum space, can be understood with the aid  of GP simulations (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2(d)-(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  To more precisely  capture the impact of gravity and the finite trap depth,  the simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2 are performed using an axially  symmetric 2D setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This numerical treatment is feasi-  ble due to the relatively short time scales over which the  spatial dynamics are studied in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Using 2D simula-  tions also enables us to capture key details of the momen-  tum kicks that 1D simulations, such as those employed  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1, miss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' At short times, the lattice kicks atoms  from the initial BEC with momentum p = 0 to the near-  resonant state with momentum p = 2ℏkL (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2(a)  and 2(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Subsequently, an additional momentum com-  ponent, referred to as the lead peak (L-peak), splits from  the p = 2ℏkL peak and decelerates as it travels away  from the minima of the relatively shallow ODT poten-  tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' As the L-peak slows, its momentum evolves until  it sweeps through the approximate resonance region cen-  tered on the line pz ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='86px + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='20ℏkL (see the gray  shaded region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3(b) and Supplemental Materi-  als [48]), where the lattice couples two nearly resonant  momentum states, corresponding to the L-peak and a  new peak labelled R1, that are separated by another  2ℏkL momentum kick (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2(b) and 2(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Similarly,  the R1-peak also decelerates until it crosses the resonance  region and the lattice generates a new peak labelled R2  (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2(c) and 2(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  This pattern continues and the  BEC fractures into a multitude of momentum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Figure 3(a) confirms our prior analysis, which suggests  a dependence on both the confining ODT potential and  moving lattice, by tracking the position of the L- and  R1-peaks in time and momentum space in a compressed  ODT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The position of the L-peak initially follows a tra-  jectory consistent with a Lissajous curve derived from a  1  0  1  2  1  0  1  pz/ħkL  px/ħkL  (b)  Red: R1-Peak  Blue: L-Peak  1  0  1  pz/ħkL  px/ħkL  1  0  1  1  2  3  4  tF (ms)  5  (a)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (a) Time evolution trajectories of the mean position  of the L-peak (circles) and R1-peak (squares) in the pz-px  plane taken in a compressed ODT with LODT,z = 33 µm [48],  extracted from experimental images similar to those shown in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2(a)-(c) at q/h = 42Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' With increasing time, markers  appear larger due to movement in the px axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Blue (red)  solid lines are the L-peak (R1-peak) trajectories correspond-  ing to Lissajous curves (see text) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (b) Trajectories of  panel (a) projected into the pz-px plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The gray shaded  region encompasses an approximate resonance region where  decelerated atoms are kicked by the lattice (to create, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=',  the R1-peak) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  simplified classical treatment of a single particle initially  moving with momentum 2ℏkL in the ODT (see Supple-  mental Materials [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' A sudden momentum kick im-  parted by the lattice couples the L- and R1-peaks as the  former crosses the approximate resonance region, shown  by the gray region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3(b) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' As time increases  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', deeper into the trajectory of each peak), the agree-  ment between the experimentally observed and the the-  oretically predicted trajectories for the L- and R1-peaks  deteriorates, as shown near the end of the time axis in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The deterioration is most clearly seen when  comparing an ideal (effectively LODT,i = ∞) harmonic  trap with our typical shallow ODT depth that features a  comparatively small curvature (see Supplemental Mate-  rials [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3, we use compressed ODTs instead of  shallow ODTs, trading reduced visibility and condensate  2  3  55  fraction, to better illustrate the bending of the trajectory  in the px − pz plane [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Conclusion – Our results open a new direction for the  exploitation of spatial dynamics as a control knob for  quantum simulations of many-body quantum spin models  with tunable, time-dependent interactions [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Tai-  lored modulation of the spatial profile could be used to  control the precise time dependence of the spin-spin inter-  actions and realize Floquet-driven spin dynamics [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  This can have immediate applications for the dynam-  ical generation of entangled spin states for quantum-  enhanced sensing [36, 39, 61–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The observed short  time dynamics also raise intriguing questions about equi-  libration of spinor BECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  For example, future studies  might utilize the time dependence of c2(t) to force these  systems along different equilibration trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Acknowledgements – D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' acknowledges support by the  National Science Foundation (NSF) through grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  PHY-2110158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' L-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' acknowledges support by NSF  through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' PHY-2110052 and the Dodge Family  College of Arts and Sciences at the University of Okla-  homa (OU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' H-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' acknowledge  support by the Noble Foundation and the NSF through  Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' PHY-1912575 and PHY-2207777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This work  used the OU Supercomputing Center for Education and  Research (OSCER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  ∗ lewisswan@ou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='edu  † yingmei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='liu@okstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Hardesty-Shaw,1 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Guan,2, 3, 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Austin,1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Blume,2, 3 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Lewis-Swan,2, 3 and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Liu1  1Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA  2Homer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Dodge Department of Physics and Astronomy,  The University of Oklahoma, 440 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Brooks Street, Norman, Oklahoma 73019, USA  3Center for Quantum Research and Technology, The University of Oklahoma,  440 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Brooks Street, Norman, Oklahoma 73019, USA  4Department of Physics and Astronomy, Washington State University, Pullman, WA 99164, USA  (Dated: January 10, 2023)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='02707v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='quant-gas]  6 Jan 2023  S2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  EXPERIMENTAL DETAILS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Experimental sequence  Our experimental sequences start by creating a S = 1 spinor BEC of up to 105 sodium (23Na) atoms in a crossed,  anisotropic harmonic optical dipole trap (ODT) at a particular quadratic Zeeman shift q tuned by external magnetic  fields, similar to our previous work [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  We apply a resonant radio-frequency (RF) pulse to prepare an initial  state with fractional population ρ0 = ⟨ˆn0⟩/N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5 in the |S = 1, m = 0⟩ state and zero magnetization, M =  ⟨ˆn1 − ˆn−1⟩/N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We then adiabatically load the initial state into a one-dimensional moving optical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The  lattice is constructed from two nearly orthogonal lattice beams originating from a single-mode laser with wavelength  1064 nm and characterized by the potential Vlat(r, t) = uL cos2[kL ·r−2π∆f(t)t/4], with lattice vector kL oriented at  approximately 40◦ from the z-axis defined by gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The resulting standing wave potential has a lattice spacing of  λL/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='81µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The time-dependent frequency difference ∆f(t) = |fH −fV |, where fH and fV are the corresponding  lattice beam frequencies, determines the velocity v of the moving lattice, v = λL(fH − fV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The velocity v is  manipulated via a linear ramping rate α = h(∆f(t2) − ∆f(t1))  t2 − t1  such that when v < 0 (v > 0) the atoms move in  the p = 2ℏkL (p = −2ℏkL) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' In the main text, the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1 were taken with positive velocities, while  the data in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2 and 3 were taken with negative velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The value of ∆f is initially set to zero and, after an  adiabatic ramp of the lattice depth uL to its final value at t = t1, is quenched to its final value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The total time the  atoms spend in the lattice is denoted by tF (for details see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(i) of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' At the conclusion of each  sequence, the trapping potentials are turned off so that the atoms can ballistically expand and be captured using a  two-step microwave imaging after a given time of flight (TOF) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Each data point in this paper is an average of at  least 8 repeated measurements and all error bars reported are estimated one standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Optical dipole trap  An essential element of our experimental setup is a harmonic confinement potential skew to the moving lattice  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The interplay of these two potentials triggers the nontrivial spatial dynamics that are key to our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The harmonic confinement is provided by a crossed optical dipole trap (ODT) constructed by two orthogonal beams  with wavelength λ = 1064 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' One ODT beam (ODT1) is orthogonal to gravity while the other (ODT2) is at a  76 degree angle relative to gravity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' ODT1 is along the xODT axis, while the projection of ODT2 into  the plane normal to gravity falls along the yODT axis (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The moving lattice lies 72 degrees horizontally  from ODT1 and is tilted at a 40 degree angle relative to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Due to experimental considerations, our theoretical  calculations therefore occur in three distinct coordinate systems that share a common z axis defined by gravity: the  coordinate systems defined by the ODT potential, the moving lattice, and the imaging plane, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  xODT  px  yODT  py  kL,x  kL,y  72°  30°  pz  & kL,z  zODT  xODT  yODT  pz  & kL,z zODT  kL,x  kL,y  px  py  14°  76°  g  a)  b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  a) The three coordinate systems used in our theoretical calculations projected into the plane spanned by px and py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Blue (red) [black] axes refer to the ODT (moving lattice, with associated lattice vector kL) [imaging plane, with associated  momentum vector p] coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' b) Similar to a) but projected into the plane spanned by yODT and zODT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The green  vector labeled by g indicates the direction of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The projections in this figure are to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  S3  The potential generated by our crossed ODT can be parameterized to a good approximation by  V3D(xODT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' yODT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' z) = −V0  �  �  �  1  1 + x2  ODT  z2  0  exp  �  �−2(y2  ODT + z2)  w2  0  �  1 + x2  ODT  z2  0  �  �  � +  1  1 + y2  ODT  z2  0  exp  �  �−2(x2  ODT + z2)  w2  0  �  1 + y2  ODT  z2  0  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (S1)  where V0 =  P0hαfsλ2λNa  2  2π3mec2w2  0(λ2−λNa2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' w0 = 33 µm is the ODT beam waist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' z0 = πw2  0  λ  is the associated Rayleigh length,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  P0 is the ODT power,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' λNa is the D2 line of sodium atoms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' λ is the wavelength of the ODT beam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' me is the mass  of an electron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' h (ℏ) is the (reduced) Planck constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' αfs ≈  1  137 is the fine structure constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' g is the gravitational  acceleration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' and c is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The ODT power, P0, can be varied to change the effective trap depth and  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Shallow ODT  Compressed ODT  LODT,z  LODT,z  40  30  20  10  0  10  60  40  20  0  20  ODT Potential (kHz)  zODT (μm)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Cross-sectional cut of the ODT potential after accounting for gravity in the zODT-direction at a local minimum in  the xODT and yODT directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The effective length of the ODT along the zODT-direction is LODT,z = 20µm for the shallow  ODT (red solid line) and LODT,z = 33µm for the compressed ODT (blue dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Similar plots can be made for the ODT  potential in the yODT-direction, giving an effective length of LODT,y = 31µm (LODT,y = 40µm) for the shallow (compressed)  ODT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S2 we utilize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (S1) and take into account the effects of gravity to generate cross-sectional cuts for the  compressed ODT trap (P0 ≈ 35 mW), which was utilized for the experiments discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3 of the main text  and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S3 of the Supplementary Information, and the shallow ODT trap (P0 ≈ 17 mW), which was utilized for all  other experimental figures and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The effective trap length LODT,i is defined as the difference between the  values for which V3D takes on a local maximum and local minimum (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S2) in the i-coordinate axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Thus, the  shallow trap (red solid line) is characterized by a smaller effective trap length LODT,i than the compressed trap (blue  dashed line), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', atoms with high momentum exit the shallow trap more easily than the compressed trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Since the  trapping extends over a larger spatial region for the larger LODT,i, the compressed trap extends the time scales over  which the complex spatial dynamics can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' However, the extended trapping times come at the cost of an  increased average atom temperature, which in turn tends to reduce the coherence and condensate fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This is  demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S3, which exhibits less coherent peaks than those shown in the analogous time-of-flight (TOF)  image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2(c) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Spin-mixing oscillations  All spin oscillation data presented in the main text are extracted from considering just the zero momentum, p = 0,  peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' For completeness, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S4 presents spin oscillation data derived from the same data sets as Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(a)-(c) of the  main text but obtained by counting all trapped atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Using this data, we obtain similar results to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(a)-(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  specifically, the respective fitted periods agree within the margins of the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The agreement between the distinct  counting methods provides further support for the assumption of a dynamical single spatial-mode approximation  (dSMA), as it indicates that coherence is retained between different momentum components as they evolve under the  dynamics driven by the lattice potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(g) of the main text we see good agreement between the experimental data and the sSMA predictions in  the Zeeman-dominated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This is because we expect the dynamics of c2(t) to be less important in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  To support this, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S5 shows experimental data for time traces of ρ0, the fractional population of the |S = 1, m = 0⟩  state, obtained for a large range of ∆f at q/h = 42 Hz, in the Zeeman-dominated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We observe no significant  S4  4  2  0  2  4  4  2  0  2  L  R1  tF = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='72 ms  px/ħkL  pz/ħkL  Optical Density  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3  R2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Typical experimental TOF images obtained with the compressed ODT (LODT,z = 33µm, LODT,y = 40µm) displayed  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Data are taken following the same experimental sequence as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2(c) of the main text, which instead used the  shallow ODT (LODT,z = 20µm, LODT,y = 31µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The colorbar scale on the right indicates the optical density of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  ρ0  tF (ms)  tF (ms)  q < q*  q ≈ q*  q > q*  tF (ms)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  35  30  25  20  15  10  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  35  30  25  20  15  10  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  35  30  25  20  15  10  5  (a)  (b)  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Example time traces of ρ0, the fractional population of the |S = 1, m = 0⟩ state, obtained by counting all trapped  atoms (see text) for: (a) q/h = 15 Hz, (b) q/h = 25 Hz, and (c) q/h = 65 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Here, q is the quadratic Zeeman energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' All  data is obtained from an experimental sequence illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(i) of the main text with final lattice depth uL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3ER  and fixed ∆f = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Sinusoidal fits (lines), which are used to extract the spin oscillation period T, are shown to guide  the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  dependence of the period or amplitude of the oscillations on the actual value of ∆f (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', the speed of the moving  lattice), and thus c2(t) associated with the resulting spatial dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  DETAILS ON THEORETICAL TREATMENT OF SPIN-1 GASES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Gross-Pitaevskii treatment of spin-1 condensates  We consider a spin-1 condensate of N sodium atoms of mass MNa under external confinement in the presence of a  moving lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The two-body interactions are characterized by the spin-independent and spin-dependent interaction  coefficients g0 and g2,  g0 = 4πℏ2(aS=0 + 2aS=2)  3MNa  (S2)  and  g2 = 4πℏ2(aS=2 − aS=0)  3MNa  ,  (S3)  where aS=0 = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='9a0 and aS=2 = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5a0 are, respectively, the s-wave scattering lengths for the S = 0 and S = 2  states with a0 the Bohr radius [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Our treatment includes the quadratic Zeeman shift term (proportional to q) but  not the linear Zeeman shift, which does not play a role for the dynamics since it is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  At the mean-field level, the coupled dynamics of the spin and spatial degrees of freedom is described by the time-  dependent spinor Gross-Pitaevskii (GP) equation,  S5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  25  20  15  10  5  tF (ms)  ρ0  Δf = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6 ER/h  Δf = 7 ER/h  Δf = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3 ER/h  q/h = 42 Hz  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Time traces of ρ0 at fixed q/h = 42Hz and a range of lattice speeds, set by ∆f = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER/h (blue), 7ER/h (green),  and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3ER/h (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' All data is for a final lattice depth uL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Sinusoidal fits (lines), which are used to extract the  spin oscillation period T, are shown to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The ∆f = 7ER/h (∆f = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3ER/h) curves are offset by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4) in the  y direction for visual clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  iℏ ∂  ∂t  �  �  ψ−1  ψ0  ψ1  �  � =  �  − ℏ2∇2  2MNa  + V (r, t) + g0(N − 1)  �  |ψ−1|2 + |ψ0|2 + |ψ1|2�� �  �  ψ−1  ψ0  ψ1  �  � +  �  �  q 0 0  0 0 0  0 0 q  �  �  �  �  ψ−1  ψ0  ψ1  �  �  (S4)  + g2(N − 1)  �  �  |ψ−1|2 + |ψ0|2 − |ψ1|2  ψ∗  1ψ0  0  ψ1ψ∗  0  |ψ−|2 + |ψ−1|2  ψ−1ψ∗  0  0  ψ∗  −1ψ0  |ψ1|2 + |ψ0|2 − |ψ−1|2  �  �  �  �  ψ−1  ψ0  ψ1  �  � ,  where V (r, t) includes contributions from the moving optical lattice Vlat(r, t), the confining potential (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' I A),  and gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Within the GP formalism, each Zeeman component of the BEC is described by a mean-field wavefunc-  tion ψm(r, t), such that both spin and spatial degrees of freedom, and their interplay, are simultaneously captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The model ignores quantum fluctuations, which are expected to contribute minimally for the regimes in which the  experiment operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  For the scenarios reported in the main text, solution of the GP equation for the full 3D system by direct numerical  integration is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This is due to both the disparate timescales for the spin and spatial degrees of freedom  as well as the large spatial region occupied by the BEC when it is kicked by the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Specifically, the repeated  momentum kicks imparted by the moving optical lattice on the fractured BEC lead to intricate spatial structure  as well as a rapidly expanding cloud that must be tracked over comparatively long time scales (36 ms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1 of  the main text), requiring a large simulation box with good spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Thus, the GP result presented in the  main text are from numerical simulations of reduced dimensionality 1D (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1) or 2D (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2) models, wherein the  interaction coefficients g0 and g2, mean spatial density of the BEC and trapping potential V (r, t) are adjusted to  capture the features of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The GP results presented are generated by numerically integrating  the coupled GP equations using the XMDS2 software package [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The dynamics shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 2-3 of the main text are restricted to a few ms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', much shorter times than those on  which the spin oscillation dynamics occurs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1 of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' These dynamics out to 3 ms are amenable  to 2D GP equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' To this end, we define a “2D simulation plane” by the direction of the px − pz imaging plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The reduced dimensionality trap V2D is obtained by projecting the 3D trap V3D onto the 2D simulation plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Our  simulations capture the role of the ODT trap (including, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', non-harmonic corrections to the confining potential)  and its interplay with gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Approximating the azimuthal and polar angles introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' I B by 30◦ and 76◦,  respectively, the 2D trap is given by  V2D(y, z) = V3D [cos(76◦) cos(30◦)y, − sin(30◦)y, sin(76◦)z] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (S5)  The projection of the moving lattice in the px − py plane is described by kL,x (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Neglecting the 12◦ angle  between the kL,x axis and the negative px axis, the moving lattice is in the px − pz plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Given the fact that the  S6  lattice has a 40◦ angle relative to the pz axis (see the discussion above Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S1), the lattice vector is described by  kL = kL[− sin(40◦), 0, cos(40◦)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The reduction from 3D to 2D does change the mean density and, correspondingly,  the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This is accounted for by introducing the effective 2D interaction strength geff  2D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' similar to the  1D case, g0(N − 1) and g2(N − 1) are replaced by geff  2D and geff  2D/28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='06, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The coupling constant geff  2D is set  by enforcing that the chemical potential of the 2D system, obtained by solving the scalar GP equation, is equal to  that of the 3D system at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The procedure for mapping the full 3D system to a reduced dimensionality model is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Figure S6(c)  illustrates this exemplary for the 1D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Since the chemical potential µ is the characteristic energy scale of the  time-independent scalar GP equation and c2 the characteristic energy scale of the effective spin Hamiltonian, it is  natural to demand that the reduced dimensionality scalar GP model reproduces these two energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The solid  and dashed horizontal lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S6(c) show the chemical potential µ and the spin-spin interaction strength c2 of  the 3D system in the presence of a static optical lattice with uL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3 ER (note that the initial state preparation  within the full 3D framework is significantly simpler than tracking the time evolution);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' this is the final lattice depth  used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Fixing the parameters of the scalar 1D GP equation requires setting the values of ℏωz/ER  and geff  1D/(ERk−1  L ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The 1D scalar GP equation is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (S7) with the r-vector replaced by z and g0(N − 1)  replaced by geff  1D, where geff  1D has units of “energy times length”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The 1D spinor GP equation is obtained analogously,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', g0(N − 1) and g2(N − 1) are replaced by geff  1D and geff  1D/28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='06, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The black and red dashed lines in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S6(c) show c2 and µ as a function of geff  1D/(ERk−1  L ) for fixed ℏωz/ER (plugging in uL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3 ER, this corresponds  to ωz = 197 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' It can be seen that the dashed lines cross the solid lines at geff  1D/(ERk−1  L ) ≈ 7 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5, respectively,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', for the ℏωz/ER value chosen there exists no unique geff  1D/(ERk−1  L ) at which the values of c2 and µ calculated within  the 1D framework agree with the respective values calculated within the 3D framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Since we do not find a unique  geff  1D/(ERk−1  L ) for other values of ℏωz/ER either, there exists—unless additional conditions are added—an arbitrariness  in the chosen 1D simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' With this in mind, we select parameters that place the divergence of T at  about the same q value as observed experimentally (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Specifically, the simulations shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(d)-  1(f) of the main text (and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S6(a)) use the same final lattice depth as the experiment (namely, uL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3 ER),  ωz = 197 Hz, and geff  1D/(ERk−1  L ) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (a) Spin oscillation period T as a function of Zeeman shift q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The black markers show T extracted from 1D spinor  GP calculations that account for the spin-independent and spin-dependent interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The spin oscillation period diverges  around q∗/h = 22 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Emulating the analysis of the experimental data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(g) of the main text, the solid line is  obtained by fitting the 1D GP data using the analytical sSMA expressions and treating the spin-spin interaction as a fitting  parameter (namely, c2,fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (b) Time evolution of the overlap of the Zeeman states (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (S10)) as a function of the Zeeman  shift q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' An overlap F of one indicates that all Zeeman states share a common spatial density profile, consistent with the  dSMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The only significant deviation from unity is observed for relatively long times (t ≳ 20 ms) when q ≈ q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (c) Parameter  calibration of reduced dimensionality 1D GP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The black and red solid horizontal lines show the interaction energy  c2 (left axis) and chemical potential µ (right axis), respectively, for a 3D scalar BEC in the presence of a static lattice with  uL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3 ER for N = 80, 000 and angular trapping frequencies ωx = 125 Hz, ωy = 125 Hz, and ωz = 155 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The red and  black dashed lines show c2 and µ, obtained by solving the 1D scalar GP equation as a function of the 1D coupling constant  geff  1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The 1D calculations use ωz = 197 Hz and uL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='3 ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Single spatial-mode approximation and effective spin model  We now motivate and introduce an approximate treatment that decouples the spin and spatial degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Typically, the energy scales associated with the spin-independent terms of the Hamiltonian (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', the energy scales  of the external harmonic and lattice confinement, the interactions that are proportional to g0, and the chemical  potential µ of the system) are of the order of kilohertz and much larger than those of the spin-dependent terms of  70  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5  (b)  60  (2H)  5  α (kHz)  50  40  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5  30  20  4  5  10  15  20  81Dk/ER60  (a)  45  (su) L  30  15  0  0  15  30  45  60  q/h (Hz)ur(b)S7  the Hamiltonian (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', the value of the Zeeman shift q and the interactions that are proportional to g2), which are of  the order of hertz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This scale separation motivates an approximate treatment wherein the spin and spatial degrees of  freedom are treated independently [6], with the spatial degree of freedom controlled solely by the spin-independent  terms of the Hamiltonian and the spin dynamics governed by the spin-dependent terms of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Following the literature [7], we make a single spatial-mode approximation (SMA) wherein the bosonic field operators  are decomposed as  ˆψm(r) = ˆamφ(r, t),  (S6)  where ˆam (ˆa†  m) is a bosonic operator that destroys (creates) a particle in Zeeman state m in a spatial mode defined  by the spatial mean-field wavefunction φ(r, t), which is normalized to one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=',  �  d3r|φ(r, t)|2 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (S6), m  labels the Zeeman states (m = 0 and ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The key assumption of the SMA is that φ(r, t) is identical for all three  Zeeman states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The mean-field wave function φ(r, t) is the solution to the time-dependent scalar Gross-Pitaevskii  (GP) equation  iℏ∂φ(r, t)  ∂t  ≈  �  − ℏ2∇2  2MNa  + V (r, t) + g0(N − 1)|φ(r, t)|2  �  φ(r, t),  (S7)  where V (r, t) is the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The spin dynamics, in turn, is governed by the effective Hamiltonian ˆHeff(t),  ˆHeff(t) = c2(t)  2N  ˆS · ˆS + q(ˆa†  1ˆa1 + ˆa†  −1ˆa−1),  (S8)  where the quantity ˆS denotes a collective spin operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The time-dependent interaction strength c2(t),  c2(t) = (N − 1)g2  �  d3r|φ(r, t)|4,  (S9)  is driven by the time dependence of the spatial mean-field wave function φ(r, t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', the spin dynamics is governed—  through the coefficient c2(t)—by the spatial dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' In our experiment, the spatial dynamics is, to a large degree,  induced by the moving optical lattice potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The time dependence of the interaction coefficient c2(t) is distinct  from prior works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' [2, 8, 9]), which assumed that the condensate is prepared in the ground-state of a static  confining potential such that subsequent spatial motion is minimal and to a good approximation |φ(r, t)|2 = |φ(r, 0)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  For this reason, we use the distinguishing nomenclature of dynamical SMA (dSMA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' time-dependent c2) and static  SMA (sSMA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' time-independent c2) for our and prior works, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The latter is recovered from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (S8) by  assuming c2(t) = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' To zeroth-order, the dynamics of the moving lattice experiments during the first few milli-  seconds is dominated by spatial dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' At later times, however, the spin degrees of freedom become increasingly  important as evidenced by the observation of spin oscillations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1 of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The dSMA is supported by our 1D spinor GP calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' First, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1 of the main text shows good agreement  between dSMA and GP predictions for ρ0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Second, we can explicitly validate the assumption that each Zeeman  state occupies a single common spatial mode by computing the overlap,  F =  |  �  dxψ∗  0(x)ψ1(x)|  ��  dx|ψ0(x)|2  ��  dx|ψ1(x)|2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (S10)  The time evolution of this quantity over a range of Zeeman shifts is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' We observe that F remains  near unity across the interaction dominated regime (q < q∗) throughout the timescales we investigate (up to 40 ms in  the GP calculations, which is longer than the 30 ms covered by the experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' In the Zeeman regime, the overlap  remains close to unity for t ≲ 20 ms before minor deviations appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' As might be expected naively, a substantial  breakdown of the single spatial-mode approximation occurs in a narrow region around the critical regime, q ≈ q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The experimental spin oscillation data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1 of the main text are analyzed using the sSMA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', the mean-  field equations associated with ˆHeff(t) for a time-independent spin-spin interaction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Specifically, the spin  oscillation period T is extracted by fitting the experimentally measured fractional population ρ0(t) for various q  with sinusoidal functions, which provide good approximations to the spin oscillation dynamics away from the critical  regime where the period diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' All fits include at least one full period of oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' To determine the spin-spin  interaction coefficient from the extracted periods, we perform a nonlinear least squares fit of the T-versus-q data using  the analytical solutions to the mean-field equations associated with ˆHeff [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The fit uses ρ0(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='5 and θ(0) = 0  and treats c2 as a free parameter (we denote the fit result by c2,fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' While ρ0(0) is measured experimentally, θ(0)  is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' However, based on the experimental sequence used to prepare the initial Zeeman populations we expect that  θ(0) is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' As reported in the main text, our fitting procedure yields c2,fit = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='7(1) Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  S8  To gain additional insights, we perform an analogous analysis for the spin oscillation data obtained by solving the  1D spinor GP equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Specifically, we obtain ρ0(t) and the associated period T by solving the spinor GP equations  (see markers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S6(a)), and then extract c2,fit from the T-versus-q data using the sSMA (solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  For the 1D spinor GP simulations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S6(a), the spin oscillation period diverges at q∗/h ≈ 22 Hz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=',  at roughly the same value of the Zeeman energy as in the experiment presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The qualitative  agreement between the experimental and theoretical analysis, particularly the consistency with the sSMA results, is  encouraging and suggests that the 1D spinor GP simulations provide a qualitatively correct description of the moving  lattice experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The GP simulations also enable us to better understand why the spin oscillation period extracted from the exper-  imental data is consistent with the predictions of sSMA, even though the time traces show significant discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  In the regime q ≤ q∗, the sSMA theory predicts that the spin oscillation period should be strongly correlated with  the spin-spin interaction strength [3, 10, 11], which the GP results (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 1(h) of the main text) predict to  decay relatively slowly, compared to the typical timescales of the spin degree of freedom, apart from initial transient  dynamics for t ≲ 3 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This suggests that the spin oscillation period obtained from the experimental data for q ≤ q∗  should be interpreted as being reflective of the characteristic scale of c2(t) over the experimental sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' On the  other hand, in the regime q ≫ q∗ the oscillation period is expected to be dominated by the quadratic Zeeman shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Thus, the experimentally observed spin oscillation periods are fit well by the sSMA predictions as the time dependence  of c2(t) is less relevant for larger q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  CLASSICAL INTERPRETATION OF SPATIAL DYNAMICS: LISSAJOUS CURVES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Model  Figure 3 of the main text presents an analysis of the experimentally observed spatial dynamics of the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Specifically, the plot tracks the positions of the L- and R1-peaks in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' These peaks are extracted  by fitting each momentum peak in a TOF image with a 2D Gaussian and extracting the fitted position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Positions  are then converted to quasimomentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The main text compared the experimental results to the predicted Lissajous  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This section introduces the classical trajectory model that generates these Lissajous curves, using a single  confined particle and initial conditions that are determined self-consistently from the properties of the experimental  optical lattice and confining potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='8  pcom,x/ℏkL  pcom,y/ℏk  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Example trajectory of the center-of-mass momentum pcom extracted from a 2D GPE simulation via pcom =  �  d2p p �  m |ψm(p, t)|2, for a lattice depth of uL = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2ER and ∆f = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Our model is underpinned by the insight that the spatial motion of the condensate atoms is predominantly controlled  by the approximately harmonic trap and the main effect of the optical lattice is to stroboscopically impart sudden  (instantaneous) kicks to the atoms when a resonance condition is satisfied dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This is well supported by  an inspection of the time evolution of the center-of-mass (COM) momentum of the entire atomic cloud according  to a 2D GP calculation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The phase-space trajectory of the COM momentum is dominated by  smooth evolution – as different momentum components move freely within the confining potential V2D(x, z) – with  the exception of a sequence of regular abrupt reversals, indicating the sudden imparting of 2ℏkL to a subset of atoms  by the moving optical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Our classical model constructs analogous Lissajous curves for an ensemble of individual  (non-interacting) momentum components – defined with respect to their initial position and velocity within the trap  S9  – that are treated as classical particles traversing the 3D confining potential V3D(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Note that our classical trajectory  simulations employ the full 3D potential since these calculations are not numerically challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The initial position  and momentum for each particle are determined by solving for the dynamically appearing resonances at which the  moving lattice couples different momentum components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  For concreteness, we introduce the specifics of our model by using it to describe the trajectory of the L-peak, which  is assumed to be instantaneously populated at t = t1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', when the lattice detuning is suddenly quenched) due to  a near-resonant coupling mediated by the moving lattice between the initial BEC at p = 0 and a second non-zero  momentum state p = 2ℏkL whose kinetic energy differs by 4ER/h ≈ ∆f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Already, we note that the assumption  that the L-peak is “born” instantaneously at t = t1 is an oversimplification, but we argue that for lattice depths uL  on the order of ER the coupling between momentum states generated by the moving lattice is much faster than any  subsequent spatial motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Based on this description, the L-peak is initially located at the central minimum of the  trap, which herein we define as the origin of our co-ordinate system, rL(t = t1) = (0, 0, 0), as it is directly outcoupled  from the stationary BEC, and has initial momentum pL(t = t1) = 2ℏkL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  We expect the L-peak to quickly separate from the stationary BEC and then to begin climbing the walls of the  confining potential and decelerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' To describe this evolution, we assume that the mean position and momentum of  the L-peak follow a trajectory defined by an equivalent classical particle subject to the confining potential V3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Such  a trajectory is described by solving the equations,  MNa  d2r(t)  dt2  = −∇V3D [cos(30◦)x − sin(30◦)y, cos(76◦) sin(30◦)x + cos(76◦) cos(30◦)y + sin(76◦)z, sin(76◦)z − cos(76◦)y]  (S11)  and  p(t) = MNa  dr(t)  dt ,  (S12)  for the mean position and momentum of the corresponding classical particle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' At this point, we highlight  that we are assuming that the moving lattice is effectively invisible for the L-peak’s motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' this is justified since  the L-peak’s motion rapidly changes its “status” from being on resonance with the p = 0 BEC at t = t1 to being  off-resonant with the p = 0 BEC for t just a bit larger than t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  The L-peak moves freely within the confining potential until it dynamically satisfies a resonance condition where  the optical lattice instantaneously couples the L-peak with momentum pL(t) to a new state – the R1-peak – with  momentum pR1(t) = pL(t)+2ℏkL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The resonance condition is defined by equating the difference between the kinetic  energies of these states (it is assumed that the potential energy is unchanged as the new momentum component is  created at the same location within the trap) with the energy imparted by the moving lattice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER,  |pL(t) + 2ℏkL|2  2MNa  − |pL(t)|2  2MNa  = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (S13)  For the lattice vector kL used in our moving-lattice experiment, this resonance condition can be written as a relation  between different momentum components of the L-peak,  pL(t) · kL = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6/4)ℏk2  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  (S14)  In the experiment, we approximately have kL = kL[− sin(40◦), 0, cos(40◦)], which leads to the resonance condition  of pL,z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='86pL,x + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='20ℏkL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The critical time when this resonance is fulfilled is estimated to be tcr = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ℏ/ER =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='67 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The time tcr also defines the time at which we assume the R1-peak is “born”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The initial condition for the R1-  peak is then straightforward to compute as rR1(tcr) = rL(tcr) = (−65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='0, 0, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1)k−1  L  and pR1(tcr) = pL(tcr) + 2ℏkL =  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='42, 0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='35)ℏkL + 2ℏkL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The computation of the Lissajous curve for the R1-peak, and subsequent creation of,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', the R2-peak, follows the same scheme as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  We reiterate that these theoretical calculations should only be relied upon as a qualitative guide to help understand  the mechanism of trap-induced resonances that lead to the experimentally observed dynamical fracturing of the BEC  in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The Lissajous curves can only provide an approximate guide to the observed trajectories of  the fractured components in momentum space for a number of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Primarily, these are that the phase-space  trajectories generated by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (S11) and (S12) ignore the role of: i) interactions and ii) time-dependent corrugation  of the confining potential due to the moving lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Lastly, we comment on the width of the approximate resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' As previously discussed, the optical lattice couples  momentum states separated by 2ℏkL over a finite energy window, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', states that do not exactly fulfill the resonance  criterion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (S14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' While quantifying the width of this energy window, or resonance region, is not straightforward,  and further complicated by the role of interactions and motion of the BEC components in the ODT, we can provide  a rough estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  S10  We use that the coupling of the initial p = 0 BEC to the p = 2ℏkL momentum state is off-resonance (the  energy difference between these states is 4ER rather than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER) to determine an estimate of a minimum width of  the resonance region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Specifically, the fact that we see atoms transfer from the zero momentum state to the 2ℏkL  momentum state at t ≈ t1 for a detuning of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER, even though the true resonance for these momentum states occurs  for a detuning of 4ER, indicates that the lattice-induced coupling of momentum states is near resonance for a window  of at least 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='6ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Correspondingly, the upper and lower boundaries of the resonance width can be derived  by replacing the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' (S13) with 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='2ER and 4ER, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' This width is what is plotted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3(b) of the main text and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S8 as gray shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Experimental comparison with Lissajous curves  pz/ħkL  px/ħkL  Red (Blue): R1-Peak (L-Peak)  Solid lines: theory  Markers: experiment  2  1  0  1  2  1  0  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Trajectories of the mean position of L-peak (blue triangles) and R1-peak (red squares), extracted similarly to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3  of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Markers represent data taken in a typical ODT (LODT,z = 20 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Blue (red) solid lines are L-peak (R1-peak)  positions according to Lissajous curves based on a calculation of a classical point particle in an ideal (effectively, LODT,i = ∞)  harmonic trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Preceding L-peak formation, atoms occupy the p = 0 state at pz = px = 0 (prepared BEC) and the p = 2ℏkL  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The gray shaded region marks the approximate resonance region where decelerated atoms are kicked by the lattice (to  create, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', the R1-peak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3 of the main text, theoretical calculations of the Lissajous trajectories qualitatively reproduce  the experimental trajectories that utilize the compressed ODT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' However, as discussed in the main text, ODT com-  pression is accompanied by lower condensate fractions and reduced visibility (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S8 we compare  the Lissajous curves with the ODT typically employed in our experiment (uncompressed trap with LODT,z = 20 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  We find stark differences between experiment and theory curves for these parameters, including an early exiting of  atoms out of the ODT trap due to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' The disagreements found in the compressed trap of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S8  illustrate that fully capturing all the aspects of the experiment is necessarily beyond the simplified classical model,  which is designed to minimally capture the dynamical appearance of resonances between different momentum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  A closer quantitative comparison would be provided by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', a full 3D GP simulation and analysis of momentum  space dynamics analogous to what is carried out in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 3 and S8 for the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  [1] Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Quantum quench and nonequilibrium dynamics in lattice-confined spinor condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  123, 113002 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' And references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  [2] Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Tang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Webb, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' & Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Dynamics in spinor condensates tuned by a microwave dressing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content='  [3] Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Tang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Webb, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' & Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Antiferromagnetic spinor condensates in a two-dimensional optical  lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content='  S11  [4] Knoop, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content=' URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content='com/science/  article/pii/S0010465512002822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='011601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  [9] Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' 102, 125301 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  [10] Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content=', Chapman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content=' URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='013602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='  [11] Kronj¨ager, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content=', Navez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=', Bongs, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' & Sengstock, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Magnetically tuned spin dynamics resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQf2AKe/content/2301.02707v1.pdf'}
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+Multi-channel Medium Access Control Protocols
+for Wireless Networks within Computing Packages
+Bernat Oll´e∗, Pau Talarn∗, Albert Cabellos-Aparicio∗, Filip Lemic†, Eduard Alarc´on∗, and Sergi Abadal∗
+∗NaNoNetworking Center in Catalunya (N3Cat), Universitat Polit`ecnica de Catalunya, 08034 Barcelona, Spain
+†AI-driven Systems Lab, i2Cat Foundation, 08034 Barcelona, Spain
+Abstract—Wireless communications at the chip scale emerge
+as a interesting complement to traditional wire-based approaches
+thanks to their low latency, inherent broadcast nature, and ca-
+pacity to bypass pin constraints. However, as current trends push
+towards massive and bandwidth-hungry processor architectures,
+there is a need for wireless chip-scale networks that exploit and
+share as many channels as possible. In this context, this work
+addresses the issue of channel sharing by exploring the design
+space of multi-channel Medium Access Control (MAC) protocols
+for chip-scale networks. Distinct channel assignment strategies
+for both random access and token passing are presented and
+evaluated under realistic traffic patterns. It is shown that, even
+with the improvements enabled by the multiple channels, both
+protocols maintain their intrinsic advantages and disadvantages.
+I. INTRODUCTION
+Efficient integrated networks at the chip scale within
+Systems-in-Package (SiPs) are a prerequisite for high perfor-
+mance in such computing systems. Currently, most systems
+incorporate a Network-in-Package (NiP) consisting of a set
+of on-chip routers and intra-/inter-chip wired links [1], [2].
+However, recent scaling [3], [4], specialization [5], [6], and
+disintegration trends [7], [8] are increasing the pressure placed
+on the interconnect, to the point that new communication
+paradigms may be required [9], [10].
+Among the emerging alternatives, wireless chip-scale com-
+munications stand as a promising contender [11]–[14]. This
+communication paradigm relies the use of modulated electro-
+magnetic waves for data transmission using the chip package
+as communications medium (Fig. 1). The resulting wireless
+in-package links provide low latency, inherent broadcast capa-
+bilities, and global reconfigurability.
+Since the communications medium is shared, wireless in-
+package communications require Medium Access Control
+(MAC) protocols to avoid or manage wasteful collisions. In
+this scenario, MAC protocols generally reduce to variants
+of multiplexing, random access, or token passing [15]–[21].
+Even though recent works have demonstrated that computing
+packages could support a few frequency [22], [23] and space
+channels [24], [25], it is still unclear how MAC protocols can
+benefit from them. This is because more than a few chan-
+nels are needed to implement truly scalable frequency/space
+multiplexing techniques [15], and most importantly, because
+multi-channel variants of random access and token passing
+have not been explored yet.
+This work is supported by the European Commission under H2020 grant
+WiPLASH (GA 863337) and HE grant WINC (GA 101042080).
+Package Substrate
+Chiplet
+Chiplet
+Package Substrate
+Chiplet
+Chiplet
+Interposer
+Package Substrate
+Chiplet
+Chiplet
+Silicon bridge
+Package Substrate
+Chiplet
+Chiplet
+Wireless Antennas
+Fig. 1. Pictorial view of a wireless chip-to-chip communication link.
+This paper aims to bridge this gap by focusing on the
+study of multi-channel versions of the two most representative
+protocol types in chip-scale scenarios, i.e. random access and
+token passing. In particular, the main contributions are as
+follows. We first describe the different ways we can extend
+random access and token passing with a small set of channels
+in Sec. II. Then, in Sec. III, we evaluate these protocol variants
+with traffic models typically used to mimic multiprocessor
+workloads [26]. This analysis sheds light on the impact of
+channel assignment on the protocol performance, as summa-
+rized in Sec. IV and concluded in Sec. V.
+II. MULTI-CHANNEL MAC PROTOCOLS
+In this work, we describe three distinct channel assignment
+strategies for random access and token passing. As baselines,
+we take BRS [18] for random access and the baseline from
+[20] for token passing. The strategies presented here are not
+provably optimal, but they are simple (as required by the
+resource constraints of the chip-scale scenario) and represen-
+tative of the potential techniques that can be used.
+A. Assignment Methods for BRS
+In random access protocols such as BRS [18], nodes con-
+tend for channel access and back off if the channel is busy or
+there is a collision. Assuming N nodes, we study three ways
+to reduce the collision probability using Nc channels, namely:
+AS1: Channels are assigned to nodes individually and ran-
+domly. When a node has a packet to transmit, the node is
+assigned a random channel. If the channel is busy or there is
+a collision, nodes undergo a random back off and also choose
+a random channel to use in the next attempt.
+AS2: Each channel is assigned to
+N
+Nc nodes statically follow-
+ing a uniform distribution, this is, assuming that all nodes have
+the same load (see Fig. 2, left). While this is not optimal for
+spatially unbalanced traffic, it serves as a baseline.
+AS3: Channels are assigned to a variable number of nodes
+following a distribution that balances the load in each channel
+(see Fig. 2, right). To that end, nodes are ordered based on
+arXiv:2301.11471v1  [cs.ET]  27 Jan 2023
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+Fig. 2.
+Graphical representations of assignment techniques AS2 (left) and
+AS3 (right) for BRS assuming 16 nodes and 4 channels.
+1
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+Fig. 3. Graphical representations of the different assignment techniques for
+token passing assuming 16 nodes and 4 channels.
+the expected normalized load and assigned to each channel in
+order following a greedy algorithm.
+B. Assignment Methods for Token Passing
+In token passing [27], typically, all N nodes are sorted
+forming a virtual ring and the token is passed in order through
+that ring. In a version with Nc channels, each channel can be
+a token. The design decisions then lie on the number of rings
+and the nodes that form each ring. For instance:
+AS1: We assume as many rings as there are channels and map
+nodes uniformly to each ring. In other words, we distribute
+them in rings of
+N
+Nc nodes, regardless of their expected load.
+AS2: We assume a single virtual ring with multiple tokens
+circulating in it. In this case, tokens can jump over other
+tokens: when node i holds a token for multiple cycles during
+a transmission, idle tokens that arrive at i-1 can jump to i+1.
+AS3: This strategy is similar to AS1, but nodes are mapped
+to rings based on their expected load. This may lead to rings
+of different sizes, but similar in the expected overall load.
+III. PERFORMANCE EVALUATION
+The architecture and application parameters are summarized
+in Table I. We implement both single-channel baselines and
+multi-channel versions of BRS and token passing as finite state
+machines in a modified version of Multi2sim that models wire-
+less links and supports collision detection [28]. The protocols
+are stressed with synthetic traffic modeling uneven injection
+distributions (through the σ parameter) and bursty temporal
+behavior (through the Hurst exponent H) [26]. The default
+values for the different parameters are N = 64 nodes, Nc = 4
+channels, H = 0.5 and σ = 1. Simulations are cycle-accurate.
+In all cases, we compare the packet latency (in cycles) and
+throughput (in packets/cycle) of the different options. Given
+the high number of protocol strategies and traffic types, instead
+of plotting the classical latency–throughput curve, we make
+use of box plots that summarize the latency and throughput
+statistics. In our plots, the X axis shows the parameters under
+study. The plots have two Y axis: the left axis represents
+the latency and corresponds to the box plot values, whereas
+TABLE I
+CHARACTERISTICS OF SIMULATED PROTOCOLS AND APPLICATIONS.
+Application
+Synthetic traffic, H=0.5–0.85, σ=0.05–100
+System
+N=64–512 cores, one antenna/core, 1-GHz clock
+Network
+80-bit packets (preamble: 20 bits), Nc=1–4 channels
+Link
+BRS [18], Token passing
+Physical
+On-Off Keying, 20 Gb/s
+101
+102
+Latency cycles/packet
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Throughput paq/cycle
+BRS
+C1
+AS1_C2
+AS2_C2
+AS3_C2
+AS1_C4
+AS2_C4
+AS3_C4
+101
+102
+Latency cycles/packet
+0
+0.2
+0.4
+0.6
+0.8
+1
+Throughput paq/cycle
+TOKEN
+C1
+AS1_C2
+AS2_C2
+AS3_C2
+AS1_C4
+AS2_C4
+AS3_C4
+Fig. 4. Performance of multi-channel BRS (top) and token passing (bottom)
+for an increasing number of channels, C1 to C4, and different assignments.
+the right axis represents the throughput and corresponds to
+single-value markers of saturation throughput. Since a single
+packet takes 4 cycles in a single channel to be transmitted, the
+maximum throughput is 0.25 packets/cycle/channel.
+A. Number of Channels
+Here, we discuss the results shown in Fig. 4 for BRS and
+token passing and an increasing number of channels.
+Latency. In general, it can be observed that BRS is less stable
+than token in terms of latency as the range of values is larger,
+with a higher number of outlier points. However, BRS has a
+much better zero-load latency than token since, in BRS, the
+protocol allows nodes to start transmitting immediately when
+the channel is sensed idle. This fact also can explain why
+independently of the parameters evaluated here (assignment,
+number of channels) the minimum latency is quite similar. The
+worst-case latency, however, clearly improves when having
+multiple channels, as the high load is distributed over multiple
+channels. On the other hand, in token passing, nodes must
+wait until they possess the token to start transmitting. For this
+reason, when the number of nodes is large, N = 64 in this
+case, the system remains idle much longer.
+Throughput. The results for token passing depict a rather
+stable increase in saturation throughput as more channels are
+added, regardless of the assignment method. This could be
+due to the use, by default, of non-bursty and non-hotspot
+traffic to evaluate scalability. On the other hand, the results
+for BRS illustrate a different behavior than in token passing.
+Firstly, BRS cannot reach a saturation throughput as high as
+token passing. The main reasons are that channel contention
+and multiple collisions lead to channel waste and, hence, to a
+reduced throughput. Furthermore, BRS is more irregular than
+token passing in terms of saturation throughput as it depends
+
+10 1
+10 2
+Latency cycles/paq
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Throughput paq/cycle
+Ncores BRS
+AS1_64
+AS1_128
+AS1_512
+AS2_64
+AS2_128
+AS2_512
+AS3_64
+AS3_128
+AS3_512
+10 1
+10 2
+Latency cycles/paq
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Throughput paq/cycle
+Sigmas BRS
+AS1_0.05
+AS1_0.1
+AS1_1
+AS1_100
+AS2_0.05
+AS2_0.1
+AS2_1
+AS2_100
+AS3_0.05
+AS3_0.1
+AS3_1
+AS3_100
+101
+102
+Latency cycles/paq
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Throughput paq/cycle
+Hs BRS
+AS1_H0.5
+AS1_H0.7
+AS1_H0.85
+AS2_H0.5
+AS2_H0.7
+AS2_H0.85
+AS3_H0.5
+AS3_H0.7
+AS3_H0.85
+Fig. 5. Performance of multi-channel BRS protocol for an increasing number of nodes, N=64–512 (left graph), different spatial concentration levels, σ=0.1–100
+(center graph), different temporal burstiness levels, H=0.5–0.85 (right graph), and different assignment techniques.
+10 1
+10 2
+Latency cycles/paq
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Throughput paq/cycle
+Ncores TOKEN
+AS1_64
+AS1_128
+AS1_512
+AS2_64
+AS2_128
+AS2_512
+AS3_64
+AS3_128
+AS3_512
+10 1
+10 2
+Latency cycles/paq
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Throughput paq/cycle
+Sigmas TOKEN
+AS1_0.05
+AS1_0.1
+AS1_1
+AS1_100
+AS2_0.05
+AS2_0.1
+AS2_1
+AS2_100
+AS3_0.05
+AS3_0.1
+AS3_1
+AS3_100
+101
+102
+Latency cycles/paq
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Throughput paq/cycle
+Hs TOKEN
+AS1_H0.5
+AS1_H0.7
+AS1_H0.85
+AS2_H0.5
+AS2_H0.7
+AS2_H0.85
+AS3_H0.5
+AS3_H0.7
+AS3_H0.85
+Fig. 6. Performance of multi-channel token passing protocol for an increasing number of nodes, N=64–512 (left graph), different spatial concentration levels,
+σ=0.1–100 (center graph), different temporal burstiness levels, H=0.5–0.85 (right graph), and different assignment techniques.
+on the percentage of collisions at high loads. As a result,
+the difference between the saturation throughput achieved for
+different assignments increases with the number of channels.
+B. Number of Nodes
+Next, we comment on the performance of BRS and token
+passing for an increasing number of nodes, with Nc = 4. The
+results are shown in the left charts of Fig. 5 and Fig. 6.
+Latency. BRS has a much lower latency than token passing
+due to its ability to transmit when the channel is idle. The
+span of the latency values differs across number of nodes and
+assignments, but in general are restrained to similar values
+because in the end, the same aggregated load ends up being
+distributed over more nodes. Static assignment of channels
+(AS2) works worse than the other alternatives. On the other
+hand, from the plot of token passing, it is clear that more nodes
+lead to much higher latency due to the increase of the token
+turnaround time. In fact, the low-load latency is proportional
+to the number of nodes in all cases. The span of the latency
+values is similar across the different system sizes.
+Throughput. In general, saturation throughput is slightly
+higher for a lower number of nodes. In our protocols, having
+more nodes means having a higher population and, hence, a
+higher chance of collisions even for the same load for BRS,
+and a higher waiting time (or lower probability of having all
+nodes backlogged) in token passing. It seems, in any case, that
+BRS is more resilient to the change in the number of nodes
+as the drop is more subtle, except for AS3, where possibly
+the load balancing algorithm is not performing well when
+such a large number of nodes has to be classified. Finally,
+all three assignments have very similar throughput in all cases
+for token passing, whereas AS1 (random channel assignment
+to individual packets) works better in BRS.
+C. Hotspot Traffic
+We next discuss the results shown in the middle plots of
+Fig. 5 and Fig. 6, which illustrate the impact of uneven spatial
+injection distribution on performance. We remind that low/high
+values of σ mean that traffic is hotspot/evenly distributed [26].
+Latency. In BRS, the hotspot behavior of traffic does not seem
+to have a large influence on the performance of the different
+assignment methods. The outlier, third quartile, and maximum
+values within the distribution seem to be mildly impacted by
+the hotspot nature of traffic. In general, BRS is resilient to
+such variations and actually could benefit from having a lower
+amount of nodes contending for the available channels. Still,
+the results show a small tendency to worse results when traffic
+is concentrated around a few nodes, possibly because of the
+nodes with higher load reaching higher backoff values. In AS3,
+this situation is avoided by proactively placing high-load nodes
+in different channels. Similarly, in token passing, latency is
+affected by the concentration of traffic around a given set of
+nodes mostly because the different assignment methods are
+able to provide tokens quickly to nodes that need it, even if
+they are spaced apart within the ring. This is clearly visible in
+the extreme case of σ = 0.05. Similarly, outlier values seem
+to be larger when traffic is more hotspot. We also observe how
+AS2 fails to provide a good performance at low loads, and this
+behavior is exacerbated for very hotspot traffic.
+Throughput. The throughput of BRS in its different im-
+plementations does not vary significantly with the type of
+
+spatial distribution of traffic, except for AS3, where a higher
+concentration of traffic around a few nodes seems to have a
+positive effect on the throughput. One reason could be that the
+most active nodes are distributed over the different channels
+so that contention is minimized. That does not happen in
+other assignment methods. Different behavior is observed in
+token passing, where the hotspot behavior of traffic modifies
+the throughput of the different assignment methods, with
+AS3 being affected a bit less. This is because if the load is
+concentrated around a small set of nodes, a large portion of the
+airtime is wasted while passing the token among these nodes.
+D. Bursty Traffic
+Finally, we present the latency and throughput results for
+an increasingly bursty traffic. The results are shown in the
+right plots of Fig. 5 and Fig. 6 for BRS and token passing,
+respectively. Temporal injection of traffic is modeled through
+the Hurst exponent [26], with higher values indicating more
+bursty behavior, i.e. longer bursts followed by longer silences.
+Latency. In BRS, it can be seen that the higher the value of
+H, the higher the latency in average and also the more un-
+predictable. This is because with an H of 0.5, the packets are
+injected following a random Poisson process, which minimizes
+the probability of collisions. However, when increasingly
+bursty traffic is considered, the probability of packets being
+injected (and nodes trying to transmit) in the same exact
+cycle increases. The effect is multiplicative with the burstiness,
+as the effect of cascading collisions leads to an exponential
+increase of the backoff time. This affects the system at all
+loads. On the other hand, token passing also suffers when
+bursty traffic is served, leading to very high latency especially
+for high values of H. The latency is a bit more stable than in
+the case of BRS, mainly because the protocol does not react
+with exponential backoffs, but rather with linear token passings
+to bursts of traffic. Still, the latency is much higher than that
+of BRS, discouraging its use for large number of nodes.
+Throughput. On one hand, it can be verified that in BRS,
+the saturation throughput remains rather constant across all
+assignments regardless of the value of the Hurst exponent. A
+possible reason could stem from the behavior of the backoff
+mechanism; bursty traffic leads to a large number of collisions
+which increases latency even for low loads, but the protocol
+may converge to a large backoff value that can accommodate
+the load even if it comes in bursts. In other works, the backoff
+mechanisms spreads out the bursts of traffic over time, until
+all nodes are backlogged. On the other hand, it can be seen
+that in the case of token passing, the saturation throughput
+seems to drop significantly for higher numbers of H, to a
+point that the achieved throughput becomes comparable with
+that of BRS. A potential reason for this behavior is the lack
+of an adaptive mechanism to react to bursts; the token has to
+still move around the ring even if bursts of traffic lead to the
+generation of multiple packets in a given node, leading to gaps
+where the wireless channel remains silent. When traffic is less
+bursty, the probability of such events is lower.
+AS2_N64
+AS3_S0.05
+AS1_N64
+AS1_H0.5
+0
+10
+20
+30
+40
+50
+60
+0
+0,2
+0,4
+0,6
+0,8
+1
+1,2
+Latency cycles/packet
+Throughput packets/cycle
+B_C
+T_C
+B_N
+T_N
+B_S
+T_S
+B_H
+T_H
+LOW LATENCY REGION
+HIGH CAPACITY REGION
+Fig. 7. Summary of the latency and throughput results over all the protocols,
+assignment methods, and traffic conditions. B and T stand for BRS (random
+access) and token passing, C and N denote number of channels and nodes,
+whereas S and H represent the different spatial and temporal injection
+distributions. For instance, the B C symbols represent the latency-throughput
+of all the assignment methods for BRS for different number of channels. Two
+desirable design spaces and a Pareto frontier are also given.
+IV. DISCUSSION
+Figure 7 plots the performance of all the compared protocols
+and assignments representing the zero-load latency (X axis)
+and saturation throughput (Y axis) of a particular protocol for
+a given number of channels and assignment method.
+In general terms, BRS is preferred over token in terms
+of zero-load latency given its ability to transmit immediately
+when the channels are idle. Hence, we see most BRS points
+located in a low latency region. Among the assignment tech-
+niques, AS1 achieves similar results than AS3 and would
+probably be preferred as it does not require prior knowledge of
+the load of each node to assign the channels. On the downside,
+the throughput is half of that of token passing, at most.
+On the other hand, token passing can reach high throughput
+levels in the high capacity region, close to the maximum
+total bandwidth of the wireless network. However, while
+putting more channels reduces the latency significantly, the
+best latency in token passing is still several cycles away
+from the BRS values. Finally, we observe that it is hard to
+provide a good channel assignment overall: AS3 requires prior
+knowledge on the traffic distribution, AS1 does not perform
+well for hotspot traffic and AS2 has high latency.
+V. CONCLUSIONS
+This paper has explored several techniques to extend ran-
+dom access and token passing MAC protocols to multiple
+channels for wireless chip-scale networks. In general, more
+channels alleviate the problems of both types of protocols, in-
+creasing the throughput of random access and cutting down the
+latency of token passing to a few tens of cycles. Additionally,
+random access is more resilient to hotspot and bursty traffic
+and more scalable to massive chip-scale networks. However,
+the higher throughput achievable with token renders the de-
+cision of the protocol (and assignment) to choose extremely
+challenging. Hence, we see a trend similar to that of single-
+channel protocols: it would be desirable to develop a multi-
+channel protocol that is able to seamlessly obtain the best of
+both paradigms. This will be explored in future work.
+
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+page_content=' Filip Lemic†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Eduard Alarc´on∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' and Sergi Abadal∗ ∗NaNoNetworking Center in Catalunya (N3Cat),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Universitat Polit`ecnica de Catalunya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 08034 Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Spain †AI-driven Systems Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' i2Cat Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 08034 Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Spain Abstract—Wireless communications at the chip scale emerge as a interesting complement to traditional wire-based approaches thanks to their low latency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' inherent broadcast nature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' and ca- pacity to bypass pin constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' However, as current trends push towards massive and bandwidth-hungry processor architectures, there is a need for wireless chip-scale networks that exploit and share as many channels as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In this context, this work addresses the issue of channel sharing by exploring the design space of multi-channel Medium Access Control (MAC) protocols for chip-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Distinct channel assignment strategies for both random access and token passing are presented and evaluated under realistic traffic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' It is shown that, even with the improvements enabled by the multiple channels, both protocols maintain their intrinsic advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' INTRODUCTION Efficient integrated networks at the chip scale within Systems-in-Package (SiPs) are a prerequisite for high perfor- mance in such computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  Currently, most systems incorporate a Network-in-Package (NiP) consisting of a set of on-chip routers and intra-/inter-chip wired links [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' However, recent scaling [3], [4], specialization [5], [6], and disintegration trends [7], [8] are increasing the pressure placed on the interconnect, to the point that new communication paradigms may be required [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Among the emerging alternatives, wireless chip-scale com- munications stand as a promising contender [11]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This communication paradigm relies the use of modulated electro- magnetic waves for data transmission using the chip package as communications medium (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The resulting wireless in-package links provide low latency, inherent broadcast capa- bilities, and global reconfigurability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Since the communications medium is shared, wireless in- package communications require Medium Access Control (MAC) protocols to avoid or manage wasteful collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In this scenario, MAC protocols generally reduce to variants of multiplexing, random access, or token passing [15]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Even though recent works have demonstrated that computing packages could support a few frequency [22], [23] and space channels [24], [25], it is still unclear how MAC protocols can benefit from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This is because more than a few chan- nels are needed to implement truly scalable frequency/space multiplexing techniques [15], and most importantly, because multi-channel variants of random access and token passing have not been explored yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  This work is supported by the European Commission under H2020 grant WiPLASH (GA 863337) and HE grant WINC (GA 101042080).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Package Substrate Chiplet Chiplet Package Substrate Chiplet Chiplet Interposer Package Substrate Chiplet Chiplet Silicon bridge Package Substrate Chiplet Chiplet Wireless Antennas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Pictorial view of a wireless chip-to-chip communication link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This paper aims to bridge this gap by focusing on the study of multi-channel versions of the two most representative protocol types in chip-scale scenarios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' random access and token passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In particular, the main contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' We first describe the different ways we can extend random access and token passing with a small set of channels in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' III, we evaluate these protocol variants with traffic models typically used to mimic multiprocessor workloads [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This analysis sheds light on the impact of channel assignment on the protocol performance, as summa- rized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' IV and concluded in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' MULTI-CHANNEL MAC PROTOCOLS In this work, we describe three distinct channel assignment strategies for random access and token passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' As baselines, we take BRS [18] for random access and the baseline from [20] for token passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The strategies presented here are not provably optimal, but they are simple (as required by the resource constraints of the chip-scale scenario) and represen- tative of the potential techniques that can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  Assignment Methods for BRS In random access protocols such as BRS [18], nodes con- tend for channel access and back off if the channel is busy or there is a collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Assuming N nodes, we study three ways to reduce the collision probability using Nc channels, namely: AS1: Channels are assigned to nodes individually and ran- domly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' When a node has a packet to transmit, the node is assigned a random channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' If the channel is busy or there is a collision, nodes undergo a random back off and also choose a random channel to use in the next attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' AS2: Each channel is assigned to N Nc nodes statically follow- ing a uniform distribution, this is, assuming that all nodes have the same load (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 2, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' While this is not optimal for spatially unbalanced traffic, it serves as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' AS3: Channels are assigned to a variable number of nodes following a distribution that balances the load in each channel (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 2, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' To that end, nodes are ordered based on arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='11471v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='ET]  27 Jan 2023  Node  Node with information  Token  16 nodes  4 channels  Node  Node with information  Token  16 nodes  4 channels  16 nodes 4 channels Node Node with information Token Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Graphical representations of assignment techniques AS2 (left) and AS3 (right) for BRS assuming 16 nodes and 4 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 1 1 1 i 1 v  Node (low load)  Node (high load)  Token  Node  Node with information  Token  16 nodes  4 channels  Node Node with information Token 16 nodes 4 channels Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Graphical representations of the different assignment techniques for token passing assuming 16 nodes and 4 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' the expected normalized load and assigned to each channel in order following a greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Assignment Methods for Token Passing In token passing [27], typically, all N nodes are sorted forming a virtual ring and the token is passed in order through that ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In a version with Nc channels, each channel can be a token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The design decisions then lie on the number of rings and the nodes that form each ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' For instance: AS1: We assume as many rings as there are channels and map nodes uniformly to each ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In other words, we distribute them in rings of N Nc nodes, regardless of their expected load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' AS2: We assume a single virtual ring with multiple tokens circulating in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In this case, tokens can jump over other tokens: when node i holds a token for multiple cycles during a transmission, idle tokens that arrive at i-1 can jump to i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' AS3: This strategy is similar to AS1, but nodes are mapped to rings based on their expected load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This may lead to rings of different sizes, but similar in the expected overall load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' PERFORMANCE EVALUATION The architecture and application parameters are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  We implement both single-channel baselines and multi-channel versions of BRS and token passing as finite state machines in a modified version of Multi2sim that models wire- less links and supports collision detection [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The protocols are stressed with synthetic traffic modeling uneven injection distributions (through the σ parameter) and bursty temporal behavior (through the Hurst exponent H) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The default values for the different parameters are N = 64 nodes, Nc = 4 channels, H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='5 and σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Simulations are cycle-accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In all cases, we compare the packet latency (in cycles) and throughput (in packets/cycle) of the different options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Given the high number of protocol strategies and traffic types, instead of plotting the classical latency–throughput curve, we make use of box plots that summarize the latency and throughput statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In our plots, the X axis shows the parameters under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The plots have two Y axis: the left axis represents the latency and corresponds to the box plot values, whereas TABLE I CHARACTERISTICS OF SIMULATED PROTOCOLS AND APPLICATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  Application Synthetic traffic, H=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='5–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='85, σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='05–100 System N=64–512 cores, one antenna/core, 1-GHz clock Network 80-bit packets (preamble: 20 bits), Nc=1–4 channels Link BRS [18], Token passing Physical On-Off Keying, 20 Gb/s 101 102 Latency cycles/packet 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='7 Throughput paq/cycle BRS C1 AS1_C2 AS2_C2 AS3_C2 AS1_C4 AS2_C4 AS3_C4 101 102 Latency cycles/packet 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='8 1 Throughput paq/cycle TOKEN C1 AS1_C2 AS2_C2 AS3_C2 AS1_C4 AS2_C4 AS3_C4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Performance of multi-channel BRS (top) and token passing (bottom) for an increasing number of channels, C1 to C4, and different assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' the right axis represents the throughput and corresponds to single-value markers of saturation throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Since a single packet takes 4 cycles in a single channel to be transmitted, the maximum throughput is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='25 packets/cycle/channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Number of Channels Here, we discuss the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 4 for BRS and token passing and an increasing number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In general, it can be observed that BRS is less stable than token in terms of latency as the range of values is larger, with a higher number of outlier points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' However, BRS has a much better zero-load latency than token since, in BRS, the protocol allows nodes to start transmitting immediately when the channel is sensed idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This fact also can explain why independently of the parameters evaluated here (assignment, number of channels) the minimum latency is quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  The worst-case latency, however, clearly improves when having multiple channels, as the high load is distributed over multiple channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' On the other hand, in token passing, nodes must wait until they possess the token to start transmitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' For this reason, when the number of nodes is large, N = 64 in this case, the system remains idle much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The results for token passing depict a rather stable increase in saturation throughput as more channels are added, regardless of the assignment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This could be due to the use, by default, of non-bursty and non-hotspot traffic to evaluate scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' On the other hand, the results for BRS illustrate a different behavior than in token passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Firstly, BRS cannot reach a saturation throughput as high as token passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The main reasons are that channel contention and multiple collisions lead to channel waste and, hence, to a reduced throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Furthermore, BRS is more irregular than token passing in terms of saturation throughput as it depends  10 1 10 2 Latency cycles/paq 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
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+page_content='7 Throughput paq/cycle Ncores BRS AS1_64 AS1_128 AS1_512 AS2_64 AS2_128 AS2_512 AS3_64 AS3_128 AS3_512 10 1 10 2 Latency cycles/paq 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
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+page_content='7 Throughput paq/cycle Sigmas BRS AS1_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='05 AS1_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='1 AS1_1 AS1_100 AS2_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='05 AS2_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='1 AS2_1 AS2_100 AS3_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='05 AS3_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='1 AS3_1 AS3_100 101 102 Latency cycles/paq 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
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+page_content='7 Throughput paq/cycle Hs BRS AS1_H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
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+page_content='85 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Performance of multi-channel BRS protocol for an increasing number of nodes, N=64–512 (left graph), different spatial concentration levels, σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
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+page_content='5–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='85 (right graph), and different assignment techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 10 1 10 2 Latency cycles/paq 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
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+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  Performance of multi-channel token passing protocol for an increasing number of nodes, N=64–512 (left graph), different spatial concentration levels, σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
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+page_content='85 (right graph), and different assignment techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' on the percentage of collisions at high loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' As a result, the difference between the saturation throughput achieved for different assignments increases with the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Number of Nodes Next, we comment on the performance of BRS and token passing for an increasing number of nodes, with Nc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The results are shown in the left charts of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' BRS has a much lower latency than token passing due to its ability to transmit when the channel is idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The span of the latency values differs across number of nodes and assignments, but in general are restrained to similar values because in the end, the same aggregated load ends up being distributed over more nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Static assignment of channels (AS2) works worse than the other alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' On the other hand, from the plot of token passing, it is clear that more nodes lead to much higher latency due to the increase of the token turnaround time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In fact, the low-load latency is proportional to the number of nodes in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The span of the latency values is similar across the different system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In general, saturation throughput is slightly higher for a lower number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  In our protocols, having more nodes means having a higher population and, hence, a higher chance of collisions even for the same load for BRS, and a higher waiting time (or lower probability of having all nodes backlogged) in token passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' It seems, in any case, that BRS is more resilient to the change in the number of nodes as the drop is more subtle, except for AS3, where possibly the load balancing algorithm is not performing well when such a large number of nodes has to be classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Finally, all three assignments have very similar throughput in all cases for token passing, whereas AS1 (random channel assignment to individual packets) works better in BRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Hotspot Traffic We next discuss the results shown in the middle plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 6, which illustrate the impact of uneven spatial injection distribution on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' We remind that low/high values of σ mean that traffic is hotspot/evenly distributed [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In BRS, the hotspot behavior of traffic does not seem to have a large influence on the performance of the different assignment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The outlier, third quartile, and maximum values within the distribution seem to be mildly impacted by the hotspot nature of traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In general, BRS is resilient to such variations and actually could benefit from having a lower amount of nodes contending for the available channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  Still, the results show a small tendency to worse results when traffic is concentrated around a few nodes, possibly because of the nodes with higher load reaching higher backoff values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In AS3, this situation is avoided by proactively placing high-load nodes in different channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Similarly, in token passing, latency is affected by the concentration of traffic around a given set of nodes mostly because the different assignment methods are able to provide tokens quickly to nodes that need it, even if they are spaced apart within the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This is clearly visible in the extreme case of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Similarly, outlier values seem to be larger when traffic is more hotspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' We also observe how AS2 fails to provide a good performance at low loads, and this behavior is exacerbated for very hotspot traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The throughput of BRS in its different im- plementations does not vary significantly with the type of  spatial distribution of traffic, except for AS3, where a higher concentration of traffic around a few nodes seems to have a positive effect on the throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' One reason could be that the most active nodes are distributed over the different channels so that contention is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' That does not happen in other assignment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Different behavior is observed in token passing, where the hotspot behavior of traffic modifies the throughput of the different assignment methods, with AS3 being affected a bit less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This is because if the load is concentrated around a small set of nodes, a large portion of the airtime is wasted while passing the token among these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Bursty Traffic Finally, we present the latency and throughput results for an increasingly bursty traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The results are shown in the right plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 6 for BRS and token passing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Temporal injection of traffic is modeled through the Hurst exponent [26], with higher values indicating more bursty behavior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' longer bursts followed by longer silences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In BRS, it can be seen that the higher the value of H, the higher the latency in average and also the more un- predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This is because with an H of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='5, the packets are injected following a random Poisson process, which minimizes the probability of collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' However, when increasingly bursty traffic is considered, the probability of packets being injected (and nodes trying to transmit) in the same exact cycle increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  The effect is multiplicative with the burstiness, as the effect of cascading collisions leads to an exponential increase of the backoff time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This affects the system at all loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' On the other hand, token passing also suffers when bursty traffic is served, leading to very high latency especially for high values of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' The latency is a bit more stable than in the case of BRS, mainly because the protocol does not react with exponential backoffs, but rather with linear token passings to bursts of traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Still, the latency is much higher than that of BRS, discouraging its use for large number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' On one hand, it can be verified that in BRS, the saturation throughput remains rather constant across all assignments regardless of the value of the Hurst exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' A possible reason could stem from the behavior of the backoff mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' bursty traffic leads to a large number of collisions which increases latency even for low loads, but the protocol may converge to a large backoff value that can accommodate the load even if it comes in bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In other works, the backoff mechanisms spreads out the bursts of traffic over time, until all nodes are backlogged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' On the other hand, it can be seen that in the case of token passing, the saturation throughput seems to drop significantly for higher numbers of H, to a point that the achieved throughput becomes comparable with that of BRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  A potential reason for this behavior is the lack of an adaptive mechanism to react to bursts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' the token has to still move around the ring even if bursts of traffic lead to the generation of multiple packets in a given node, leading to gaps where the wireless channel remains silent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' When traffic is less bursty, the probability of such events is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' AS2_N64 AS3_S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='05 AS1_N64 AS1_H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='5 0 10 20 30 40 50 60 0 0,2 0,4 0,6 0,8 1 1,2 Latency cycles/packet Throughput packets/cycle B_C T_C B_N T_N B_S T_S B_H T_H LOW LATENCY REGION HIGH CAPACITY REGION Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Summary of the latency and throughput results over all the protocols, assignment methods, and traffic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' B and T stand for BRS (random access) and token passing, C and N denote number of channels and nodes, whereas S and H represent the different spatial and temporal injection distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' For instance, the B C symbols represent the latency-throughput of all the assignment methods for BRS for different number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Two desirable design spaces and a Pareto frontier are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' DISCUSSION Figure 7 plots the performance of all the compared protocols and assignments representing the zero-load latency (X axis) and saturation throughput (Y axis) of a particular protocol for a given number of channels and assignment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In general terms, BRS is preferred over token in terms of zero-load latency given its ability to transmit immediately when the channels are idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  Hence, we see most BRS points located in a low latency region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Among the assignment tech- niques, AS1 achieves similar results than AS3 and would probably be preferred as it does not require prior knowledge of the load of each node to assign the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' On the downside, the throughput is half of that of token passing, at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' On the other hand, token passing can reach high throughput levels in the high capacity region, close to the maximum total bandwidth of the wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' However, while putting more channels reduces the latency significantly, the best latency in token passing is still several cycles away from the BRS values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Finally, we observe that it is hard to provide a good channel assignment overall: AS3 requires prior knowledge on the traffic distribution, AS1 does not perform well for hotspot traffic and AS2 has high latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' CONCLUSIONS This paper has explored several techniques to extend ran- dom access and token passing MAC protocols to multiple channels for wireless chip-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' In general, more channels alleviate the problems of both types of protocols, in- creasing the throughput of random access and cutting down the latency of token passing to a few tens of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' Additionally, random access is more resilient to hotspot and bursty traffic and more scalable to massive chip-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' However, the higher throughput achievable with token renders the de- cision of the protocol (and assignment) to choose extremely challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  Hence, we see a trend similar to that of single- channel protocols: it would be desirable to develop a multi- channel protocol that is able to seamlessly obtain the best of both paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content=' This will be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='  REFERENCES [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
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+page_content=' Ogras, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FJT4oBgHgl3EQfMCwG/content/2301.11471v1.pdf'}
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+CURSIVE CAPTION TEXT DETECTION IN VIDEOS
+Ali Mirza
+Center of Artificial Intelligence
+Mantalus
+Melbourne, Australia
+ali.mirza@mantalus.com
+Imran Siddiqi
+Center of Excellence in Artificial Intelligence
+Bahria University
+Islamabad, Pakistan
+imran.siddiqi@bahria.edu.pk
+Keywords Text Detection · Script Identification · Deep Neural Networks (DNNs) · Convolutional Neural Networks
+(CNNs) · Video Text · Video Frames Dataset
+ABSTRACT
+Textual content appearing in videos represents an interesting index for semantic retrieval of videos
+(from archives), generation of alerts (live streams) as well as high level applications like opinion
+mining and content summarization. One of the key components of such systems is the detection
+of textual content in video frames and the same makes the subject of our present study. This
+paper presents a robust technique for detection of textual content appearing in video frames. More
+specifically we target text in cursive script taking Urdu text as a case study. Detection of textual
+regions in video frames is carried out by fine-tuning object detectors based on deep convolutional
+neural networks for the specific case of text detection. Since it is common to have videos with caption
+text in multiple-scripts, cursive text is distinguished from Latin text using a script-identification
+module. Finally, detection and script identification are combined in a single end-to-end trainable
+system. Experiments on a comprehensive dataset of around 11,000 video frames report an F-measure
+of 0.91.
+1
+Introduction
+In the recent years, there has been a tremendous increase in the amount of digital multimedia data, especially the video
+content, both in the form of video archives and live streams. According to statistics [6], 300 hours of video is being
+uploaded every minute on the YouTube. A key factor responsible for this enormous increase is the availability of
+low-cost smart phones equipped with cameras. With such huge collections of data, there is a need to have efficient
+as well as effective retrieval techniques allowing users retrieve the desired content. Traditionally, videos are mostly
+stored with user assigned annotations or keywords which are called tags. When a content is to be searched, a keyword
+provided as query is matched with these tags to retrieve the relevant content. The assigned tags, naturally, cannot
+encompass the rich video content leading to a constrained retrieval. A better and more effective strategy is to search
+within the actual content rather than simply matching the tags i.e. Content based Image or Video Retrieval. CBVR
+systems have been researched and developed for long a time and allow a smarter way of retrieving the desired content.
+The term content may refer to the visual content (for example objects or persons in the video), audio content (the
+spoken keywords for instance) or the textual content (News tickers, anchor names, score cards etc.). Among these, the
+focus of our current study lies on textual content. More specifically, we target a smart retrieval system that exploits the
+textual content in videos as an index.
+The textual content in video can be categorized into two broad classes, scene text and caption text. Scene text (Figure 1)
+is captured through camera during the video recording process and may not always be correlated with the content.
+Examples of scene text include advertisement banners, sign boards, text on a T-shirt etc. Scene text is commonly
+employed for applications like robot navigation and assistance systems for the visually impaired. Artificial or caption
+text (Figure 2) is superimposed on video and typical examples include News tickers, movie credits, score cards, names
+of anchors etc. Caption text is generally correlated with the video content and is mostly applied for semantic retrieval of
+arXiv:2301.03164v1  [cs.CV]  9 Jan 2023
+
+Cursive Caption Text Detection in Videos
+videos.
+Figure 1: Examples of Scene Text
+Figure 2: Examples of Caption Text
+The key components of a textual content based indexing and retrieval system include detection of text regions [36],
+extraction of text (segmentation from background) [37], identification of script (for multi-script videos) [25] and
+finally recognition of text (through a video OCR) [16]. Among these, we focus on detection of text in the current
+study. Detection of text can be carried out using unsupervised [4, 8, 3], supervised [63, 18, 32] or hybrid [36, 53]
+approaches.
+Unsupervised text detection employs image analysis techniques to discriminate between text and
+non-text regions. Supervised methods, on the other hand, involve training a learning algorithm with examples of text
+and non-text regions to discriminate between the two. In some cases, a combination of the two techniques is em-
+ployed where the candidate text regions identified by unsupervised methods are validated through a supervised approach.
+This paper presents a comprehensive framework for video text detection in a multi-script environment. Though we
+primarily target cursive caption text, since video frames frequently contain text in more than one script, text in the
+Roman script is also detected by the proposed technique. The key highlights of this study are outlined in the following.
+• Development of a comprehensive dataset of video images with ground truth information supporting evaluation
+of detection and recognition tasks.
+• Adaptation of various deep learning based object detectors for detection of textual content.
+• Combination of text detection and script identification in a single end-to-end system.
+• Validation of proposed technique through an extensive series of experiments and a comprehensive performance
+comparison of various detectors.
+The paper is organized as follows. In the next section, we present an overview of the current state-of-the-art on detection
+of textual content in videos. In Section 3, we introduce the dataset developed in our study along with the ground
+truth information. Section 4 presents the details of the proposed framework while Section 5 presents the experimental
+protocol, the realized results and the corresponding discussion. Finally, Section 6 concludes the paper with a discussion
+on open challenges on this subject.
+2
+
+EIVE YOUR
+SFORTUNE
+ELECTIONCOMMISS
+LOF
+PAKISTAN
+02
+www.samaa.ty
+O92NEWSHD
+O92NEWSCHANNELWWW.92NEWSHD.TV
+NEWS
+SAMAA
+11SWAT
+4200
+ja
+www.samaa.ty
+SAMAA
+SundayCursive Caption Text Detection in Videos
+2
+Background
+Detection of textual content in videos, images, documents and natural scenes has been investigated for more than four
+decades. The domain has matured progressively over the years starting with trivial image analysis based systems to
+complex end-to-end learning based systems. We discuss notable contributions to text detection in the following while
+detailed surveys on the problem (and related problems) can be found in [40, 60, 68, 72, 48].
+Text detection refers to localization of textual content in images. Techniques proposed for detection of text are typically
+categorized into unsupervised and supervised approaches. While unsupervised approaches primarily rely on image
+analysis techniques to segment text from background, supervised methods involve training a learning algorithm to
+discriminate between text and non-text regions.
+Unsupervised text detection techniques include edge-based methods [4, 8, 3, 20, 24] which (assume and) exploit
+the high contrast between text and its background; connected component based methods [27, 28, 31, 39] which
+mostly rely on the color/intensity of text pixels and texture-based methods [2, 22] which consider textual content
+in the image as a unique texture that distinguishes itself from the non-text regions. Texture based methods have
+remained a popular choice of researchers and features based on Gabor filters [11], wavelets [64], curvelets [14], local
+binary patterns (LBP) [1], discrete cosine transformation (DCT) [75], histograms of oriented gradients (HoG) [9] and
+Fourier transformation [51] have been investigated in the literature. Another common category of techniques includes
+color-based methods [65, 66, 50] which are similar in many aspects to the component-based methods and employ color
+information of text pixels to distinguish it from non-text regions.
+Supervised approaches for detection of textual content typically employ state-of-the-art learning algorithms which
+are trained on examples of text and non-text blocks either using pixel values or by first extracting relevant features.
+Classifiers like naive Bayes [52], Support Vector Machine [74], Artificial Neural Network [67, 36] and Deep Neural
+Networks [30] have been investigated for this problem over the years.
+In the recent years, deep learning based solutions have been widely applied to a variety of recognition problems and
+have outperformed the traditional techniques. Among deep learning based techniques adapted for text detection, Huang
+et al. [21] employed sliding windows with CNNs to detect textual regions in low resolution scene images. Likewise,
+fully convolutional networks are explored for detection of textaul regions in [73] and the technique is evaluated on
+various ICDAR datasets. A similar work is presented by Gupta et al. [15] where CNNs are trained using synthetic data
+for detection of text at multiple scales from natural images. Another method called ‘SegLink’, is proposed in [49] that
+relies on decomposing the text into segments (oriented boxes of words or lines) and links (connecting two adjacent
+segments). The segments and links are detected using fully convolutional networks at multiple scales and combined
+together to detect the complete text line. In [57] vertical anchor based method is reported that predicts text and non-text
+scores of fixed size regions and reports high detection performance on the ICDAR 2013 and ICDAR 2015 datasets. In
+another notable work, Wang et al. [61] present a framework based on conditional random field (CRF) to detect text in
+scene images. Authors define a cost function by considering the color, stroke, shape and spatial features with CNN for
+effective detection of textual regions.
+Among other end-to-end trainable deep neural networks based systems, Liao et al. [32] present a system called
+‘TextBoxes’ which detects text in natural images in a single forward pass network. The technique was later extended to
+‘TextBoxes++’ and was evaluated on four public databases outperforming the state-of-the-art. He et al. [18] improved
+the convolutional layer of CNNs to detect text with arbitrary orientation. EAST [76], is another well-known scene text
+detector that provides promising results in challenging scenarios. In another study [63], an ensemble of Convolutional
+Neural Networks (CNNs) is trained on synthetic data to detect video text in East Asian languages.
+The literature is relatively limited once it comes to detection of cursive caption text. Among one of the preliminary
+works, Jamil et al. [26] exploit edge based features with morphological processing to detect Urdu caption text from
+a small set of 150 video frames. The same study was extended by Raza et al. [54] and evaluated on a larger set of
+1000 video frames reporting a recall of 0.80. The dataset of images termed as IPC Artificial Text dataset [54] was
+also made publicly available. In a later study [41] by the same group, the authors proposed a cascaded framework
+of spatial transforms to detect caption text in five different scripts including Arabic and Urdu. In a relatively recent
+work on detection of Arabic caption text, Zayene et al. [70] employ a combination of stroke width transform with a
+convolutional auto-encoder and evaluate the technique on a publicly available dataset AcTiV-DB [71]. In one of our
+3
+
+Cursive Caption Text Detection in Videos
+previous works [36], we investigated a combination of image analysis techniques with textural features to detect textual
+regions in video frames and realized an F-measure of 0.80 on 1000 images.
+Summarizing, it can be concluded that the problem of text detection has been dominated by the application of different
+deep learning based techniques in the recent years. The availability of benchmark datasets has also contributed to
+the rapid developments in this area. While detection of text in languages based on the Latin alphabet has received
+significant research attention and is very much mature, detection of cursive text still remains a relatively less addressed
+and challenging issue. Development of a (generic) text detector that could work in multi-script environments also
+remains an open problem.
+In the next section, we introduce the dataset that has been collected and labeled as a part of this study.
+3
+Dataset
+Availability of labeled datasets is of utmost importance for algorithmic development and evaluation of any computerized
+system. From the perspective of Urdu caption text detection, a dataset of 1000 labeled video frames has been made
+publicly available [54]. A collection of 1000 frames, however, seems to be very small to generalize the findings for
+practical applications. We, therefore, collected and labeled a comprehensive dataset of video frames allowing evaluation
+of text detection and text recognition tasks. We collected a set of 46 videos from four different News channels in
+Pakistan. All videos are recorded at a resolution of 900 × 600 and a frame rate of 25 fps. Frames in these video
+contain textual content in two languages, (cursive) Urdu and English. The collected video frames are labeled from two
+perspectives, detection and recognition. For detection, the bounding rectangle of all text regions in a frame is labeled
+and stored. Similarly, for recognition, the transcription of each text line is stored as ground truth.
+In the literature, several evaluation metrics have been proposed to evaluate the performance of text detection
+systems [26, 35, 62]. In our system, for evaluation of the text detection module, we employ the most commonly used
+area based precision and recall measures reported in [26] and defined as follows.
+Let AE be the estimated text area given by the system and AT be the ground truth text area, then the precision P and
+recall R are defined as:
+P =
+AE ∩ AT
+AE
+(1)
+R =
+AE ∩ AT
+AT
+(2)
+The precision and recall measures can be combined in a single F-measure as follows.
+F =
+2 × Precision × Recall
+Precision + Recall
+(3)
+The same idea can be extended to multiple images by simply summing up area of intersection and dividing by
+the total ground truth area (in N images) for recall and the total detected area for precision. To compute these
+measures, for each frame, we need to store the actual location of the textual content. The text detected automatically
+by the system can then be compared with the ground truth text regions to compute precision, recall and F-measure.
+The idea is illustrated in Figure 3.
+Figure 3-a illustrates an example where the text regions detected by the
+system are shown while Figure 3-b illustrates the ground truth text locations for the given frame. The detected and
+ground truth text regions can be compared to compute the metrics defined earlier and quantify the detection performance.
+To facilitate the labeling process and standardize the ground truth data, a comprehensive labeling tool has been developed
+that allows storing the location of each textual region in a frame along with its ground truth transcription. The location
+is stored in terms of the x and y coordinates of the top left of the bounding box along its width and height. The ground
+truth information of each frame is stored as an XML file that comprises frame meta data and the information on textual
+regions. A screen shot of the labeling tool is presented in Figure 4 while the ground truth information of an example
+frame is illustrated in Figure 5.
+It is known that videos typically contain 25–30 frames per second; consequently, successive frames in a video contain
+redundant information (both visual and textual content). From the view point of automatic analysis systems, frames
+4
+
+Cursive Caption Text Detection in Videos
+Figure 3: Text regions in an image and the corresponding ground truth image
+Figure 4: Screen shot of ground truth labeling tool for text data
+Figure 5: An XML file containing ground truth information of a frame
+with unique content are of interest. Hence, each single video frame does not need to be labeled as major proportions of
+such frames will have exactly the same textual information. In our study, we have extracted more than 11,000 frames
+5
+
+BREAKINGNEWS
+BREAKI
+BREAK
+NEW
+e
+NATIONALOASPAKISTANWILLNEVERLEAVEITSKASHMIRI BRETHRENALONE:COAS
+NATIONAL OAS  PAKISTAN WILL NEVER LEAVEITS KASHMIRIBRETHREN ALONE:
+COAS Video Images Groud Truth Labeling Tool
+十
+ Attributes
+Text Type
+Frames
+Frame No
+O Atficial
+O Scene
+Browse
+C:NUsers VAliNDesktop \Frames
+Samaa_News_2017041
+Language
+Samaa_News_20170413_11^
+O Urdu
+Samaa_News_20170413_11
+X: 891 Y : 358
+Urdu Text
+Samaa_News_20170413_11
+Rectangle Location
+X84Y522
+Samaa_News_20170413_11
+Samaa_News_20170413_1
+Rectangle Siz
+Width
+Heighi
+4
+5
+6
+7
+8
+Samaa_News_20170413_11
+588
+54
+c
+‘s
+上
+Samaa News 20170413 1
+Rectangle Area 
+Samaa_News_20170413_11
+31752
+3
+n
+Samaa_News_20170413_11
+Image Size 
+Samaa_News_20170413_11
+900 × 600
+Samaa_News_20170413_11
+Grids
+Samaa_News_20170413_11
+ Cross Hair
+Space
+Samaa News 20170413 11
+cill
+Samaa_News_20170413_11
+Channe
+O Ary Default
+Samaa_News_20170413_11
+O Samaa
+ Store Text Feed
+Samaa_News_20170413_11~
+ekung O
+REPEAT
+O Express
+eLaw
+SAMAA
+REPEAT
+SAMAA
+ Done Generate XML
+215
+ Zoom Selected
+Zoomed Preview
+Add
+Remove Feed
+Remaining Images : 34<?xml version="1.o" encoding="utf-8"?>
+<VideoLabel>
+<FrameMetaData>
+<Video>Frames</Video>
+<Channel>Samaa News</Channel>
+<FrameNo>Samaa_News_20170413_113759_10701</FrameNo>
+</FrameMetaData>
+<TextFeeds TotalFeeds="7">
+<UrduFeeds TotalUrduFeeds="5"
+<TextLine ID="2" Text Type="Artificial" X="710" Y="461" Width="130" Height="29" Text=",l-"/>
+/..**!..-+xa1 ..9.-tah .0..-+pm ..ss..- ..9..-x ..Tett++..-d x1 ....- aux1>
+<TextLine ID="4" Text Type="Artificial" X="711" Y="555" Width="123" Height="25" Text="15_2
+（"/>
+c:1 -x .s..-tah ..88s..-m ....- .8..-x ..ett+..=d x .S..- aux>
+</UrduFeeds>
+<English Feeds TotalEnglishFeeds="2">
+<TextLine ID-"1" TextType="Artificial" X-"720" Y="441" Width-"106" Height="20" Text="repeat" />
+<TextLine ID-"2" TextType="Artificial" X="71o" Y="495" Width="131" Height="25" Text="samaa" />
+</EnglishFeeds>
+</TextFeeds>
+</VideoLabel>Cursive Caption Text Detection in Videos
+from videos with an attempt to have as much unique text as possible. The statistics of videos, frames and text lines of
+our dataset are presented in Table 1. Inspired from the Arabic caption text dataset AcTiV-DB [71], we have named the
+our dataset UTiV (Urdu Text in Video). The dataset along with its ground truth has also been made publicly available1
+to support quantitative evaluation of text detection and recognition tasks.
+Table 1: Statistics of labeled video frames
+S#
+Channel
+Videos
+Labeled Images
+Urdu Lines
+English Lines
+1
+Ary News
+7
+3,206
+10,250
+3,605
+2
+Samaa News
+13
+2,503
+10,961
+4,411
+3
+Dunya News
+16
+3,059
+10,723
+8,861
+4
+Express News
+10
+2,424
+8,536
+6,755
+Total
+46
+11,203
+40,470
+23,632
+4
+Methods
+This section presents the details of detecting textual content from video frames. Detection relies on adapting object
+detectors based on deep convolutional neural networks for text regions. Once the text is detected, script of the detected
+text is identified by employing the ConvNets in a classification framework. Subsequently, text detection and script
+identification are combined in a single end-to-end system that detects the textual content along with its script. Details
+are presented in the following sections.
+4.1
+Deep Learning based Object Detectors
+Deep neural networks enjoy a renewed interest of the machine learning community thanks primarily to the availability
+of high performance computing hardware (GPUs) as well as large data sets to train these systems. A major development
+contributing to the current fame of deep learning was the application of ConvNets by Krizhevsky et al. [29] on the
+ImageNet Large Scale Visual Recognition competition [46], which greatly reduced the error rates. Since then, CNNs
+are considered to be state-of-the-art feature extractors and classifiers [55, 56] and have been applied to a variety of
+recognition tasks [5, 58, 10].
+While traditional CNNs are typically employed for object classification, Region-based Convolutional Networks
+(R-CNN) [13] and their further enhancements Fast R-CNN [12] and Faster R-CNN [45] adapt CNNs for object
+detection. In addition to different variants of R-CNN, a number of new architectures have also been proposed in the
+recent years for real time object detection. The most notable of these include YOLO (You Only Look Once) [42] and
+SSD (Single Shot Detector) [34]. Each of these object detectors can be trained to detect C object classes (plus one for
+the background). The output of the detector is the location of the bounding box (four coordinates) containing one of the
+C classes as well as the class confidence score.
+In our study, for detection of textual content in a given frame, we investigated a number of CNN based object detectors.
+Although, many object detectors are trained with thousands of class examples and provide high accuracy in detection
+and recognition of different objects, these object detectors can not be directly applied to identify text regions in images.
+These models have to be tuned to the specific problem of discrimination of text from non-text regions. The convolutional
+base of these models can be trained from scratch or, known pre-trained models can be fine-tuned by training them on
+text and non-text regions. In our study, we investigated the following object detectors for localization of text regions.
+• Faster R-CNN
+• Region-Based Fully Convolutional Networks (R-FCN)
+• Single Shot Detector (SSD)
+• You Only Look Once (YOLO)
+For completeness, we provide a brief overview of these object detectors in the following sections.
+1http://cbvir.media-tics.net/
+6
+
+Cursive Caption Text Detection in Videos
+4.1.1
+Faster R-CNN
+Faster R-CNN [45] is an enhanced version of its predecessors R-CNN [13] and Fast R-CNN [12]. Each of these
+detectors exploits the powerful features of ConvNets for object localization as well as classification. R-CNN was one
+of the first attempts to apply ConvNets for object detection. An R-CNN scans the input image for potential objects
+using Selective Search [58] that generates around 2,000 region proposals. Each of these region proposals is then fed
+to a CNN for feature extraction. The output of the CNN is finally employed by an SVM to classify the object and a
+linear regressor to tighten the bounding box. R-CNN was enhanced in terms of training efficiency by extending it to
+Fast R-CNN [12]. In Fast R-CNN, rather than separately feeding each region proposal to the ConvNet, convolution is
+performed only once on the complete image and the region proposals are projected on the feature maps. Furthermore,
+the SVM in R-CNN was replaced by a softmax layer extending the network to predict the class labels rather than
+using a separate model. While Fast R-CNN significantly reduced the time complexity of the basic R-CNN, a major
+bottleneck was the selective search algorithm to generate the region proposals. This was addressed through Region
+Proposal Network (RPN) in Faster R-CNN [45] which shares convolutional features with the detection network. RPN
+predicts region proposals which are then fed to the detection network to identify the object class and refine the bounding
+boxes produced by the RPN. A summary of various R-CNN models in presented in Figure 6.
+Figure 6: Summary of R-CNN Family based Object Detectors
+4.1.2
+You Only Look Once (YOLO)
+YOLO [42] takes a different approach to object detection primarily focusing on improving the detection speed (rather
+than accuracy). As the name suggests, YOLO employs a single pass of the convolutional network for localization and
+classification of objects from the the input images. The input image is divided into a grid and an object is expected
+to be detected by the grid which holds the center of the object. Each cell in the grid predicts up to two bounding
+boxes (and class probabilities). The network comprises 24 convolutional and fully connected layers. YOLO works in
+real time but in terms of accuracy, it is known to make significant localization errors in comparison to region based
+object detectors (Faster R-CNN for instance). YOLO was later enhanced to YOLO9000 [43] by introducing batch
+normalization, increasing the resolution of the input image (by a factor of 2) and introducing the concept of anchor boxes.
+YOLO9000 employs Darknet 19 architecture with 19 convolutional layers, 5 max pooling layers and a softmax layer for
+classification objects. Incremental improvements in YOLO v2 resulted in YOLO v3 [44] that uses logistic regerssion to
+predict the score of objectness for each bounding box. Furthermore, it employs class-wise logistic classifiers (rather
+than softmax) allowing multi-label classification.
+7
+
+Box offset
+SVM object
+Box offset
+Softmax
+Box offset
+Softmax
+Regressor
+Classifier
+Regressor
+Classifier
+Regressor
+Classifier
+Independent
+Joint
+Joint
+Region CNN
+Region CNN
+Region CNN
+Fine-tuned
+Fine-tuned
+Features
+Features
+Features
+ROl Pooling
+Pre-trained
+RPN
+Pre-trained
+ROl Pooling
+Region
+Region
+Deep CNN
+Deep CNN
+Proposal
+Proposal
+Deep CNN
+Independent
+Independent
+Independent
+MP
+GMP
+R-CNN
+Fast R-CNN
+Faster R-CNNCursive Caption Text Detection in Videos
+4.1.3
+Single Shot Detector (SSD)
+Unlike the R-CNN series object detectors which require two shots to detect objects in an image, Single Shot Multi-box
+Detector [34], as the name suggests, requires a single shot to detect objects (similar to YOLO). SSD relies on the idea
+of default boxes and multi-scale predictions and directly applies bounding box regression to the default boxes without
+generating the region proposals. Detection at multiple scales are handled by exploiting the feature maps of different
+convolutional layers corresponding to different receptive fields in the input image. The architecture has an input size
+of 300 × 300 × 3 and primarily builds on the VGG-16 architecture discarding the fully connected layers. VGG-16 is
+used as base network mainly due to its robust performance of image classification tasks. The bounding box regression
+technique of SSD is inspired by [56] while the MultiBox relies on priors, the pre-computed fixed size bounding boxes.
+The priors are selected in such a way that their Intersection over Union ratio (with ground truth objects) is greater than
+0.5. The MultiBox starts with the priors as predictions and attempt to regress closer to the ground truth bounding boxes.
+SSD works in real time but requires images of fixed square size and is known to miss small objects in the image.
+4.1.4
+Region-Based Fully Convolutional Networks (R-FCN)
+R-FCN [7] builds on the idea of increasing the detection accuracy by maximizing the shared calculations. R-FCN
+generates position-sensitive score maps to represent different relative positions of an object. An object is represented by
+k2 relative positions dividing it into a grid of size k × k. A ConvNet (ResNet in the original R-FCN paper) sweeps the
+input image and an additional fully convoltional layer produces the position-sensitive scores in k2 × (C + 1) score maps
+where C is the number of classes plus 1 class for the background. A fully convolutional proposal network generates
+regions of interest which are divided in k2 bins and the corresponding class probabilities are obtained from the score
+maps. The scores are averaged to convert the k2 × (C + 1) values into a one dimensional (C + 1) sized vector which is
+finally fed to the softmax layer for classification. Localization is carried out using the bounding box regression similar
+to other object detectors. R-FCN speeds up the detection in comparison to Faster R-CNN but compared to other Single
+Shot methods, it requires more computational resources.
+4.2
+Adapting Object Detectors for Text Detection
+In the context of object detection, the problem of text detection can be formulated as a two class problem. The text
+regions represent object of interest while the non-text regions need to be ignored. The object detectors discussed in
+the previous sections are adapted for text detection using two pre-trained models, ResNet 101 [17] and Inception
+v2 [23]. These models are trained on the large scale Microsoft COCO (Common Objects in Context) database [33]. The
+database contains images of 91 different object types with a total of 2.5 million labeled instances in 328K images. The
+pre-trained network serves as starting point rather than random weight initialization and the network is made to learn the
+specific class labels (text or non-text) by continuing back propagation. The ground truth localization information of the
+textual regions in the video frames is employed for training the models, the overall workflow being illustrated in Figure 7.
+A critical aspect in employing object detectors for text detection is the choice of anchor boxes.
+The anchor
+boxes in all the detectors have been designed to detect general object categories. Text appearing in videos has
+specific geometric properties in terms of size and aspect ratio hence the default anchor boxes of the detectors
+need to be adapted to detect textual regions. We carried out a comprehensive analysis of the textual regions in
+terms of width, height and aspect ratios of the bounding boxes. As a result of this analysis we have chosen a
+base anchor of size 256 × 256. To each anchor box we apply three scales (1.0, 2.0, 5.0) and five aspect ratios
+(0.125, 0.1875, 0.25, 0.375, 0.50) as illustrated in Figure 8. Models are fine-tuned using the proposed anchor boxes
+and the effectiveness of these anchor boxes is validated through experimental study as presented in Section 5 of the paper.
+4.3
+Script Identification
+As discussed earlier, we primarily target detection of cursive caption text. However, like many practical scenarios, video
+frames in our case contain bilingual textual content (Urdu & English). Consequently, once the text is detected, we
+need to identify the script of each detected region (Figure 9) so that the subsequent processing of each type of script
+can be carried out by the respective recognition engine. For script identification, we employ CNNs in a classification
+framework (rather than detection). Urdu and English text lines are employed to fine-tune CNNs to discriminate between
+the two classes. Once trained, the model is able to separate text lines as a function of the script. Similar to detection,
+rather than training the networks from scratch, we fine-tune known pre-trained models (Inception and ResNet) to solve
+the two-class classification problem.
+8
+
+Cursive Caption Text Detection in Videos
+Figure 7: Overview of adapting object detectors for text detection
+Figure 8:
+Anchor boxes (base size 256 × 256) at three scales (1.0, 2.0, 5.0) and five aspect ratios
+(0.125, 0.1875, 0.25, 0.375, 0.50)
+Figure 9: Script identification of detected text lines
+9
+
+Test Image
+Frames
+VIDEO
+Output Image
+Ground Truth Data
+Object
+Fine-Tuning
+Text Detector
+Inference
+Detector
+CNN Model
+Pre-trained
+CNN
+ModelsScale: 1.0
+(a)
+Scale: 2.0
+(b)
+Scale: 5.0
+(c)Urdu Text Regions
+Bilingual Text Regions
+SHAZIA ZEESHAN
+ZAFAR HILALY
+Script
+Identification
+English Text Regions
+AHMAD AWAIS
+ORYAMAQBOOL JAN
+SHAZIA ZEESHAN
+ZAFAR HILALY
+AHMAD AWAIS
+ORYAMAQBOOLJANCursive Caption Text Detection in Videos
+4.4
+Hybrid Text Detector & Script Identifier
+Detection of text and identification of script, as discussed previously, can be implemented in a cascaded framework
+where the output of text detector is fed to the script identifier. A deep learning framework can be tuned to discriminate
+between text and non-text regions and the extracted text regions can be fed to a separate script recognition model that
+identifies the script of the detected text. This, however, introduces a bottleneck of training two separate networks.
+Furthermore, the cascaded solution also implies that errors in detection are propagated to the next step as well. We,
+therefore, propose to combine the text detector and script identifier in a single hybrid model. Rather than treating
+detection as a two-class problem (text and non-text), we consider it as a three class problem, i.e. non-text regions,
+English text and Urdu text. This not only avoids training two separate models but also eliminates the accumulation
+of errors in a cascaded solution. The superiority of the combined text detector and script identifier is also supported
+through quantitative evaluations as discussed in the next section.
+All detectors are trained in an end-to-end manner with a multi-task objective function that combines the classification
+and regression losses. The evolution of training loss for the investigated detectors (with Inception and ResNet) is
+illustrated in Figure 10 where it can be seen that the loss begins to stabilize from 30 epochs on wards.
+Figure 10: Training loss of various detectors – Hybrid text detector and script identifier
+5
+Experiments and Results
+The detection performance is evaluated through a series of experiments carried out on the collected set of video frames.
+We first present the experimental protocol followed by the detection results of various object detectors. We then present
+the script identification results and the performance of the combined text detector and script identifier. Furthermore,
+performance sensitivity of the system as well as a comparison with state-of-the-art is also presented.
+5.1
+Experimental Settings
+As introduced in Section 3, we collected a total of 11,203 video frames from four different News channel videos. The
+localization information of text regions in these frames is used to train and subsequently evaluate the text detection and
+script identification performance. The distribution of frames into training and test sets along with the number of text
+lines in each set is summarized in Table 2 while the details of detection performance are presented in the next section.
+5.2
+Text Detection Results
+Object detectors including Faster R-CNN, YOLO, SSD and R-FCN are adapted to detect textual content by fine-tuning
+the Inception and ResNet pre-trained models and changing the anchor boxes as discussed previously. Performance of
+10
+
+7
++FasterRCNN-Inception
+6.5
+Faster RCNN-ResNet
+6
+SSD-Inception
+SSD-ResNet101
+5.5 
+米RFCN-Inception
+5
+RFCN-ResNet
+TRANING LOSS
+4.5
+YoloV3
+3.5
+3
+2.5
+2
+1.5
+1
+0.5 
+0
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+EPOCHSCursive Caption Text Detection in Videos
+Table 2: Data distribution for text detection experiments
+Train
+Test
+Frames
+Lines
+Frames
+Line
+Urdu
+8,500
+31,321
+2,703
+9,149
+English
+16,207
+7,425
+Total
+49,046
+11,056
+each of these detectors in terms of precision, recall and F-measure is summarized in Table 3. It can be seen that in all
+cases, detectors pre-trained on Inception outperform those trained on ResNet. Among various detectors, Faster R-CNN
+reports the highest F-measure of 0.90. The lowest performance is reported by Yolo reading an F-measure of 0.66. A
+comprehensive study on the trade-off between speed and accuracy of various object detectors is presented in [19] and
+our findings on detection of text are consistent with those of [19]. It is also important to recap that precision and recall
+are computed using area based metrics. As a result, if the detected bounding box is larger (smaller) than the ground
+truth, it results in penalizing the precision (recall) of the detector as illustrated in Figure 11. The output of the Faster
+R-CNN based text detector for few sample frames in our dataset is illustrated in Figure 12.
+Table 3: Text Detection Results
+RestNet
+Inception
+Model
+Precision
+Recall
+F-Measure
+Precision
+Recall
+F-Measure
+SSD
+0.83
+0.71
+0.77
+0.82
+0.77
+0.80
+R-FCN
+0.79
+0.86
+0.82
+0.84
+0.89
+0.86
+Faster R-CNN
+0.82
+0.90
+0.85
+0.86
+0.95
+0.90
+Yolo
+-
+-
+-
+0.63
+0.69
+0.66
+Figure 11: Computation of precision and recall (a):Ground Truth Bounding Box (b): Detected region is larger than
+ground truth (c):Detected region is smaller than ground truth (d):Detected region overlaps perfectly with the ground
+truth
+In an attempt to provide an insight into the detection errors, few of the errors are illustrated in Figure 13. It can be seen
+that in most cases, the detector is able to detect the textual region but the localization is not perfect i.e. in some cases
+the bounding box is larger (shorter) than the actual content leading to a reduced precision (recall).
+11
+
+SAMAA
+SAMAA
+(a)
+(b)
+SAMAA
+SAMAA
+(c)
+(d)Cursive Caption Text Detection in Videos
+ 
+ 
+ 
+ 
+Figure 12: Text detection results on sample images (Faster R-CNN with Inception)
+Figure 13: Imperfect Localization of Text Regions
+5.3
+Script Identification Results
+For script identification, we employ the same distribution of frames into training and test sets as that of the detection
+protocol. Text lines from the video frames in the training set are employed to fine-tune the pre-trained ConvNets while
+the identification rates are computed on text lines from the frames in the test set. A total of 31, 321 Urdu and 16, 207
+English text lines are used in the training set while the test set comprises 9, 9149 and 7, 425 text lines in Urdu and
+English respectively. The resulting confusion matrix is presented in Table 4 while the precision, recall and F-measure
+are summarized in Table 5. It can be seen that the model was able to correctly identify the scripts with an accuracy of
+more than 94%.
+12
+
+text: 99%
+text:
+/O0U
+text: 99%
+了02912.
+text: 99%
+text: 96%
+text: 99%
+text: 92%
+CAN BECOMELEADING EXPORTINGNATIOR
+000
+text: 77%
+text: 59%
+77.7370nua
+text: 97%
+text: 92%
+text: 96%
+text: 53%
+INYA
+FACEBOOK.COM/DUNYANEWS
+0007
+text: 
+85%
+GWLC
+58
+-0.78text: 96%
+text: 88%
+text: 99%
+text: 91%
+JEWS
+WITTER:@DUNYANEWSI FACEBOOK.CO
+text: 81%
+text: 53%
+8.45-0.30text: 84%
+text: 98%
+text:
+1000:
+text: 95%
+Wewill pressurize PMto resig
+text: 83%
+:ImranKhan
+text: 65%
+text: 98%
+text: 80%
+1021PN43300
+2YearsWarrant
+Its including 21 each in Puniab and Sindh will beCursive Caption Text Detection in Videos
+Table 4: Script identification confusion matrix
+Urdu
+English
+Urdu
+8763
+386
+English
+551
+6874
+Table 5: Performance of Script Identification
+Precision
+Recall
+F–Measure
+Urdu
+0.940
+0.957
+0.95
+English
+0.946
+0.925
+0.94
+5.4
+Hybrid Text Detection & Script Identification Results
+As discussed previously, text detection and script identification can be combined in a single model treating detection as
+a three (rather than two) class problem. The results of these experiments are summarized in Table 6 keeping the same
+distribution of training and test frames as in the previous experiments. Many interesting observations can be drawn from
+the results in Table 6. Similar to the script independent detectors, models pre-trained on Inception outperform those
+trained on ResNet and the observation is consistent for all four detectors. Likewise, Faster R-CNN reports the highest
+F-measure both for detection of Urdu and English text reading 0.91 and 0.87 respectively. In all cases, the performances
+on detection of Urdu text are better than those on Engish text. This can be attributed to the fact that the data is collected
+primarily from Urdu News channels which have limited amount of English text. It is also interesting to note that by
+combining text detection and script identification in a single model, not only the cascaded solution is avoided, the
+detection F-measures have also improved (in most cases). Though the improvement is marginal, eliminating the separate
+processing of detected text regions to identify the script offers a much simplified (yet effective) solution. Detection
+outputs on sample frames for the four detectors are illustrated in Figure 14.
+Table 6: Performance of hybrid text detector and script identifier
+RestNet
+Inception
+Method
+Script
+Precision
+Recall
+F-Measure
+Precision
+Recall
+F-Measure
+SSD
+Urdu
+0.83
+0.72
+0.77
+0.82
+0.78
+0.80
+English
+0.80
+0.63
+0.70
+0.82
+0.70
+0.75
+R-FCN
+Urdu
+0.80
+0.87
+0.83
+0.85
+0.90
+0.87
+English
+0.73
+0.81
+0.77
+0.77
+0.84
+0.81
+Faster R-CNN
+Urdu
+0.82
+0.92
+0.86
+0.87
+0.95
+0.91
+English
+0.80
+0.81
+0.80
+0.81
+0.94
+0.87
+Yolo
+Urdu
+-
+-
+-
+0.64
+0.70
+0.67
+English
+-
+-
+-
+0.62
+0.67
+0.64
+In an attempt to carry out an in-depth analysis of the detection performance and its evolution with respect to important
+system parameters, we carried out another series of experiments using Faster R-CNN (with Inception). In the first such
+experiment, we study the performance sensitivity to the amount of training data. We train the model by varying the
+number of text line images (from 10K to 49K) and compute the detector F-measure. Naturally, the detector performance
+enhances with the increase in the amount of training data (Figure 15) and begins to stabilize from around 30K-35K
+training lines.
+Resolution of input video frames is an important parameter that might affect the detector performance. To study the
+detector sensitivity to image resolution, we varied the image resolution from 256 × 144 to 1920 × 1080. The resolution
+was varied only in the test set and all sets of images were evaluated on the detector trained on a single resolution
+(900 × 600). The F-measures in Figure 16 are more less consistent for varied image resolutions reflecting the robustness
+of the detector. The proposed anchor boxes adapted for textual content play a key role in achieving this scale invariance.
+5.5
+Performance Comparison
+In an attempt to compare the performance of our detector with those reported in the literature, we present a comparative
+overview of various text detectors targeting cursive caption text in Table 7. It is important to note that since different
+studies are evaluated on different datasets, a direct comparison of these techniques is difficult. Most of the listed studies
+employ a small set of images (≤ 1000). Moradi et al. [38] and Zayene et al. [70] report results on relatively larger
+datasets with F-measures of 0.89 and 0.84 respectively. In comparison to other studies, we employ a significantly
+13
+
+Cursive Caption Text Detection in Videos
+ 
+ 
+(a) 
+ 
+ 
+ 
+(b) 
+ 
+ 
+ 
+(c) 
+ 
+ 
+ 
+(d) 
+ 
+Figure 14: Detection output of hybrid text detection and script identification for different detectors (a): SSD (b): R-FCN
+(c): Faster RCNN (d): Yolo
+14
+
+urdu:95%
+BBEAKING
+NEWS
+%86nJn
+urdu:80%
+urdu:91%
+urdu:97%
+urdu:96%
+Lrdu:99%
+english:85%
+cnglish:87%
+07:53urdu:97%
+urdu:89%
+%86.npin
+urdu:89%
+english:78%
+urdu:97%
+JUSTIN
+enalish:81%
+BRR
+9.21-0.23urdu:60%
+EIREAKING
+NEWS
+urdu:96%
+urdu:92%
+7
+%96:npJn
+urdu:96%
+urdu:91%
+uluu:98%
+english:98%
+english:95%
+AG
+urdu:99
+ITIESAIMINGATDESTABILIZINGPAKISTAN:
+english:88%
+07:53%66npin
+urdu:82%
+urdr:97%
+%66npin
+urdu:94%
+english:80%
+%66npn
+JUSTIN
+english:72%
+english:65%
+9210.23Benglish:93%
+english:80%
+urdu:99%
+urdu:97%
+urdu:94%
+urdu:97%
+urdu:99%
+urdu:72%
+urdu:94%
+english:84%
+english:L97%
+english:67%
+AGE
+TIESAIMINGATDESTABILIZINGPAKISTAN:
+LIVEI
+17:53urdu:98%
+%66npJn
+urdu:99%
+urdu:98%
+urdu:78%
+english:9tenglish:91%
+urdu:99%
+DUSTIN
+english:95%
+english:73%
+9.21-0.23english0.14
+HIEAKING
+Fdu18.508.15
+urdu 9.15
+english 0.32
+english 0.37
+AGEACTIVITIESAIMINGATDESTABILIZINGPAKISTAN:
+english 0.27english e.1l
+urdu0.18
+nduo.1
+english e.15
+nalish e.13
+englibinc.
+JUISTIN
+english0:11
+BRR
+3210.23Cursive Caption Text Detection in Videos
+Figure 15: Impact of size of training data on text detection performance (Faster R-CNN with Inception)
+Figure 16: Impact of video resolution on text detection performance (Faster R-CNN with Inception)
+larger set of images with an F-measure of 0.91. Furthermore, for a fair comparison, we also evaluated our system on
+the set 1000 images in the publicly available IPC dataset [54], the corresponding F-measure reads 0.92 validating the
+effectiveness of our detection technique,
+6
+Conclusion
+This paper presented a system for detection of caption text appearing in video frames. The developed technique relies
+on exploiting deep learning based object detectors and adapting them for text detection. Since it is common in videos to
+have text in more than one script, we presented, as a case study, video frames with text in cursive (Urdu) and Roman
+(English) scripts. Since each script requires different processing, the detection is combined with script identification in
+an end-to-end fashion so that the system is able to not only localize the text but also identify its script. Among various
+investigated object detectors, Faster R-CNN with our proposed set of anchor boxes reported the highest detection rates.
+The presented work is a part of a comprehensive video indexing and retrieval system and the current study focused on
+the detection of text. In our other [77, 78] work, the detection module is integrated with the video OCR module so that
+15
+
+0.95
+0.9
+0.9
+0.89
+0.89
+0.9
+0.88
+0.86
+F-Measure
+0.85
+0.85
+0.82
+0.8
+0.76
+0.75
+0.7
+10k
+15k
+20k
+25k
+30k
+35k
+40k
+45k
+49k
+Number of Tranining Lines1.000
+0.950
+-Urdu
+MEASURE
+English
+0.900
+0.850
+0.800
+0.750
+256 X 144
+426 X 240
+640 X 360
+900 X 600
+1280 X 720 1920 X 1080
+VIDEO RESOLUTIONCursive Caption Text Detection in Videos
+Table 7: Performance comparison with other techniques
+Study
+Method
+Dataset
+Script
+Video Frames
+Precision
+Recall
+F-Measure
+Jamil et al.(2011) [26]
+Edge-based Features
+IPC
+Urdu
+150
+0.77
+0.81
+0.79
+Siddiqi and Raza(2012) [54]
+Image Analysis
+IPC
+Urdu
+1,000
+0.71
+0.80
+0.75
+Moradi et al.(2013) [38]
+LBP with SVM
+-
+Farsi/Arabic
+4971
+0.91
+0.87
+0.89
+Raza et al.(2013) [41]
+Cascade of Transforms
+IPC
+Urdu
+1,000
+0.80
+0.89
+0.84
+Raza et al.(2013) [41]
+Cascade of Transforms
+IPC
+Arabic
+300
+0.81
+0.93
+0.86
+Yousfi et al.(2014) [69]
+ConvNet
+-
+Arabic
+201
+0.75
+0.80
+0.77
+Zayene et al.(2015) [71]
+SWT
+AcTiV
+Arabic
+425
+0.67
+0.73
+0.70
+Zayene et al.(2016) [70]
+SWT&Conv Autoencoders
+AcTiV
+Arabic
+1843
+0.83
+0.85
+0.84
+Shahzad et al.(2017) [47]
+Image Analysis
+-
+Urdu/Arabic
+240
+0.83
+0.93
+0.88
+Mirza et al.(2018) [36]
+Textural Features
+UTiV
+Urdu
+1,000
+0.72
+0.89
+0.80
+Unar et al.(2018) [59]
+Image Analysis+SVM
+IPC
+Urdu
+1,000
+0.83
+0.88
+0.85
+Proposed Method
+Deep ConvNets
+UTiV
+Urdu
+11,203
+0.87
+0.95
+0.91
+IPC
+Urdu
+1,000
+0.91
+0.93
+0.92
+detected text is recognized. Once recognized, videos are indexed based on keywords and retrieved on user provided
+queries. In addition to retrieval, automatic News summarization from ticker text and comparison of News across various
+News channels is also planned to be implemented. Furthermore, the textual content based retrieval will be combined
+with audio as well as visual objects appearing in videos.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf,len=967
+page_content='CURSIVE CAPTION TEXT DETECTION IN VIDEOS  Ali Mirza  Center of Artificial Intelligence  Mantalus  Melbourne, Australia  ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='mirza@mantalus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='com  Imran Siddiqi  Center of Excellence in Artificial Intelligence  Bahria University  Islamabad, Pakistan  imran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='siddiqi@bahria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='pk  Keywords Text Detection · Script Identification · Deep Neural Networks (DNNs) · Convolutional Neural Networks  (CNNs) · Video Text · Video Frames Dataset  ABSTRACT  Textual content appearing in videos represents an interesting index for semantic retrieval of videos  (from archives), generation of alerts (live streams) as well as high level applications like opinion  mining and content summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' One of the key components of such systems is the detection  of textual content in video frames and the same makes the subject of our present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' This  paper presents a robust technique for detection of textual content appearing in video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' More  specifically we target text in cursive script taking Urdu text as a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Detection of textual  regions in video frames is carried out by fine-tuning object detectors based on deep convolutional  neural networks for the specific case of text detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Since it is common to have videos with caption  text in multiple-scripts, cursive text is distinguished from Latin text using a script-identification  module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Finally, detection and script identification are combined in a single end-to-end trainable  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Experiments on a comprehensive dataset of around 11,000 video frames report an F-measure  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  1  Introduction  In the recent years, there has been a tremendous increase in the amount of digital multimedia data, especially the video  content, both in the form of video archives and live streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' According to statistics [6], 300 hours of video is being  uploaded every minute on the YouTube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A key factor responsible for this enormous increase is the availability of  low-cost smart phones equipped with cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' With such huge collections of data, there is a need to have efficient  as well as effective retrieval techniques allowing users retrieve the desired content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Traditionally, videos are mostly  stored with user assigned annotations or keywords which are called tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' When a content is to be searched, a keyword  provided as query is matched with these tags to retrieve the relevant content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The assigned tags, naturally, cannot  encompass the rich video content leading to a constrained retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A better and more effective strategy is to search  within the actual content rather than simply matching the tags i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Content based Image or Video Retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' CBVR  systems have been researched and developed for long a time and allow a smarter way of retrieving the desired content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  The term content may refer to the visual content (for example objects or persons in the video), audio content (the  spoken keywords for instance) or the textual content (News tickers, anchor names, score cards etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Among these, the  focus of our current study lies on textual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' More specifically, we target a smart retrieval system that exploits the  textual content in videos as an index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  The textual content in video can be categorized into two broad classes, scene text and caption text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Scene text (Figure 1)  is captured through camera during the video recording process and may not always be correlated with the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Examples of scene text include advertisement banners, sign boards, text on a T-shirt etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Scene text is commonly  employed for applications like robot navigation and assistance systems for the visually impaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Artificial or caption  text (Figure 2) is superimposed on video and typical examples include News tickers, movie credits, score cards, names  of anchors etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Caption text is generally correlated with the video content and is mostly applied for semantic retrieval of  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='03164v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='CV]  9 Jan 2023  Cursive Caption Text Detection in Videos  videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Figure 1: Examples of Scene Text  Figure 2: Examples of Caption Text  The key components of a textual content based indexing and retrieval system include detection of text regions [36],  extraction of text (segmentation from background) [37], identification of script (for multi-script videos) [25] and  finally recognition of text (through a video OCR) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Among these, we focus on detection of text in the current  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Detection of text can be carried out using unsupervised [4, 8, 3], supervised [63, 18, 32] or hybrid [36, 53]  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Unsupervised text detection employs image analysis techniques to discriminate between text and  non-text regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Supervised methods, on the other hand, involve training a learning algorithm with examples of text  and non-text regions to discriminate between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In some cases, a combination of the two techniques is em-  ployed where the candidate text regions identified by unsupervised methods are validated through a supervised approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  This paper presents a comprehensive framework for video text detection in a multi-script environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Though we  primarily target cursive caption text, since video frames frequently contain text in more than one script, text in the  Roman script is also detected by the proposed technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The key highlights of this study are outlined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Development of a comprehensive dataset of video images with ground truth information supporting evaluation  of detection and recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Adaptation of various deep learning based object detectors for detection of textual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Combination of text detection and script identification in a single end-to-end system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Validation of proposed technique through an extensive series of experiments and a comprehensive performance  comparison of various detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In the next section, we present an overview of the current state-of-the-art on detection  of textual content in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In Section 3, we introduce the dataset developed in our study along with the ground  truth information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Section 4 presents the details of the proposed framework while Section 5 presents the experimental  protocol, the realized results and the corresponding discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Finally, Section 6 concludes the paper with a discussion  on open challenges on this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  2  EIVE YOUR  SFORTUNE  ELECTIONCOMMISS  LOF  PAKISTAN  02  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='samaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='ty  O92NEWSHD  O92NEWSCHANNELWWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='92NEWSHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='TV  NEWS  SAMAA  11SWAT  4200  ja  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='samaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='ty  SAMAA  SundayCursive Caption Text Detection in Videos  2  Background  Detection of textual content in videos, images, documents and natural scenes has been investigated for more than four  decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The domain has matured progressively over the years starting with trivial image analysis based systems to  complex end-to-end learning based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' We discuss notable contributions to text detection in the following while  detailed surveys on the problem (and related problems) can be found in [40, 60, 68, 72, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Text detection refers to localization of textual content in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Techniques proposed for detection of text are typically  categorized into unsupervised and supervised approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' While unsupervised approaches primarily rely on image  analysis techniques to segment text from background, supervised methods involve training a learning algorithm to  discriminate between text and non-text regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Unsupervised text detection techniques include edge-based methods [4, 8, 3, 20, 24] which (assume and) exploit  the high contrast between text and its background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' connected component based methods [27, 28, 31, 39] which  mostly rely on the color/intensity of text pixels and texture-based methods [2, 22] which consider textual content  in the image as a unique texture that distinguishes itself from the non-text regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Texture based methods have  remained a popular choice of researchers and features based on Gabor filters [11], wavelets [64], curvelets [14], local  binary patterns (LBP) [1], discrete cosine transformation (DCT) [75], histograms of oriented gradients (HoG) [9] and  Fourier transformation [51] have been investigated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Another common category of techniques includes  color-based methods [65, 66, 50] which are similar in many aspects to the component-based methods and employ color  information of text pixels to distinguish it from non-text regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Supervised approaches for detection of textual content typically employ state-of-the-art learning algorithms which  are trained on examples of text and non-text blocks either using pixel values or by first extracting relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Classifiers like naive Bayes [52], Support Vector Machine [74], Artificial Neural Network [67, 36] and Deep Neural  Networks [30] have been investigated for this problem over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  In the recent years, deep learning based solutions have been widely applied to a variety of recognition problems and  have outperformed the traditional techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Among deep learning based techniques adapted for text detection, Huang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [21] employed sliding windows with CNNs to detect textual regions in low resolution scene images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Likewise,  fully convolutional networks are explored for detection of textaul regions in [73] and the technique is evaluated on  various ICDAR datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A similar work is presented by Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [15] where CNNs are trained using synthetic data  for detection of text at multiple scales from natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Another method called ‘SegLink’, is proposed in [49] that  relies on decomposing the text into segments (oriented boxes of words or lines) and links (connecting two adjacent  segments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The segments and links are detected using fully convolutional networks at multiple scales and combined  together to detect the complete text line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In [57] vertical anchor based method is reported that predicts text and non-text  scores of fixed size regions and reports high detection performance on the ICDAR 2013 and ICDAR 2015 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In  another notable work, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [61] present a framework based on conditional random field (CRF) to detect text in  scene images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Authors define a cost function by considering the color, stroke, shape and spatial features with CNN for  effective detection of textual regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Among other end-to-end trainable deep neural networks based systems, Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [32] present a system called  ‘TextBoxes’ which detects text in natural images in a single forward pass network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The technique was later extended to  ‘TextBoxes++’ and was evaluated on four public databases outperforming the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [18] improved  the convolutional layer of CNNs to detect text with arbitrary orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' EAST [76], is another well-known scene text  detector that provides promising results in challenging scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In another study [63], an ensemble of Convolutional  Neural Networks (CNNs) is trained on synthetic data to detect video text in East Asian languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  The literature is relatively limited once it comes to detection of cursive caption text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Among one of the preliminary  works, Jamil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [26] exploit edge based features with morphological processing to detect Urdu caption text from  a small set of 150 video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The same study was extended by Raza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [54] and evaluated on a larger set of  1000 video frames reporting a recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The dataset of images termed as IPC Artificial Text dataset [54] was  also made publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In a later study [41] by the same group, the authors proposed a cascaded framework  of spatial transforms to detect caption text in five different scripts including Arabic and Urdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In a relatively recent  work on detection of Arabic caption text, Zayene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [70] employ a combination of stroke width transform with a  convolutional auto-encoder and evaluate the technique on a publicly available dataset AcTiV-DB [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In one of our  3  Cursive Caption Text Detection in Videos  previous works [36], we investigated a combination of image analysis techniques with textural features to detect textual  regions in video frames and realized an F-measure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='80 on 1000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Summarizing, it can be concluded that the problem of text detection has been dominated by the application of different  deep learning based techniques in the recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The availability of benchmark datasets has also contributed to  the rapid developments in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' While detection of text in languages based on the Latin alphabet has received  significant research attention and is very much mature, detection of cursive text still remains a relatively less addressed  and challenging issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Development of a (generic) text detector that could work in multi-script environments also  remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  In the next section, we introduce the dataset that has been collected and labeled as a part of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  3  Dataset  Availability of labeled datasets is of utmost importance for algorithmic development and evaluation of any computerized  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' From the perspective of Urdu caption text detection, a dataset of 1000 labeled video frames has been made  publicly available [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A collection of 1000 frames, however, seems to be very small to generalize the findings for  practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' We, therefore, collected and labeled a comprehensive dataset of video frames allowing evaluation  of text detection and text recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' We collected a set of 46 videos from four different News channels in  Pakistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' All videos are recorded at a resolution of 900 × 600 and a frame rate of 25 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Frames in these video  contain textual content in two languages, (cursive) Urdu and English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The collected video frames are labeled from two  perspectives, detection and recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' For detection, the bounding rectangle of all text regions in a frame is labeled  and stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Similarly, for recognition, the transcription of each text line is stored as ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  In the literature, several evaluation metrics have been proposed to evaluate the performance of text detection  systems [26, 35, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In our system, for evaluation of the text detection module, we employ the most commonly used  area based precision and recall measures reported in [26] and defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Let AE be the estimated text area given by the system and AT be the ground truth text area, then the precision P and  recall R are defined as:  P =  AE ∩ AT  AE  (1)  R =  AE ∩ AT  AT  (2)  The precision and recall measures can be combined in a single F-measure as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  F =  2 × Precision × Recall  Precision + Recall  (3)  The same idea can be extended to multiple images by simply summing up area of intersection and dividing by  the total ground truth area (in N images) for recall and the total detected area for precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' To compute these  measures, for each frame, we need to store the actual location of the textual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The text detected automatically  by the system can then be compared with the ground truth text regions to compute precision, recall and F-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  The idea is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Figure 3-a illustrates an example where the text regions detected by the  system are shown while Figure 3-b illustrates the ground truth text locations for the given frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The detected and  ground truth text regions can be compared to compute the metrics defined earlier and quantify the detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  To facilitate the labeling process and standardize the ground truth data, a comprehensive labeling tool has been developed  that allows storing the location of each textual region in a frame along with its ground truth transcription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The location  is stored in terms of the x and y coordinates of the top left of the bounding box along its width and height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The ground  truth information of each frame is stored as an XML file that comprises frame meta data and the information on textual  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A screen shot of the labeling tool is presented in Figure 4 while the ground truth information of an example  frame is illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  It is known that videos typically contain 25–30 frames per second;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' consequently, successive frames in a video contain  redundant information (both visual and textual content).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' From the view point of automatic analysis systems, frames  4  Cursive Caption Text Detection in Videos  Figure 3: Text regions in an image and the corresponding ground truth image  Figure 4: Screen shot of ground truth labeling tool for text data  Figure 5: An XML file containing ground truth information of a frame  with unique content are of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Hence, each single video frame does not need to be labeled as major proportions of  such frames will have exactly the same textual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In our study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' we have extracted more than 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='000 frames  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='BREAKINGNEWS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='BREAKI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='BREAK  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content=' The statistics of videos, frames and text lines of  our dataset are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Inspired from the Arabic caption text dataset AcTiV-DB [71], we have named the  our dataset UTiV (Urdu Text in Video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The dataset along with its ground truth has also been made publicly available1  to support quantitative evaluation of text detection and recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Table 1: Statistics of labeled video frames  S#  Channel  Videos  Labeled Images  Urdu Lines  English Lines  1  Ary News  7  3,206  10,250  3,605  2  Samaa News  13  2,503  10,961  4,411  3  Dunya News  16  3,059  10,723  8,861  4  Express News  10  2,424  8,536  6,755  Total  46  11,203  40,470  23,632  4  Methods  This section presents the details of detecting textual content from video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Detection relies on adapting object  detectors based on deep convolutional neural networks for text regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Once the text is detected, script of the detected  text is identified by employing the ConvNets in a classification framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Subsequently, text detection and script  identification are combined in a single end-to-end system that detects the textual content along with its script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Details  are presented in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='1  Deep Learning based Object Detectors  Deep neural networks enjoy a renewed interest of the machine learning community thanks primarily to the availability  of high performance computing hardware (GPUs) as well as large data sets to train these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A major development  contributing to the current fame of deep learning was the application of ConvNets by Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [29] on the  ImageNet Large Scale Visual Recognition competition [46], which greatly reduced the error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Since then, CNNs  are considered to be state-of-the-art feature extractors and classifiers [55, 56] and have been applied to a variety of  recognition tasks [5, 58, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  While traditional CNNs are typically employed for object classification, Region-based Convolutional Networks  (R-CNN) [13] and their further enhancements Fast R-CNN [12] and Faster R-CNN [45] adapt CNNs for object  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In addition to different variants of R-CNN, a number of new architectures have also been proposed in the  recent years for real time object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The most notable of these include YOLO (You Only Look Once) [42] and  SSD (Single Shot Detector) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Each of these object detectors can be trained to detect C object classes (plus one for  the background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The output of the detector is the location of the bounding box (four coordinates) containing one of the  C classes as well as the class confidence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  In our study, for detection of textual content in a given frame, we investigated a number of CNN based object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Although, many object detectors are trained with thousands of class examples and provide high accuracy in detection  and recognition of different objects, these object detectors can not be directly applied to identify text regions in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  These models have to be tuned to the specific problem of discrimination of text from non-text regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The convolutional  base of these models can be trained from scratch or, known pre-trained models can be fine-tuned by training them on  text and non-text regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In our study, we investigated the following object detectors for localization of text regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Faster R-CNN  Region-Based Fully Convolutional Networks (R-FCN)  Single Shot Detector (SSD)  You Only Look Once (YOLO)  For completeness, we provide a brief overview of these object detectors in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  1http://cbvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='media-tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='net/  6  Cursive Caption Text Detection in Videos  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='1  Faster R-CNN  Faster R-CNN [45] is an enhanced version of its predecessors R-CNN [13] and Fast R-CNN [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Each of these  detectors exploits the powerful features of ConvNets for object localization as well as classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' R-CNN was one  of the first attempts to apply ConvNets for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' An R-CNN scans the input image for potential objects  using Selective Search [58] that generates around 2,000 region proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Each of these region proposals is then fed  to a CNN for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The output of the CNN is finally employed by an SVM to classify the object and a  linear regressor to tighten the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' R-CNN was enhanced in terms of training efficiency by extending it to  Fast R-CNN [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In Fast R-CNN, rather than separately feeding each region proposal to the ConvNet, convolution is  performed only once on the complete image and the region proposals are projected on the feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Furthermore,  the SVM in R-CNN was replaced by a softmax layer extending the network to predict the class labels rather than  using a separate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' While Fast R-CNN significantly reduced the time complexity of the basic R-CNN, a major  bottleneck was the selective search algorithm to generate the region proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' This was addressed through Region  Proposal Network (RPN) in Faster R-CNN [45] which shares convolutional features with the detection network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' RPN  predicts region proposals which are then fed to the detection network to identify the object class and refine the bounding  boxes produced by the RPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A summary of various R-CNN models in presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Figure 6: Summary of R-CNN Family based Object Detectors  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='2  You Only Look Once (YOLO)  YOLO [42] takes a different approach to object detection primarily focusing on improving the detection speed (rather  than accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' As the name suggests, YOLO employs a single pass of the convolutional network for localization and  classification of objects from the the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The input image is divided into a grid and an object is expected  to be detected by the grid which holds the center of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Each cell in the grid predicts up to two bounding  boxes (and class probabilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The network comprises 24 convolutional and fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' YOLO works in  real time but in terms of accuracy, it is known to make significant localization errors in comparison to region based  object detectors (Faster R-CNN for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' YOLO was later enhanced to YOLO9000 [43] by introducing batch  normalization, increasing the resolution of the input image (by a factor of 2) and introducing the concept of anchor boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  YOLO9000 employs Darknet 19 architecture with 19 convolutional layers, 5 max pooling layers and a softmax layer for  classification objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Incremental improvements in YOLO v2 resulted in YOLO v3 [44] that uses logistic regerssion to  predict the score of objectness for each bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Furthermore, it employs class-wise logistic classifiers (rather  than softmax) allowing multi-label classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Box offset  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='SVM object  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Box offset  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Softmax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Box offset  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Softmax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Regressor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Regressor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Regressor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Independent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Joint  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Joint  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Region CNN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Region CNN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Region CNN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Fine-tuned  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Fine-tuned  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Features  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Features  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Features  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='ROl Pooling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Pre-trained  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='RPN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Pre-trained  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='ROl Pooling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Region  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Region  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Deep CNN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Deep CNN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Proposal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Proposal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Deep CNN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Independent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Independent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Independent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='MP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='GMP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='R-CNN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Fast R-CNN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='Faster R-CNNCursive Caption Text Detection in Videos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='3  Single Shot Detector (SSD)  Unlike the R-CNN series object detectors which require two shots to detect objects in an image, Single Shot Multi-box  Detector [34], as the name suggests, requires a single shot to detect objects (similar to YOLO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' SSD relies on the idea  of default boxes and multi-scale predictions and directly applies bounding box regression to the default boxes without  generating the region proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Detection at multiple scales are handled by exploiting the feature maps of different  convolutional layers corresponding to different receptive fields in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The architecture has an input size  of 300 × 300 × 3 and primarily builds on the VGG-16 architecture discarding the fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' VGG-16 is  used as base network mainly due to its robust performance of image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The bounding box regression  technique of SSD is inspired by [56] while the MultiBox relies on priors, the pre-computed fixed size bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  The priors are selected in such a way that their Intersection over Union ratio (with ground truth objects) is greater than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The MultiBox starts with the priors as predictions and attempt to regress closer to the ground truth bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  SSD works in real time but requires images of fixed square size and is known to miss small objects in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='4  Region-Based Fully Convolutional Networks (R-FCN)  R-FCN [7] builds on the idea of increasing the detection accuracy by maximizing the shared calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' R-FCN  generates position-sensitive score maps to represent different relative positions of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' An object is represented by  k2 relative positions dividing it into a grid of size k × k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A ConvNet (ResNet in the original R-FCN paper) sweeps the  input image and an additional fully convoltional layer produces the position-sensitive scores in k2 × (C + 1) score maps  where C is the number of classes plus 1 class for the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A fully convolutional proposal network generates  regions of interest which are divided in k2 bins and the corresponding class probabilities are obtained from the score  maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The scores are averaged to convert the k2 × (C + 1) values into a one dimensional (C + 1) sized vector which is  finally fed to the softmax layer for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Localization is carried out using the bounding box regression similar  to other object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' R-FCN speeds up the detection in comparison to Faster R-CNN but compared to other Single  Shot methods, it requires more computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='2  Adapting Object Detectors for Text Detection  In the context of object detection, the problem of text detection can be formulated as a two class problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The text  regions represent object of interest while the non-text regions need to be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The object detectors discussed in  the previous sections are adapted for text detection using two pre-trained models, ResNet 101 [17] and Inception  v2 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' These models are trained on the large scale Microsoft COCO (Common Objects in Context) database [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The  database contains images of 91 different object types with a total of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5 million labeled instances in 328K images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The  pre-trained network serves as starting point rather than random weight initialization and the network is made to learn the  specific class labels (text or non-text) by continuing back propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The ground truth localization information of the  textual regions in the video frames is employed for training the models, the overall workflow being illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  A critical aspect in employing object detectors for text detection is the choice of anchor boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  The anchor  boxes in all the detectors have been designed to detect general object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Text appearing in videos has  specific geometric properties in terms of size and aspect ratio hence the default anchor boxes of the detectors  need to be adapted to detect textual regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' We carried out a comprehensive analysis of the textual regions in  terms of width, height and aspect ratios of the bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' As a result of this analysis we have chosen a  base anchor of size 256 × 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' To each anchor box we apply three scales (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='0) and five aspect ratios  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='1875, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='375, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='50) as illustrated in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Models are fine-tuned using the proposed anchor boxes  and the effectiveness of these anchor boxes is validated through experimental study as presented in Section 5 of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='3  Script Identification  As discussed earlier, we primarily target detection of cursive caption text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' However, like many practical scenarios, video  frames in our case contain bilingual textual content (Urdu & English).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Consequently, once the text is detected, we  need to identify the script of each detected region (Figure 9) so that the subsequent processing of each type of script  can be carried out by the respective recognition engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' For script identification, we employ CNNs in a classification  framework (rather than detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Urdu and English text lines are employed to fine-tune CNNs to discriminate between  the two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Once trained, the model is able to separate text lines as a function of the script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Similar to detection,  rather than training the networks from scratch, we fine-tune known pre-trained models (Inception and ResNet) to solve  the two-class classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  8  Cursive Caption Text Detection in Videos  Figure 7: Overview of adapting object detectors for text detection  Figure 8:  Anchor boxes (base size 256 × 256) at three scales (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='0) and five aspect ratios  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='1875, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='375, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='50)  Figure 9: Script identification of detected text lines  9  Test Image  Frames  VIDEO  Output Image  Ground Truth Data  Object  Fine-Tuning  Text Detector  Inference  Detector  CNN Model  Pre-trained  CNN  ModelsScale: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='0  (a)  Scale: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='0  (b)  Scale: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='0  (c)Urdu Text Regions  Bilingual Text Regions  SHAZIA ZEESHAN  ZAFAR HILALY  Script  Identification  English Text Regions  AHMAD AWAIS  ORYAMAQBOOL JAN  SHAZIA ZEESHAN  ZAFAR HILALY  AHMAD AWAIS  ORYAMAQBOOLJANCursive Caption Text Detection in Videos  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='4  Hybrid Text Detector & Script Identifier  Detection of text and identification of script, as discussed previously, can be implemented in a cascaded framework  where the output of text detector is fed to the script identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A deep learning framework can be tuned to discriminate  between text and non-text regions and the extracted text regions can be fed to a separate script recognition model that  identifies the script of the detected text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' This, however, introduces a bottleneck of training two separate networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Furthermore, the cascaded solution also implies that errors in detection are propagated to the next step as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' We,  therefore, propose to combine the text detector and script identifier in a single hybrid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Rather than treating  detection as a two-class problem (text and non-text), we consider it as a three class problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' non-text regions,  English text and Urdu text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' This not only avoids training two separate models but also eliminates the accumulation  of errors in a cascaded solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The superiority of the combined text detector and script identifier is also supported  through quantitative evaluations as discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  All detectors are trained in an end-to-end manner with a multi-task objective function that combines the classification  and regression losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The evolution of training loss for the investigated detectors (with Inception and ResNet) is  illustrated in Figure 10 where it can be seen that the loss begins to stabilize from 30 epochs on wards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Figure 10: Training loss of various detectors – Hybrid text detector and script identifier  5  Experiments and Results  The detection performance is evaluated through a series of experiments carried out on the collected set of video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  We first present the experimental protocol followed by the detection results of various object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' We then present  the script identification results and the performance of the combined text detector and script identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Furthermore,  performance sensitivity of the system as well as a comparison with state-of-the-art is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='1  Experimental Settings  As introduced in Section 3, we collected a total of 11,203 video frames from four different News channel videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The  localization information of text regions in these frames is used to train and subsequently evaluate the text detection and  script identification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The distribution of frames into training and test sets along with the number of text  lines in each set is summarized in Table 2 while the details of detection performance are presented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='2  Text Detection Results  Object detectors including Faster R-CNN, YOLO, SSD and R-FCN are adapted to detect textual content by fine-tuning  the Inception and ResNet pre-trained models and changing the anchor boxes as discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Performance of  10  7  +FasterRCNN-Inception  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5  Faster RCNN-ResNet  6  SSD-Inception  SSD-ResNet101  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5  米RFCN-Inception  5  RFCN-ResNet  TRANING LOSS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5  YoloV3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5  0  0  5  10  15  20  25  30  35  40  45  50  EPOCHSCursive Caption Text Detection in Videos  Table 2: Data distribution for text detection experiments  Train  Test  Frames  Lines  Frames  Line  Urdu  8,500  31,321  2,703  9,149  English  16,207  7,425  Total  49,046  11,056  each of these detectors in terms of precision, recall and F-measure is summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' It can be seen that in all  cases, detectors pre-trained on Inception outperform those trained on ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Among various detectors, Faster R-CNN  reports the highest F-measure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The lowest performance is reported by Yolo reading an F-measure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A  comprehensive study on the trade-off between speed and accuracy of various object detectors is presented in [19] and  our findings on detection of text are consistent with those of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' It is also important to recap that precision and recall  are computed using area based metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' As a result, if the detected bounding box is larger (smaller) than the ground  truth, it results in penalizing the precision (recall) of the detector as illustrated in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The output of the Faster  R-CNN based text detector for few sample frames in our dataset is illustrated in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Table 3: Text Detection Results  RestNet  Inception  Model  Precision  Recall  F-Measure  Precision  Recall  F-Measure  SSD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='80  R-FCN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='86  Faster R-CNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='90  Yolo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='66  Figure 11: Computation of precision and recall (a):Ground Truth Bounding Box (b): Detected region is larger than  ground truth (c):Detected region is smaller than ground truth (d):Detected region overlaps perfectly with the ground  truth  In an attempt to provide an insight into the detection errors, few of the errors are illustrated in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' It can be seen  that in most cases, the detector is able to detect the textual region but the localization is not perfect i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' in some cases  the bounding box is larger (shorter) than the actual content leading to a reduced precision (recall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  11  SAMAA  SAMAA  (a)  (b)  SAMAA  SAMAA  (c)  (d)Cursive Caption Text Detection in Videos  Figure 12: Text detection results on sample images (Faster R-CNN with Inception)  Figure 13: Imperfect Localization of Text Regions  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='3  Script Identification Results  For script identification, we employ the same distribution of frames into training and test sets as that of the detection  protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Text lines from the video frames in the training set are employed to fine-tune the pre-trained ConvNets while  the identification rates are computed on text lines from the frames in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A total of 31, 321 Urdu and 16, 207  English text lines are used in the training set while the test set comprises 9, 9149 and 7, 425 text lines in Urdu and  English respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The resulting confusion matrix is presented in Table 4 while the precision, recall and F-measure  are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' It can be seen that the model was able to correctly identify the scripts with an accuracy of  more than 94%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  12  text: 99%  text:  /O0U  text: 99%  了02912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  text: 99%  text: 96%  text: 99%  text: 92%  CAN BECOMELEADING EXPORTINGNATIOR  000  text: 77%  text: 59%  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='7370nua  text: 97%  text: 92%  text: 96%  text: 53%  INYA  FACEBOOK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='COM/DUNYANEWS  0007  text:  85%  GWLC  58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='78text: 96%  text: 88%  text: 99%  text: 91%  JEWS  WITTER:@DUNYANEWSI FACEBOOK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='CO  text: 81%  text: 53%  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='45-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='30text: 84%  text: 98%  text:  1000:  text: 95%  Wewill pressurize PMto resig  text: 83%  :ImranKhan  text: 65%  text: 98%  text: 80%  1021PN43300  2YearsWarrant  Its including 21 each in Puniab and Sindh will beCursive Caption Text Detection in Videos  Table 4: Script identification confusion matrix  Urdu  English  Urdu  8763  386  English  551  6874  Table 5: Performance of Script Identification  Precision  Recall  F–Measure  Urdu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='95  English  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='925  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='94  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='4  Hybrid Text Detection & Script Identification Results  As discussed previously, text detection and script identification can be combined in a single model treating detection as  a three (rather than two) class problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The results of these experiments are summarized in Table 6 keeping the same  distribution of training and test frames as in the previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Many interesting observations can be drawn from  the results in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Similar to the script independent detectors, models pre-trained on Inception outperform those  trained on ResNet and the observation is consistent for all four detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Likewise, Faster R-CNN reports the highest  F-measure both for detection of Urdu and English text reading 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='91 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='87 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In all cases, the performances  on detection of Urdu text are better than those on Engish text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' This can be attributed to the fact that the data is collected  primarily from Urdu News channels which have limited amount of English text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' It is also interesting to note that by  combining text detection and script identification in a single model, not only the cascaded solution is avoided, the  detection F-measures have also improved (in most cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Though the improvement is marginal, eliminating the separate  processing of detected text regions to identify the script offers a much simplified (yet effective) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Detection  outputs on sample frames for the four detectors are illustrated in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Table 6: Performance of hybrid text detector and script identifier  RestNet  Inception  Method  Script  Precision  Recall  F-Measure  Precision  Recall  F-Measure  SSD  Urdu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='80  English  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='75  R-FCN  Urdu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='87  English  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='81  Faster R-CNN  Urdu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='91  English  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='87  Yolo  Urdu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='67  English 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='64  In an attempt to carry out an in-depth analysis of the detection performance and its evolution with respect to important  system parameters, we carried out another series of experiments using Faster R-CNN (with Inception).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In the first such  experiment, we study the performance sensitivity to the amount of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' We train the model by varying the  number of text line images (from 10K to 49K) and compute the detector F-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Naturally, the detector performance  enhances with the increase in the amount of training data (Figure 15) and begins to stabilize from around 30K-35K  training lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  Resolution of input video frames is an important parameter that might affect the detector performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' To study the  detector sensitivity to image resolution, we varied the image resolution from 256 × 144 to 1920 × 1080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The resolution  was varied only in the test set and all sets of images were evaluated on the detector trained on a single resolution  (900 × 600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The F-measures in Figure 16 are more less consistent for varied image resolutions reflecting the robustness  of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' The proposed anchor boxes adapted for textual content play a key role in achieving this scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='5  Performance Comparison  In an attempt to compare the performance of our detector with those reported in the literature, we present a comparative  overview of various text detectors targeting cursive caption text in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' It is important to note that since different  studies are evaluated on different datasets, a direct comparison of these techniques is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Most of the listed studies  employ a small set of images (≤ 1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Moradi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [38] and Zayene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' [70] report results on relatively larger  datasets with F-measures of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='84 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='23Cursive Caption Text Detection in Videos  Figure 15: Impact of size of training data on text detection performance (Faster R-CNN with Inception)  Figure 16: Impact of video resolution on text detection performance (Faster R-CNN with Inception)  larger set of images with an F-measure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content=' Furthermore, for a fair comparison, we also evaluated our system on  the set 1000 images in the publicly available IPC dataset [54], the corresponding F-measure reads 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='92 validating the  effectiveness of our detection technique,  6  Conclusion  This paper presented a system for detection of caption text appearing in video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content=' Since it is common in videos to  have text in more than one script, we presented, as a case study, video frames with text in cursive (Urdu) and Roman  (English) scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content=' Among various  investigated object detectors, Faster R-CNN with our proposed set of anchor boxes reported the highest detection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  The presented work is a part of a comprehensive video indexing and retrieval system and the current study focused on  the detection of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='750  256 X 144  426 X 240  640 X 360  900 X 600  1280 X 720 1920 X 1080  VIDEO RESOLUTIONCursive Caption Text Detection in Videos  Table 7: Performance comparison with other techniques  Study  Method  Dataset  Script  Video Frames  Precision  Recall  F-Measure  Jamil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='80  Unar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' (2018) [59]  Image Analysis+SVM  IPC  Urdu  1,000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='85  Proposed Method  Deep ConvNets  UTiV  Urdu  11,203  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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+page_content='92  detected text is recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Once recognized, videos are indexed based on keywords and retrieved on user provided  queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' In addition to retrieval, automatic News summarization from ticker text and comparison of News across various  News channels is also planned to be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Furthermore, the textual content based retrieval will be combined  with audio as well as visual objects appearing in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content='  References  [1] Marios Anthimopoulos, Basilis Gatos, and Ioannis Pratikakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' A two-stage scheme for text detection in video  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
+page_content=' Image and Vision Computing, 28(9):1413–1426, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE1T4oBgHgl3EQfawQW/content/2301.03164v1.pdf'}
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diff --git a/UNE3T4oBgHgl3EQfzwvz/content/tmp_files/2301.04732v1.pdf.txt b/UNE3T4oBgHgl3EQfzwvz/content/tmp_files/2301.04732v1.pdf.txt
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+arXiv:2301.04732v1  [math.QA]  11 Jan 2023
+SEMI-INFINITE CONSTRUCTION FOR THE DOUBLE YANGIAN OF
+TYPE A(1)
+1
+MARIJANA BUTORAC, NAIHUAN JING, SLAVEN KOˇZI´C AND FAN YANG
+Abstract. We consider certain infinite dimensional modules of level 1 for the double
+Yangian DY(gl2) which are based on the Iohara–Kohno realization. We show that they
+possess topological bases of Feigin–Stoyanovsky-type, i.e. the bases expressed in terms
+of semi-infinite monomials of certain integrable operators which stabilize and satisfy
+the difference two condition. Finally, we give some applications of these bases to the
+representation theory of the corresponding quantum affine vertex algebra.
+1. Introduction
+The integrable highest weight modules present one of the most fundamental notions
+in the representation theory of affine Kac–Moody Lie algebras; see, e.g., the book by
+Kac [14]. The problem of constructing different types of bases for such modules and
+their various substructures, especially those which establish connection with Rogers–
+Ramanujan-type identities via character formulae, has been extensively studied since the
+pioneering paper of Lepowsky and Milne [16]. Our paper is motivated by the well known
+Feigin–Stoyanovsky construction [6] of semi-infinite monomial bases for certain integrable
+highest weight modules for the affine Lie algebra �sl2. The construction relies on the fact
+that these modules can be obtained from their principal subspaces using the Weyl trans-
+lation operator. At the level 1, the resulting bases consist of semi-infinite monomials
+xα(r1)xα(r2) . . . in coefficients of the vertex operator xα(z) = �
+r∈Z xα(r)z−r−1 associ-
+ated with the positive simple root α of sl2. Their degrees r1, r2, . . . satisfy the difference
+two condition rj+1 ⩾ rj + 2 for all j = 1, 2, . . . , which comes from the integrability
+relation xα(z)2 = 0 of Lepowsky and Primc [17]. Moreover, these monomials stabilize,
+i.e. for a sufficiently large n all degrees rn, rn+1, . . . are consecutive odd or even integers,
+depending on the choice of the highest weight. Later on, the semi-infinite construction
+was generalized to the case of quantum affine algebra Uq(�sl2) by Ding and B. Feigin [1]
+using the realization of its integrable highest weight modules found by I. Frenkel and the
+second author in [4].
+The goal of this paper is to give a semi-infinite construction for certain infinite di-
+mensional modules of level 1 for the centrally extended double Yangian DY(gl2) defined
+over the commutative ring C[[h]]. Their bosonic realization, which resembles the famous
+Frenkel–Kac–Segal construction [5, 20] for affine Lie algebras, was given by Iohara and
+Kohno in [11] for DY(gl2) and then generalized to the higher rank case by Iohara [10].
+We slightly modify the Iohara–Kohno realization as the action of the original translation
+operator does not appear to be in tune with the semi-infinite construction. However, the
+action of the double Yangian is still given on the same C[[h]]-module, which we denote by
+Fi, i = 0, 1. In contrast with the aforementioned setting of affine Lie algebras and quan-
+tum affine algebras, the general theory of integrable representations for double Yangians
+has not yet been sufficiently developed. Thus, in our construction, we often need to use
+1
+
+different and more technical arguments which rely on the explicit formulae for the action
+of the double Yangian generators on Fi.
+Motivated by Ding–Feigin’s approach [1], we start by defining an auxiliary commutative
+operator X(z) on Fi, i = 0, 1, which can be regarded as a Yangian counterpart of the
+level 1 affine vertex operator xα(z). In particular, it satisfies the h-adic integrability
+relation X(z)X(z ± h) = 0. We use its coefficients in parallel with [6, 9] to introduce
+the notion of principal submodule Wi ⊂ Fi and, furthermore, to obtain the topological
+basis for Wi which provides an interpretation of the sum-sides of Rogers–Ramanujan
+identities. Next, we employ the action of translation operator on Wi to recover irreducible
+modules Li(sl2) ⊂ Fi for the double Yangian DY(sl2), such that their classical limits
+are exactly the level 1 integrable highest weight �sl2-modules L(Λi) of highest weight Λi
+with i = 0, 1. Finally, we construct the Feigin–Stoyanovsky-type semi-infinite monomial
+bases for Li(sl2), which is the main result of this paper. In addition, we generalize this
+construction to the corresponding modules Li(gl2) for the double Yangian DY(gl2) by
+using the action of its Heisenberg subalgebra, which is generated by the coefficients of
+the quantum determinant and commutes with DY(sl2).
+At the end of the paper, we obtain some applications of the semi-infinite construction
+to the quantum vertex algebra theory. In particular, by employing the Iohara–Kohno
+isomorphism [11] between two realizations of the double Yangian, we show that Li(gl2)
+are naturally equipped with the structure of irreducible modules for the corresponding
+Etingof–Kazhdan quantum affine vertex algebra of level 1 from [3].
+2. Preliminaries
+In this section, we recall the double Yangian for the general linear Lie algebra gl2 and
+the Iohara–Kohno bosonic realization of its level 1 modules.
+2.1. Double Yangian for gl2. We follow the paper of Iohara and Kohno [11] to intro-
+duce the centrally extended double Yangians for the Lie algebras gl2 and sl2 and recover
+some of their properties. Let I be the identity and P the permutation operator on C2⊗C2.
+Consider the (normalized) Yang R-matrix over the commutative ring C[[h]],
+R(u) =
+1
+1 + h/u
+�
+I + h
+uP
+�
+∈ End C2 ⊗ End C2[[h/u]].
+The double Yangian DY(gl2) is defined as the associative algebra over the ring C[[h]]
+generated by the central element C and the elements t(r)
+ij , where i, j = 1, 2 and r ∈ Z. Its
+defining relations are given by
+R12(u − v)T ±
+13(u)T ±
+23(v) = T ±
+23(v)T ±
+13(u)R12(u − v),
+(2.1)
+R12(u − v − hC/2)T +
+13(u)T −
+23(v) = T −
+23(v)T +
+13(u)R12(u − v + hC/2).
+(2.2)
+The generator matrices T ±(u) are defined by
+T ±(u) =
+�
+i,j=1,2
+eij ⊗ t±
+ij(u),
+(2.3)
+where eij ∈ End C2 denote the matrix units and the power series t±
+ij(u) are given by
+t+
+ij(u) = δij − h
+�
+r⩾0
+t(r)
+ij u−r−1
+and
+t−
+ij(u) = δij + h
+�
+r⩾1
+t(−r)
+ij
+ur−1.
+(2.4)
+2
+
+In (2.1) and (2.2) we use the subscripts to indicate the tensor factors, i.e. we have
+T ±
+13(u) =
+�
+i,j=1,2
+eij ⊗ 1 ⊗ t±
+ij(u)
+and
+T ±
+23(u) =
+�
+i,j=1,2
+1 ⊗ eij ⊗ t±
+ij(u).
+Let us discuss the classical limit of double Yangian. Consider the affine Lie algebra
+�gl2 = gl2 ⊗ C[t±1] ⊕ CK, where K is the central element and the Lie brackets are
+[eij(r), ekl(s)] = δkj eil(r + s) − δilekj(r + s) + rδr+s0K (δkj δil − δij δkl)
+(2.5)
+for eij(r) = eij ⊗ tr. Introduce the ascending filtration over DY(gl2) by setting
+deg t(r)
+ij = r
+for
+i, j = 1, 2, r ∈ Z
+and
+deg C = 0.
+(2.6)
+The images ¯t(r)
+ij and ¯C of the double Yangian generators t(r)
+ij and C in the corresponding
+graded algebra grDY(gl2) satisfy (2.5). Thus, the assignments eij(r) �→ ¯t(r)
+ij and K �→ ¯C
+define the algebra homomorphism
+U(�gl2) ⊗ C[[h]] → grDY(gl2).
+(2.7)
+Theorem 2.1. The map (2.7) is an algebra isomorphism.
+Proof. The surjectivity of the map (2.7) is clear. On the other hand, the injectivity is a
+consequence of the Poincar´e–Birkhoff–Witt theorem [12, Thm. 2.2], which states that the
+suitably ordered monomials in the double Yangian generators form its basis; see also [13,
+Prop. 3.1] and [19, Thm. 15.3]. More specifically, although we define the double Yangian
+using the normalization of the Yang R-matrix which differs from [12], the arguments from
+the corresponding part of the proof of [12, Thm. 2.2] can be still carried out analogously.
+It is worth noting that they rely on the Iohara–Kohno realization [11], which provides
+level 1 representations of the double Yangian; see Theorem 2.3 below.
+□
+Remark 2.2. The correspondence similar to (2.7), which employs the universal envelop-
+ing algebra over C, can be also established by taking the classical limit DY(gl2)/hDY(gl2)
+of the double Yangian. Indeed, by extracting the coefficients of the matrix entries in its
+defining relations (2.1) and (2.2), one observes that the resulting top degree terms with
+respect to (2.6) coincide with the terms which contain the lowest power of h.
+From now on, we shall assume that the double Yangian for gl2 is h-adically completed.
+Using the series (2.4) one obtains its Drinfeld generators [2] as follows:
+k±
+1 (u) = t±
+11(u),
+k±
+2 (u) = t±
+22(u) − t±
+21(u)t±
+11(u)−1t±
+12(u),
+(2.8)
+X+(u) = t+
+11(u − hC/4)−1t+
+12(u − hC/4) − t−
+11(u + hC/4)−1t−
+12(u + hC/4),
+(2.9)
+X−(u) = t+
+21(u + hC/4)t+
+11(u + hC/4)−1 − t−
+21(u − hC/4)t−
+11(u − hC/4)−1.
+(2.10)
+The commutation relations for these generators can be found in [11, Thm. 2.1].
+The double Yangian DY(gl2) can be decomposed into two subalgebras: the double
+Yangian DY(sl2), which is generated by the central element C1 := C and all coefficients
+of the power series
+E(u) = 1
+hX+(u + h/2),
+F(u) = 1
+hX−(u + h/2),
+H±(u) = k±
+2 (u + h/2)k±
+1 (u + h/2)−1,
+(2.11)
+3
+
+and the Heisenberg subalgebra H, which is generated by the coefficients of the series
+K±(u) = k±
+1 (u − h/2)k±
+2 (u + h/2)
+and the central element C2 = −2C. The generators of the Heisenberg subalgebra commute
+with all elements of DY(sl2) and satisfy the relations from [11, Cor. 2.2],
+�
+K±(u), K±(v)
+�
+= 0,
+u − v − h + hC2/4
+u − v + h + hC2/4K+(u)K−(v) = K−(v)K+(u)u − v − h − hC2/4
+u − v + h − hC2/4.
+The generator series of H are of the form K±(u) = 1 ∓ hκ±(u), where
+κ+(u) =
+�
+r⩾0
+κ(r)u−r−1
+and
+κ−(u) =
+�
+r⩾1
+κ(−r)ur−1.
+Consider the Heisenberg Lie subalgebra �h = h ⊗ C[t±1] ⊗ CK2 ⊂ �gl2, where K2 = −2K
+is the central element and h = CI is the one-dimensional commutative subalgebra of gl2
+spanned by the identity matrix I = e11 + e22. By (2.5), its Lie brackets are given by
+[I(r), I(s)] = rδr+s0K2,
+where I(r) = I ⊗tr. Note that the restriction of the map (2.7) produces the isomorphism
+U(�h) ⊗ C[[h]] → grH,
+(2.12)
+where grH is the corresponding graded algebra of H. It is given by I(r) �→ ¯κ(r) for all
+r ∈ Z and K2 �→ ¯C2, where ¯κ(r) and ¯C2 stand for the images of Heisenberg subalgebra
+generators κ(r) and C2 in the corresponding component of grH.
+2.2. Iohara–Kohno realization. We follow [11] to introduce a certain realization of
+level 1 modules for the double Yangian DY(gl2). Let s = Ce11 ⊕ Ce22 be the Cartan
+subalgebra of gl2. Denote by εj, j = 1, 2, the linear maps s → C given by εj(ekk) = δjk.
+The dual s∗ is equipped with the standard bilinear form (·, ·) determined by (εj, εk) = δjk.
+Let Q = Zα with α = ε1 − ε2 be the root lattice of the simple Lie algebra sl2 and
+λi = Λi − Λ0 the classical part of the i-th fundamental affine weight Λi.
+Consider the Lie algebra �s generated by the central element C and the elements aj(r),
+where j = 1, 2 and r ∈ Z, r ̸= 0, subject to the relations
+[aj(r), ak(s)] = rδjkδr+s0C.
+(2.13)
+Let �s− be its (commutative) subalgebra generated by all aj(r) with j = 1, 2 and r ∈ Z<0.
+Denote by C [Q] the group algebra of the root lattice Q. For any i = 0, 1 and s ∈ C set
+Fis = U(�s−)[[h]] ⊗ C[Q][[h]]eλi+ s
+2(ε1+ε2).
+(2.14)
+The tensor product in (2.14) is understood in the h-adically completed sense, so that the
+C[[h]]-module Fis is topologically free. Define the action of C, aj(r), ∂εj and eεj, where
+j = 1, 2 and r ̸= 0, on Fis so that for any f ⊗ eµ ∈ Fis we have
+C · f ⊗ eµ = f ⊗ eµ,
+aj(r) · f ⊗ eµ =
+�
+aj(r)f ⊗ eµ,
+if r < 0,
+[aj(r), f] ⊗ eµ,
+if r > 0,
+∂εj · f ⊗ eµ = (εj, µ)f ⊗ eµ,
+eεj · f ⊗ eµ = f ⊗ eεj+µ.
+4
+
+Let us write X±α(u) = h−1X±(u). Introduce the following operator series on Fis:
+E−(u) = exp
+��
+r>0
+�
+−a1(−r)
+r
+�
+u − 3
+4h
+�r
++ a2(−r)
+r
+�
+u + h
+4
+�r��
+,
+E+(u) = exp
+��
+r>0
+�
+a1(r) − a2(r)
+r
+�
+u + h
+4
+�−r��
+,
+E0(u) = eα
+�
+u + h
+4
+�∂α
+.
+Finally, we recall the Iohara–Kohno realization [11, Thm. 3.1].
+Theorem 2.3. The following assignments define a structure of DY(gl2)-module on Fis:
+Xα(u) �→ E−(u)E+(u)E0(u),
+X−α(u) �→ E−(u + h/2)−1E+(u − h/2)−1E0(u − h/2)−1,
+k+
+j (u) �→ exp
+�
+−
+�
+r>0
+aj(r)
+r
+��
+u + h
+2
+�−r
+−
+�
+u − h
+2
+�−r�� �
+1 −
+h
+u + h
+2
+�∂εj
+, j = 1, 2,
+k−
+j (u) �→ exp
+��
+r>0
+�δ1j a2(−r)
+r
+((u + h)r − ur) + δ2j a1(−r)
+r
+(ur − (u − h)r)
+��
+, j = 1, 2.
+3. Semi-infinite construction
+Throughout the rest of the paper, we assume that s = 0 in (2.14) and we write Fi = Fi0
+for i = 0, 1. Furthermore, from now on, we assume that the double Yangian DY(sl2) and
+the Heisenberg subalgebra H, as given in Subsection 2.1, are h-adically completed. Our
+goal is to construct the topological bases for certain DY(gl2)-modules in parallel with the
+semi-infinite construction [6] for the affine Kac–Moody Lie algebra �sl2.
+3.1. Normalized Iohara–Kohno realization. We now introduce another structure of
+level 1 DY(gl2)-module over Fi by modifying the original formulae from Theorem 2.3.
+Theorem 3.1. The following assignments define a structure of DY(gl2)-module on Fi:
+Xα(u) �→ E−(0)−1E−(u)E+(u)E0(u)u∂α/2(u + h)∂α/2
+(u + h/4)∂α
+,
+X−α(u) �→ E−(0)E−(u + h/2)−1E+(u − h/2)−1E0(u − h/2)−1
+(u − h/4)∂α
+(u − h/2)∂α/2(u + h/2)∂α/2 ,
+k+
+1 (u) �→ exp
+�
+−
+�
+r>0
+a1(r)
+r
+��
+u + h
+2
+�−r
+−
+�
+u − h
+2
+�−r�� � u + h/4
+u + 5h/4
+�∂α/2
+,
+k+
+2 (u) �→ exp
+�
+−
+�
+r>0
+a2(r)
+r
+��
+u + h
+2
+�−r
+−
+�
+u − h
+2
+�−r�� � u + h/4
+u − 3h/4
+�∂α/2
+and the action of k−
+1 (u), k−
+2 (u) is given by Theorem 2.3.
+Proof. As with the original Iohara–Kohno realization from Theorem 2.3, one verifies by
+a direct calculation that the given operators satisfy the defining relations for double
+Yangian, which are established in [11, Thm. 2.1]. As a demonstration, we shall prove that
+the operators satisfy the level 1 defining relation
+[Xα(u), X−α(v)] = 1
+h
+�
+δ(u − v − h/2)k+
+2 (u − h/4)k+
+1 (u − h/4)−1
+5
+
+−δ(u − v + h/2)k−
+2 (v − h/4)k−
+1 (v − h/4)−1�
+,
+(3.1)
+cf. [11, Thm. 2.1], where δ(u − v) = �
+r∈Z u−r−1vr is the δ-function. Throughout the
+proof, we denote by X′
+±α(u) and k
+′±
+j (u) the corresponding operators from Theorem 2.3
+in order to distinguish them from those given by Theorem 3.1.
+First, we rewrite the left hand-side of (3.1) as
+[Xα(u), X−α(v)] =
+�
+X′
+α(u), X′
+−α(v)
+�
+f(u, v),
+(3.2)
+where
+f(u, v) = u∂α/2(u + h)∂α/2
+(u + h/4)∂α
+(v − h/4)∂α
+(v − h/2)∂α/2(v + h/2)∂α/2 .
+Next, consider the right hand-side of (3.1). Using the properties of δ-function, we find
+δ(u − v + h/2) = f(u, u + h/2)δ(u − v + h/2) = f(u, v)δ(u − v + h/2)
+since f(u, u + h/2) = 1. Furthermore, we have k
+′−
+j (u) = k−
+j (u) for j = 1, 2. Therefore, the
+second summand in (3.1) can be written as
+− δ(u − v + h/2)k−
+2 (v − h/4)k−
+1 (v − h/4)−1
+= − f(u, v)δ(u − v + h/2)k
+′−
+2 (v − h/4)k
+′−
+1 (v − h/4)−1.
+(3.3)
+Finally, we consider the first summand on the right hand-side of (3.1). Observe that the
+operators k+
+j (u) and k
+′+
+j (u) are connected by the identities
+k+
+1 (u) = k
+′+
+1 (u)
+�u + h/2
+u − h/2
+�∂ε1 � u + h/4
+u + 5h/4
+�∂α/2
+,
+k+
+2 (u) = k
+′+
+2 (u)
+�u + h/2
+u − h/2
+�∂ε2 � u + h/4
+u − 3h/4
+�∂α/2
+.
+Thus, we have the equality
+δ(u − v − h/2)k+
+2 (u − h/4)k+
+1 (u − h/4)−1
+= δ(u − v − h/2)
+�u − 3h/4
+u + h/4
+�∂α �u + h
+u − h
+�∂α/2
+k
+′+
+2 (u − h/4)k
+′+
+1 (u − h/4)−1.
+(3.4)
+However, the properties of δ-function imply
+δ(u − v − h/2)
+�u − 3h/4
+u + h/4
+�∂α �u + h
+u − h
+�∂α/2
+= f(u, u − h/2)δ(u − v − h/2) = f(u, v)δ(u − v − h/2),
+so we conclude by (3.4) that the first summand on the right hand-side of (3.1) equals
+f(u, v)δ(u − v − h/2)k
+′+
+2 (u − h/4)k
+′+
+1 (u − h/4)−1.
+(3.5)
+As the operators from Theorem 2.3 satisfy (3.1), the commutation relation (3.1) now
+follows by comparing the expressions in (3.2), (3.3) and (3.5).
+□
+The formula for the action of Xα(u) from Theorem 3.1 implies
+Xα(u)1 ⊗ 1 ∈ 1 ⊗ eα + uF0[[u]]
+(3.6)
+on F0. On the other hand, (3.6) does not hold for the action of Xα(u) given by Theorem
+2.3. As this property is required for the semi-infinite construction in Subsection 3.5, in the
+6
+
+rest of the paper we only consider the DY(gl2)-action from Theorem 3.1 and, furthermore,
+we write X±α(u) and k±
+j (u) for the corresponding vertex operators.
+3.2. Commutative operators. In this subsection, we associate with the operators
+Xα(u) from Theorem 3.1 certain commutative operators X(u) which satisfy an h-adic
+version of the integrability relations which go back to Lepowsky and Primc [17]. As with
+the construction of commutative operators of Ding and B. Feigin [1], which are associ-
+ated with the Frenkel–Jing realization [4] of certain Uq(�sl2)-modules, we modify the term
+E+(u) consisting of annihilation operators aj(r) with r > 0. However, in contrast with
+the quantum affine algebra case, the term E0(u) needs to be renormalized as well.
+Consider the operator series on Fi, i = 0, 1, defined by
+X(u) = Xα(u)E
++(u)E
+0(u),
+where
+E
+0(u) = (1 − h/u)∂α/2,
+(3.7)
+the action of Xα(u) is given by Theorem 3.1 and the term E
++(u) is
+E
++(u) = exp
+��
+r>0
+�
+a1(r)
+r
+�
+u − 7
+4h
+�−r
++ a2(r)
+r
+�
+u + 1
+4h
+�−r��
+.
+Proposition 3.2. The following identities hold for operators on Fi, i = 0, 1:
+X(u)X(v) = X(v)X(u),
+(3.8)
+X(u)X(u − h) = X(u)X(u + h) = 0.
+(3.9)
+Proof. By using (2.13) one easily verifies the following relations:
+E+(u)E−(v) = (u − v)(u − v + h)
+(u + h/4)2
+E−(v)E+(u),
+(3.10)
+E0(u)E0(v) = (u + h/4)2
+(v + h/4)2 E0(v)E0(u).
+(3.11)
+The commutativity (3.8) can be verified using (3.10), (3.11) and the identities
+E
++(u)E−(v) = u − v − h
+u − v
+u + h/4
+u − 7h/4E−(v)E
++(u),
+(3.12)
+E
+0(u)E0(v) = (1 − h/u) E0(v)E
+0(u),
+(3.13)
+which also follow from (2.13). As for the equality (3.9), it is sufficient to observe that,
+due to relations (3.10), (3.11), (3.12) and (3.13), we have the decomposition
+X(u)X(v) = f(u, v)X(u, v),
+where
+f(u, v) = (u − v − h)(u − v + h),
+X(u, v) = E−(0)−2E−(u)E−(v)E+(u)E
++(u)E+(v)E
++(v)e2α(u2 − h2)∂α/2 (v2 − h2)∂α/2.
+By applying X(u, v) to an arbitrary element of Fi we get only finitely many negative
+powers of the variables u and v modulo hn for any n ⩾ 1. Hence the formal limit v → u±h
+of X(u, v) is well-defined. Therefore, the limit v → u ± h of the product X(u)X(v) is
+well-defined as well. Furthermore, since f(u, u ± h) = 0, it equals zero, as required.
+□
+7
+
+3.3. Principal submodule W0. Let us write the operator series Xα(u) and X(u) as
+Xα(u) =
+�
+r∈Z
+x(r)u−r−1
+and
+X(u) =
+�
+r∈Z
+x(r)u−r−1.
+Let 1 = 1 ⊗ 1 ∈ F0. Motivated by the notion of principal subspaces of affine Lie algebra
+modules from [6,9], we define the principal submodule W0 of F0 as the h-adic completion
+of the C[[h]]-module which is spanned by all
+x(rn) . . . x(r1) 1,
+where
+n ⩾ 0
+and
+r1, . . . , rn ∈ Z.
+(3.14)
+In this subsection, we use the monomials (3.14) to construct a topological basis for W0.
+Remark 3.3. Note that W0 coincides with the h-adically completed C[[h]]-span of all
+x(rn) . . . x(r1) 1,
+where
+n ⩾ 0
+and
+r1, . . . , rn ∈ Z.
+(3.15)
+Indeed, the terms in (3.15) are the coefficients of u−rn−1
+n
+. . . u−r1−1
+1
+in X(un) . . . X(u1) 1.
+However, by using (3.12) and (3.13), along with E
++(u) 1 = E
+0(u) 1 = 1, we find
+X(un) . . . X(u1) 1 = F · Xα(un) . . . Xα(u1) 1 for F =
+�
+1⩽r<s⩽n
+�
+1 −
+h
+us − ur
+�
+.
+(3.16)
+As F is invertible in C((un)) . . . ((u1))[[h]] and both sides of (3.16) belong to the h-adic
+completion of F0((un)) . . . ((u1)), we conclude that each element in (3.14) can be expressed
+in terms of elements of the form (3.15) and vice versa. Moreover, by rewriting (3.16) as
+h−1 �
+X(un) . . . X(u1) 1 −Xα(un) . . . Xα(u1) 1
+�
+= h−1(F − 1)Xα(un) . . . Xα(u1) 1 = h−1(1 − F −1)X(un) . . . X(u1) 1,
+we conclude that the expressions of the form h−1 (x(rn) . . . x(r1) 1 −x(rn) . . . x(r1) 1) can
+be also written as C[[h]]-linear combinations of elements (3.14) or (3.15). Observe that
+such linear combinations can be infinite, but they are well-defined as they are finite
+modulo hm for all m ⩾ 1 and the C[[h]]-module F0 is h-adically complete.
+In the following lemma, we consider the monomials of the form (3.15). In comparison
+with (3.14), they are easier to handle since their factors x(r) commute; recall (3.8). Let
+BW0 be the set of all elements (3.15) with n ⩾ 0 such that their indices satisfy
+r1 ⩽ −1
+and
+rj+1 ⩽ rj − 2
+for all
+j = 1, . . . , n − 1.
+(3.17)
+Lemma 3.4. The h-adic completion of the C[[h]]-span of BW0 coincides with W0.
+Proof. First, note that X(u) 1 possesses only nonnegative powers of the variable u, i.e.
+we have x(r) 1 = 0 for r ⩾ 0. Thus, by commutativity (3.8), the terms
+x(rn) . . . x(r1) 1
+such that
+r1, . . . , rn ∈ Z<0
+and
+rn ⩽ . . . ⩽ r1.
+(3.18)
+span an h-adically dense C[[h]]-submodule of W0. Next, we use the h-adic integrability
+relation to establish the difference two condition (3.17) for the terms in (3.18). Although
+this is just an adjustment of the well-known argument, originating from the (integrable)
+representation theory of �sl2, to the h-adic setting, we give some details for completeness.
+Extracting the coefficients of u−2r−3 and u−2r−2 in (3.9) and using (3.8) we find
+x(r)x(r + 1) = −
+�
+l⩾1
+x(r − l)x(r + l + 1)
+mod h,
+(3.19)
+8
+
+x(r)x(r) = −2
+�
+l⩾1
+x(r − l)x(r + l)
+mod h,
+(3.20)
+respectively. Suppose that we have two adjacent operators x(rj+1)x(rj) in the element
+x(rn) . . . x(r1) 1 of (3.18) which do not satisfy the difference two condition rj+1 ⩽ rj − 2.
+Then, they can be replaced modulo h by a sum of those which satisfy this condition
+using the identity (3.19) (resp. (3.20)) if rj+1 = rj − 1 (resp. rj+1 = rj). Due to (3.9),
+the difference of the left and the right hand-side in (3.19) and in (3.20) belongs to hW0.
+In other words, it is a C[[h]]-linear combination of monomials as in (3.18) such that its
+coefficients belong to hC[[h]]. Thus, for any m ⩾ 1, by repeating such procedure sufficient
+number of times one can express the original element x(rn) . . . x(r1) 1 as a C[[h]]-linear
+combination of elements which satisfy the difference two condition (3.17) modulo hm. As
+W0 is h-adically complete, this implies the lemma.
+□
+In the definition (3.7) of the series X(u), we multiplied Xα(u) by the term E
++(u), thus
+changing the classical limit of the resulting operator. Therefore, we can not instantly
+establish the linear independence of the spanning set from the previous lemma via classical
+limit. Instead, we now turn to the monomials of operators (3.14). Let BW0 be the set of
+all elements (3.14) with n ⩾ 0 which satisfy the conditions imposed by (3.17).
+Lemma 3.5. The set BW0 is linearly independent over C[[h]].
+Proof. The classical limit h → 0 of the operators X±α(u) from Theorem 3.1 with i = 0
+produces the action of �sl2 on its level 1 integrable highest weight module L(Λ0) of the
+highest weight Λ0, as given by the Frenkel–Kac–Segal realization [5,20]. Thus, the classical
+limit of BW0 produces the well-known quasi-particle basis for the principal subspace of
+L(Λ0); see [6,9]. Hence, in particular, BW0 is linearly independent over C[[h]].
+□
+Finally, we combine Lemmas 3.4 and 3.5 to obtain a basis of W0.
+Theorem 3.6. The set BW0 forms a topological basis of W0.
+Proof. By Lemma 3.5, it is sufficient to check that the C[[h]]-span of BW0 is h-adically
+dense in W0. Choose any w ∈ W0. By Lemma 3.4, it can be expressed as w = �
+j ajbj,
+where aj ∈ C[[h]] and bj ∈ BW0. Next, by Remark 3.3, we have w = �
+j ajbj mod h,
+where bj ∈ BW0 denote the elements corresponding to bj, i.e., more precisely, if bj =
+x(rn) . . . x(r1) 1, then we set bj = x(rn) . . . x(r1) 1. In addition, the remark implies that
+h−1(w − �
+j ajbj) is again a C[[h]]-linear combination of the elements of BW0. Hence, for
+any m ⩾ 1, we can repeat this procedure until we express w as a C[[h]]-linear combination
+of the elements of BW0 modulo hm, so the theorem now follows.
+□
+3.4. DY(sl2)-modules Li(sl2). Consider the set
+BL0(sl2) =
+�
+ekαb : k ∈ Z, b ∈ BW0
+�
+⊂ F0.
+Let L0(sl2) be the h-adically completed C[[h]]-span of BL0(sl2). In this subsection, we show
+that L0(sl2) is closed with respect to the DY(sl2)-action from Theorem 3.1.
+Lemma 3.7. For any integer m we have
+E+(u)mL0(sl2) ⊂ L0(sl2)[[u−1]].
+(3.21)
+9
+
+Proof. Suppose m = 1. It is sufficient to show that E+(u)ekαb belongs to L0(sl2)[[u−1]]
+for all ekαb ∈ BL0(sl2). Using the formulae from Theorem 3.1 and E+(u) 1 = 1 we find
+E+(u)ekαXα(vn) . . . Xα(v1) 1 = F · ekαXα(vn) . . . Xα(v1) 1,
+(3.22)
+where, due to relation (3.10), the factor F ∈ C[u−1][[vn, . . . , v1, h]] is given by
+F =
+n
+�
+r=1
+�
+1 − vr
+u
+� �
+1 −
+vr
+u + h
+�
+.
+For any integers s1, . . . , sn, by extracting the coefficients of vs1
+1 . . . vsn
+n from the right hand-
+side of (3.22) one obtains a power series in the variable u−1. Clearly, its coefficients are
+C[[h]]-linear combinations of the elements of the form ekαx(rn) . . . x(r1) 1. Moreover, by
+Theorem 3.6, every x(rn) . . . x(r1) 1 can be expressed in terms of elements of BW0, so
+that every ekαx(rn) . . . x(r1) 1 belongs to L0(sl2). On the other hand, by extracting the
+coefficients of vs1
+1 . . . vsn
+n from the left hand-side of (3.22) for a suitable choice of s1, . . . , sn,
+we obtain E+(u)ekαb with ekαb ∈ BL0(sl2). Thus, we conclude that all E+(u)ekαb such that
+ekαb ∈ BL0(sl2) belong to L0(sl2)[[u−1]], as required.
+If m = −1, we can use E+(u)−1 1 = 1 and verify the inclusion in (3.21) analogously.
+Finally, if m is an arbitrary integer, one easily proves by induction over |m| that we have
+E+(u1)±1 . . . E+(u|m|)±1L0(sl2) ⊂ L0(sl2)[[u−1
+1 , . . . , u−1
+|m|]].
+(3.23)
+Hence the statement of the lemma follows by setting u1 = . . . = u|m| = u in (3.23).
+□
+The proof of Lemma 3.7 relies on the fact that the coefficients of E+(u) are C[[h]]-linear
+combinations of monomials of annihilation operators aj(r) with r > 0, which implies
+E+(u)± 1 = 1. As k+
+j (u)±1 1 = 1 for j = 1, 2, one can prove the next lemma analogously.
+Lemma 3.8. For any integer m and j = 1, 2 we have
+k+
+j (u)mL0(sl2) ⊂ L0(sl2)[[u−1]].
+By combining Lemma 3.8 and the identity (2.11) we get
+Corollary 3.9. H+(u)L0(sl2) ⊂ L0(sl2)[[u−1]].
+Let us consider the remaining operators from Theorem 3.1 whose action employs the
+creation operators aj(r) with r < 0 as well. First, observe that
+Xα(u)ekα = u2k(u+h)2kekαXα(u)
+implies
+Xα(u)L0(sl2) ⊂ L0(sl2)[[u±1]]. (3.24)
+Next, write E−(u) = E−(0)−1E−(u), so that, by Theorem 3.1, we have
+Xα(u) = E−(u)E+(u)eαu∂α/2(u + h)∂α/2.
+(3.25)
+Lemma 3.10. For any integer m we have
+E−(u)mL0(sl2) ⊂ L0(sl2)[[u]].
+(3.26)
+Proof. Let m = 1. We can use the formulae from Theorem 3.1 and (3.10) to prove
+E−(u)ekαXα(vn) . . . Xα(v1) 1 = F · ekαXα(vn) . . . Xα(v1)E−(u) 1,
+(3.27)
+where the factor F ∈ C[v−1
+n , . . . , v−1
+1 ][[u, h]] is given by
+F =
+n
+�
+r=1
+�
+1 − u
+vr
+�−1 �
+1 −
+u
+vr + h
+�−1
+.
+10
+
+Since E+(u)eαu∂α/2(u + h)∂α/2 1 = eα 1, by (3.25) the rightmost term in (3.27) equals
+E−(u) 1 = e−αE−(u)E+(u)eαu∂α/2(u + h)∂α/2 1 = e−αXα(u) 1 .
+Therefore, the right hand-side of (3.27) is equal to
+F · ekαXα(vn) . . . Xα(v1)e−αXα(u) 1 = G · e(k−1)αXα(vn) . . . Xα(v1)Xα(u) 1 .
+(3.28)
+The right hand-side of (3.28) was found by moving the operator e−α all the way to the
+left, which produced the term G ∈ C[v−1
+n , . . . , v−1
+1 ][[u, h]] given by
+G = F ·
+n
+�
+r=1
+v−1
+r
+(vr + h)−1 =
+n
+�
+r=1
+(vr − u)−1 (vr − u + h)−1 .
+Thus, we proved that the left hand-side in (3.27) coincides with the right hand-side in
+(3.28). However, it is clear that all coefficients of the right hand-side of (3.28) belong
+to L0(sl2). Hence, the inclusion (3.26) for m = 1 follows as in the proof of Lemma 3.7.
+Furthermore, the general case m > 1 can be again verified by induction over m.
+Let m = −1. Note that E−(u) can be written as E−(u) = 1 − e−(u), where e−(u)
+belongs to uU(�s−)[[h, u]]. Hence E−(u) is invertible and its inverse takes the form
+E−(u)−1 =
+�
+1 − e−(u)
+�−1 =
+�
+l⩾0
+e−(u)l =
+�
+l⩾0
+�
+1 − E−(u)
+�l .
+(3.29)
+Each summand in (3.29) is given in terms of nonnegative powers of E−(u), which satisfy
+(3.26), so we conclude that E−(u)−1L0(sl2) ⊂ L0(sl2)[[u]], as required. Finally, as before,
+this is generalized to any negative integer m by induction.
+□
+By using (2.11) and the action of k−
+j (u) from Theorem 3.1 one easily verifies the identity
+H−(u − h/4) = E−(u)E−(u + h)−1 for operators on F0. Hence, by Lemma 3.10 we have
+Corollary 3.11. H−(u)L0(sl2) ⊂ L0(sl2)[[u]].
+Let us turn our attention to the operator X−α(u), which takes the form
+X−α(u) = E−(u + h/2)−1E+(u − h/2)−1e−α(u − h/2)−∂α/2(u + h/2)−∂α/2.
+(3.30)
+Lemma 3.12. X−α(u)L0(sl2) ⊂ L0(sl2)[[u±1]].
+Proof. The lemma follows by an argument which goes in parallel with the proof of Lemma
+3.7. More specifically, it relies on the identity
+X−α(u)ekαXα(vn) . . . Xα(v1) 1 = F · E−(u + h/2)−1e(k−1)αXα(vn) . . . Xα(v1) 1,
+(3.31)
+where the series F ∈ C[u−1][[vn, . . . , v1, h]] is given by
+F = (u − h/2)−k (u + h/2)−k
+n
+�
+r=1
+(u − vr − h/2)−1 (u − vr + h/2)−1 .
+In addition, it employs Lemma 3.10, which implies that all coefficients of the right hand-
+side of (3.31) belong to L0(sl2). As for the equality in (3.31), it is verified using the
+expressions (3.25) and (3.30) for the operators X±α(u) and the relation (3.10).
+□
+Finally, Corollaries 3.9 and 3.11, the inclusion in (3.24) and Lemma 3.12 imply
+Theorem 3.13. The C[[h]]-module L0(sl2) is a module for the double Yangian DY(sl2).
+Consider the C[[h]]-module L1(sl2) =
+�
+eλ1v : v ∈ L0(sl2)
+�
+. The next lemma is verified
+by a direct calculation.
+11
+
+Lemma 3.14. The following commutation relations hold:
+Xα(u)eλ1 = u1/2(u + h)1/2eλ1 Xα(u),
+X−α(u)eλ1 = (u − h/2)−1/2(u + h/2)−1/2eλ1X−α(u),
+H+(u)eλ1 = (u + 7h/4)1/2(u − h/4)−1/2eλ1 H+(u),
+H−(u)eλ1 = eλ1 H−(u).
+By using Lemma 3.14, one easily checks that L1(sl2) is closed under the action of the
+double Yangian DY(sl2) as well, so that we have
+Theorem 3.15. The C[[h]]-module L1(sl2) is a module for the double Yangian DY(sl2).
+3.5. Semi-infinite monomial bases for Li(sl2). In this subsection, we construct topo-
+logical bases for L0(sl2) and L1(sl2). Let us start with L0(sl2). Introduce the operator
+�
+X(u) =
+�
+r∈Z
+�x(r)u−r−1 = Xα(u) (1 + h/u)−∂α/2 ,
+(3.32)
+where the action of Xα(u) is given by Theorem 3.1. It satisfies the identity
+�
+X(un) . . . �X(u1) 1 = F · Xα(un) . . . Xα(u1) 1
+for
+F =
+�
+r=2,...,n
+(1 + h/ur)1−r .
+(3.33)
+Let �BW0 be the set of all elements �x(rn) . . . �x(r1) 1, where n ⩾ 0 and the integers r1, . . . , rn
+satisfy the conditions given by (3.17).
+Corollary 3.16. The set �BW0 forms a topological basis of W0.
+Proof. The given set is linearly independent as its classical limit coincides with the classical
+limit of the basis BW0 from Theorem 3.6. The fact that the C[[h]]-span of �BW0 is h-adically
+dense in W0 is established by arguing as in the proof of Theorem 3.6. However, while the
+argument therein relies on the identity (3.16), here one uses (3.33) instead.
+□
+For any integer m let �BL0(sl2),m be the set of all elements emαb such that b ∈ �BW0.
+Clearly, the union ∪m∈Z �BL0(sl2),m spans an h-adically dense C[[h]]-submodule of L0(sl2).
+We now employ the Feigin–Stoyanovsky-type construction [6] to reduce it to a linearly
+independent set. By (3.6) and (3.32) we have
+�x(−1) 1 = eα 1 .
+(3.34)
+Furthermore, by extracting the coefficients in the relation �
+X(u)e−α = e−α �
+X(u)u−2, which
+can be easily verified by a direct calculation, we find
+�x(r)e−α = e−α �x(r − 2)
+for all
+r ∈ Z.
+(3.35)
+Let L0(sl2)m be the h-adically completed C[[h]]-span of �BL0(sl2),m. Observe that the union
+∪m∈ZL0(sl2)m coincides with L0(sl2). Also, we have L0(sl2)m ⊂ L0(sl2)m−1 for all integers
+m. Indeed, this is an immediate consequence of the inclusion �BL0(sl2),m ⊂ �BL0(sl2),m−1,
+which can be proved by using the identities (3.34) and (3.35) as follows:
+emα �x(rn) . . . �x(r1) 1 = emα �x(rn) . . . �x(r1)e−αeα 1 = emα �x(rn) . . . �x(r1)e−α �x(−1) 1
+= e(m−1)α �x(rn − 2) . . . �x(r1 − 2)�x(−1) 1 ∈ �BL0(sl2),m−1.
+Finally, we obtain the topological basis for L0(sl2).
+12
+
+Theorem 3.17. The direct limit
+�BL0(sl2) = lim
+−→
+�BL0(sl2),m
+forms a topological basis of L0(sl2).
+Proof. It is clear that the elements of �BL0(sl2) span the h-adically dense C[[h]]-submodule
+of L0(sl2). On the other hand, their classical limits produce the semi-infinite basis for the
+integrable highest weight �sl2-module L(Λ0) of the highest weight Λ0, established by B.
+Feigin and Stoyanovsky [6], so that they are linearly independent.
+□
+Consider the DY(sl2)-module L1(sl2). For any integer m let �BL1(sl2),m be the set of
+all elements eλ1+mαb with b ∈ �BW0. The union ∪m∈Z �BL1(sl2),m spans an h-adically dense
+C[[h]]-submodule of L1(sl2). The semi-infinite basis for L1(sl2) is established in parallel
+with the case of L0(sl2). Naturally, in this case, its linear independence follows from
+the observation that its classical limit produces the semi-infinite basis for the integrable
+highest weight �sl2-module L(Λ1) of the highest weight Λ1 from [6]. Hence, we have
+Theorem 3.18. The direct limit
+�BL1(sl2) = lim
+−→
+�BL1(sl2),m
+forms a topological basis of L1(sl2).
+Remark 3.19. In this remark, we discuss an interpretation of the bases �BLi(sl2) in terms
+of semi-infinite monomials [6]. First, let us write vi,m = eλi+mα 1 for m ∈ Z and i = 0, 1
+so that, in particular, we have vi,0 = eλi 1. We can express the elements vi,0 = eλi 1 as
+vi,0 = eλi 1 = eλi−αeα 1 = eλi−α�x(−1) 1 = �x(1 − i)eλi−α 1 = �x(1 − i)vi,−1.
+Next, by a similar calculation, which starts with vi,−1, we get
+vi,−1 = eλi−α 1 = eλi−2αeα 1 = eλi−2α�x(−1) 1 = �x(3 − i)eλi−2α 1 = �x(3 − i)vi,−2.
+Combining the above equalities we find that vi,0 = �x(1 − i)�x(3 − i)vi,−2. By repeating
+such calculations we find
+vi,0 = �x(1 − i)�x(3 − i) . . . �x(2m − 1 − i)vi,−m
+for any
+m > 0.
+(3.36)
+Note that by (3.36) the direct limits in Theorems 3.17 and 3.18 correspond with taking
+the limit m → ∞ of the elements of the form
+emαbvi,0 = emαb�x(1 − i)�x(3 − i) . . . �x(2m − 1 − i)vi,−m,
+where
+b ∈ �BW0
+Hence, arguing as above to move the term emα to the right, we see that the elements of
+the bases �BLi(sl2) can be represented as the semi-infinite monomials
+�x(r1)�x(r2) . . . vi,−∞
+such that
+(1) Their degrees r1, r2, . . . ∈ Z satisfy the difference two condition
+rj ⩽ rj+1 − 2
+for all
+j = 1, 2, . . . ,
+(2) For each monomial there exists an index n such that all degrees rn, rn+1, . . . are
+consecutive odd (resp. even) integers if i = 0 (resp. i = 1).
+13
+
+3.6. Semi-infinite monomial bases for Li(gl2). Consider the DY(gl2)-modules
+L0(gl2) = DY(gl2) · 1
+and
+L1(gl2) = DY(gl2) · eλ1,
+where, as before, the action of the double Yangian is given by Theorem 3.1. Recall that
+the Heisenberg subalgebra H commutes with all elements of DY(sl2), so that we have
+L0(gl2) = H·DY(sl2)·1 = DY(sl2)·H·1
+and
+L1(gl2) = H·DY(sl2)·eλ1 = DY(sl2)·H·eλ1.
+Consider the subalgebra H− ⊂ H generated by 1 and all elements κ(−r) with r ⩾ 1.
+The algebra H− possesses the Poincar´e–Birkhoff–Witt basis
+BH− =
+�
+κ(−rn) . . . κ(−r1) : n ⩾ 0, rn ⩾ . . . ⩾ r1 ⩾ 1
+�
+.
+(3.37)
+Indeed, the elements of BH− span H− as all κ(−r) with r ⩾ 1 commute. On the other hand,
+their linear independence is established by the map (2.12) and the Poincar´e–Birkhoff–
+Witt theorem for U(�h−), where �h− stands for the Lie subalgebra of �h generated by all
+I(−r) = I ⊗ t−r with r ⩾ 1. For i = 0, 1 introduce the sets
+�BLi(gl2) =
+�
+hb : h ∈ BH−, b ∈ �BLi(sl2)
+�
+.
+Theorem 3.20. Let i = 0, 1. The set �BLi(gl2) forms a topological basis for Li(gl2).
+Proof. Theorems 3.17 and 3.18, along with the discussion preceding this theorem, imply
+that the set �BLi(gl2) spans an h-adically dense C[[h]]-submodule of Li(gl2). On the other
+hand, its linear independence follows from the observation that the classical limit of
+Li(gl2) is equal to U(�h−) ⊗ L(Λi).
+□
+4. Modules for the quantum vertex algebra Vc(gl2)
+In this section, we study the underlying (quantum) vertex algebraic framework of the
+semi-infinite construction. In particular, we obtain examples of irreducible modules for
+the quantized universal affine vertex algebra of gl2.
+4.1. Quantum affine vertex algebra Vc(gl2). The dual Yangian Y+(gl2) for gl2 is
+defined as the associative algebra over the ring C[[h]] generated by the elements t(−r)
+ij
+,
+where i, j = 1, 2 and r = 1, 2 . . . . Its defining relations are given by
+R12(u − v)T −
+13(u)T −
+23(v) = T −
+23(v)T −
+13(u)R12(u − v),
+where the notation is as in (2.3) and (2.4). We use the same symbols for the generators
+of the dual Yangian and the corresponding generators of the double Yangian as Y+(gl2)
+can be naturally regarded as a subalgebra of DY(gl2) due to the Poincar´e–Birkhoff–Witt
+theorem. Moreover, for any c ∈ C the h-adic completion of the dual Yangian is naturally
+equipped with the DY(gl2)-action so that the central element C acts as the scalar multipli-
+cation by c. We shall denote this DY(gl2)-module by Vc(gl2). Furthermore, Vc(gl2) can be
+equipped with the quantum vertex algebra structure via Etingof–Kazhdan’s construction
+as follows; see [3, Thm. 2.3] for more information.
+Theorem 4.1. There exists a unique quantum vertex algebra structure on Vc(gl2) so
+that the unit 1 is the vacuum vector and the vertex operator map satisfies
+Y (T −
+1n(u1) . . . T −
+n−1n(un−1), z) = T −
+1n(z + u1) . . . T −
+n−1n(z + un−1)
+× T +
+n−1n(z + un−1 − hc/2)−1 . . . T +
+1n(z + u1 − hc/2)−1.
+14
+
+Proof. In comparison with the original result [3, Thm. 2.3], we use a different normaliza-
+tion of the Yang R-matrix, which governs the relation (2.2) between the operators T ±(u).
+However, the theorem can be again proved by the analogous arguments, which can be
+also recovered from the proofs of [9, Thm. 2.3.8] and [12, Thm. 4.1].
+□
+Remark 4.2. The map Y (·, z) can be expressed modulo h as follows. Write
+t±(u) = ±h−1(I − T ±(u))
+and
+t(u) = t−(u) + t+(u).
+One can prove by induction over n the identity
+Y (t−
+1n(u1) . . . t−
+n−1n(un−1), z) = :t1n(u1) . . . tn−1n(un−1):
+mod h,
+where the normal-ordered product of operators is defined in a usual way:
+:t(u1)t(u2): = t−(u1)t(u2) + t(u2)t+(u1)
+and
+:t(u1) . . . t(ur): = :t(u1) :t(u2) . . . t(ur)::.
+Thus, it is easy to see that the classical limit h → 0 of the quantum vertex algebra Vc(gl2)
+coincides with the universal affine vertex algebra V c(gl2) which goes back to the papers
+of I. Frenkel and Zhu [7] and Lian [18]; recall Remark 2.2.
+Let c2 = −2c. Denote by Vc2(h) the level c2 module for the Heisenberg subalgebra
+H ⊂ DY(gl2) defined over the h-adic completion of H−. Clearly, we have Vc2(h) ⊂ Vc(gl2)
+and the topological basis of Vc2(h) is given by (3.37). The generator series K±(u) of H
+can be expressed as the quantum determinants of the matrices T ±(u) from the RTT-
+realization of DY(gl2), so that we have K±(u) = qdetT ±(u); see [10, Thm. B.15]. Using
+this observation, one obtains the following simple consequence of Theorem 4.1.
+Corollary 4.3. The C[[h]]-module Vc2(h) is a quantum vertex subalgebra of Vc(gl2) for
+c = −c2/2. Its vacuum vector is 1 ∈ Vc(gl2) and its vertex operator map satisfies
+Y (K−(u1) . . . K−(un), z) = K−(z + u1) . . . K−(z + un)
+× K+(z + un + hc2/4)−1 . . . K+(z + u1 + hc2/4)−1.
+Remark 4.4. The results of this subsection are presented for gl2 in order to better fit the
+setting of the paper. However, their generalization to glN with N > 2 is straightforward.
+This is also true for the notion of restricted module and Theorem 4.6 which we give below.
+4.2. Restricted modules for the double Yangian. A DY(gl2)-module V is said to
+be restricted if it is a topologically free C[[h]]-module such that for any v ∈ V and n ⩾ 1
+the expression T +(u)v possesses only finitely many negative powers of u modulo hn. The
+last requirement can be equivalently expressed as
+T +(u)v ∈ End C2 ⊗ V [u−1]h
+for all
+v ∈ V,
+(4.1)
+where V [u−1]h stands for the h-adic completion of the C[[h]]-module V [u−1]. Extending
+this notation, we shall also write V ((u))h for the h-adic completion of V ((u)).
+Proposition 4.5. The DY(gl2)-modules Fis and Fi established by Theorems 2.3 and 3.1
+respectively are restricted.
+Proof. It is clear that the underlying C[[h]]-module structure is topologically free. More-
+over, by examining the explicit formulae in the aforementioned theorems we find
+X±(u) ∈ Hom(F, F((u))h),
+k−
+j (u) ∈ Hom(F, F[[u]]),
+k+
+j (u) ∈ Hom(F, F[u−1]h),
+15
+
+where F = Fis, Fi. We now employ the identities in (2.8)–(2.10) to prove the proposition.
+First, we note that by the first equality in (2.8) we have t+
+11(u) ∈ Hom(F, F[u−1]h). Next,
+from (2.10) we obtain
+t+
+21(u) = X−(u − hC/4)t+
+11(u) + t−
+21(u − hC/2)t−
+11(u − hC/2)−1t+
+11(u),
+which implies t+
+21(u) ∈ Hom(F, F[u−1]h). As for t+
+12(u), we rewrite the relation (2.9) as
+t+
+11(u)−1t+
+12(u) = X+(u + hC/4) + t−
+11(u + hC/2)−1t−
+12(u + hC/2)
+to conclude that the product t+
+11(u)−1t+
+12(u) belongs to Hom(F, F[u−1]h). Multiplying this
+product by t+
+11(u) ∈ Hom(F, F[u−1]h), we obtain t+
+12(u) ∈ Hom(F, F[u−1]h), as required.
+Finally, the second equality in (2.8) implies
+t+
+22(u) = k+
+2 (u) + t+
+21(u)t+
+11(u)−1t+
+12(u).
+As t+
+11(u)−1 ∈ Hom(F, F[u−1]h), we conclude that the remaining matrix entry t+
+22(u) of
+T +(u) belongs to Hom(F, F[u−1]h), thus verifying the requirement imposed by (4.1).
+□
+The next theorem was our main motivation for introducing the notion of restricted
+module. Its proof relies on the RTT-realization of the double Yangian and goes in parallel
+with the proof of Theorem 4.1.
+Theorem 4.6. Let V be a restricted DY(gl2)-module of level c. Then there exists a
+unique structure of Vc(gl2)-module over V such that for all n ⩾ 1 we have
+YV (T −
+1n(u1) . . . T −
+n−1n(un−1), z) = T −
+1n(z + u1)V . . . T −
+n−1n(z + un−1)V
+× T +
+n−1n(z + un−1 − hc/2)−1
+V . . . T +
+1n(z + u1 − hc/2)−1
+V .
+The next corollary is an immediate consequence of Proposition 4.5 and Theorem 4.6.
+Corollary 4.7. There exists a unique structure of V1(gl2)-module over Li = Li(gl2) with
+i = 0, 1 such that for all n ⩾ 1 we have
+YLi(T −
+1n(u1) . . . T −
+n−1n(un−1), z) = T −
+1n(z + u1)Li . . . T −
+n−1n(z + un−1)Li
+× T +
+n−1n(z + un−1 − h/2)−1
+Li . . . T +
+1n(z + u1 − h/2)−1
+Li .
+Consider the standard structure of bosonic Fock �h-module M(−2) of level −2. In
+particular, recall that M(−2) is an irreducible �h-module which, as a vector space, coincides
+with U(�h−). The tensor product Li = M(−2) ⊗ L(Λi) for i = 0, 1, where L(Λi) is the
+integrable highest weight �sl2-module of highest weight Λi, is naturally equipped with the
+structure of irreducible module for �gl2. Thus, Li possesses the structure of irreducible
+module for the universal affine vertex algebra V 1(gl2) of level 1. On the other hand, the
+classical limit of the V1(gl2)-module Li(gl2) coincides with the V 1(gl2)-module Li.
+In the next corollary, we use the notion of irreducible module in the following sense.
+The topologically free C[[h]]-module V is said to be irreducible with respect to the action
+of associative algebra (resp. quantum vertex algebra) if it does not possess any nontrivial
+topologically free C[[h]]-submodule W which is invariant under the corresponding action
+of associative algebra (resp. quantum vertex algebra) and satisfies that hnv ∈ W for
+n ⩾ 1 and v ∈ V implies v ∈ W.
+Corollary 4.8. The C[[h]]-modules Li(gl2) are irreducible both as modules for the quan-
+tum vertex algebra V1(gl2) and as level 1 modules for the double Yangian DY(gl2).
+16
+
+Proof. The irreducibility with respect to the action of V1(gl2) follows by the discussion
+preceding the corollary. As for the action of the double Yangian, if the C[[h]]-submodule
+W ⊂ Li(gl2) is DY(gl2)-invariant, by Corollary 4.7 we have
+YLi(T −
+1n(u1) . . . T −
+n−1n(un−1), z)W ⊂ (End CN)⊗(n−1) ⊗ W[[z±1, u1, . . . , un−1]]
+for all n ⩾ 1. As the coefficients of matrix entries of T −
+1n(u1) . . . T −
+n−1n(un−1) with n ⩾ 1
+span an h-adically dense C[[h]]-submodule of V1(gl2), it is clear that W is V1(gl2)-invariant
+as well, so that the second assertion follows.
+□
+Remark 4.9. Ding–Feigin’s construction of semi-infinite monomial bases [1] in the case of
+quantum affine algebra Uq(�sl2) suggests that the suitable analogues of Corollaries 4.7 and
+4.8 can be established for Etingof–Kazhdan’s quantum vertex algebra [3] associated with
+the trigonometric R-matrix of type A(1)
+1 . On the other hand, by using different approach,
+the h-adic quantum vertex algebra structure over certain wide class of irreducible modules
+for untwisted quantum affinization algebras was recently obtained by Kong [15].
+Acknowledgement
+M.B. and S.K. would like to thank Mirko Primc for helpful discussions. This work
+has been supported in part by Chinese National Natural Science Foundation grant nos.
+12101261 and 12171303 and by Croatian Science Foundation under the project UIP-2019-
+04-8488.
+References
+[1] J. Ding, B. Feigin, Commutative quantum current operators, semi-infinite construction and functional
+models, Represent. Theory 4 (2000), 330–341; arXiv:q-alg/9612009.
+[2] V. G. Drinfeld, A new realization of Yangians and quantized affine algebras, Soviet. Math. Dokl. 36
+(1988), 212–216.
+[3] P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, V, Selecta Math. (N.S.) 6 (2000), 105–130;
+arXiv:math/9808121 [math.QA].
+[4] I. B. Frenkel, N. Jing, Vertex representations of quantum affine algebras, Proc. Natl. Acad. Sci. USA
+85 (1988), 9373–9377.
+[5] I. Frenkel, V. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent.
+Math. 62 (1980/81), 23–66.
+[6] A. V. Stoyanovsky, B. L. Feigin, Functional models of the representations of current algebras and semi-
+infinite Schubert cells, Funktsional Anal. i Prilozhen. 28 (1994), 68–90, 96 (in Russian); translation
+in Funct. Anal. Appl. 28 (1994), 55–72; arXiv:hep-th/9308079.
+[7] I. B. Frenkel and Y.-C. Zhu, Vertex operator algebras associated to representations of affine and
+Virasoro algebras, Duke Math. J. 66 (1992), 123–168.
+[8] M. Gardini, Quantum vertex algebras, Ph.D. thesis, Sapienza – University of Rome, 2018.
+[9] G. Georgiev, Combinatorial constructions of modules for infinite-dimensional Lie algebras, I. Prin-
+cipal subspace, J. Pure Appl. Algebra 112 (1996), 247–286; arXiv:hep-th/9412054.
+[10] K. Iohara, Bosonic representations of Yangian double DYℏ(g) with g = glN, slN, J. Phys. A 29
+(1996), 4593–4621; arXiv:q-alg/9603033.
+[11] K. Iohara, M. Kohno, A central extension of DYh(gl2) and its vertex representations, Lett. Math.
+Phys. 37 (1996), 319–328; arXiv:q-alg/9603032.
+[12] N. Jing, S. Koˇzi´c, A. Molev, F. Yang, Center of the quantum affine vertex algebra in type A, J.
+Algebra 496 (2018), 138–186; arXiv:1603.00237 [math.QA].
+[13] N. Jing, F. Yang, Center of the Yangian double in type A, arXiv:2207.01712 [math.QA].
+[14] V. G. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge,
+1990.
+[15] F. Kong, Quantum affine vertex algebras associated to untwisted quantum affinization algebras,
+arXiv:2212.04888 [math.QA].
+17
+
+[16] J. Lepowsky, S. Milne, Lie algebraic approaches to classical partition identities, Adv. Math. 29
+(1978), 15–59.
+[17] J. Lepowsky, M. Primc, Structure of the standard modules for the affine Lie algebra A(1)
+1 , Contemp.
+Math. 46, Amer. Math. Soc., Providence, 1985.
+[18] B.-H. Lian, On the classification of simple vertex operator algebras, Comm. Math. Phys. 163 (1994),
+307–357.
+[19] M. Nazarov, Double Yangian and the universal R-matrix, Jpn. J. Math 15 (2020), 169–221;
+arXiv:1904.02517 [math.QA].
+[20] G. Segal, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys. 80
+(1981), 301–342.
+Marijana Butorac:
+Faculty of Mathematics, University of Rijeka,
+Radmile Matejˇci´c 2, 51000 Rijeka, Croatia
+Email address: mbutorac@math.uniri.hr
+Naihuan Jing:
+Department of Mathematics, North Carolina State University,
+Raleigh, NC 27695, USA
+Email address: jing@ncsu.edu
+Slaven Koˇzi´c:
+Department of Mathematics, Faculty of Science, University of Zagreb,
+Bijeniˇcka cesta 30, 10000 Zagreb, Croatia
+Email address: kslaven@math.hr
+Fan Yang:
+Department of Mathematics, Jiaying University,
+Meizhou, Guangdong 514000, China
+Email address: 1329491781@qq.com
+18
+
diff --git a/UNE3T4oBgHgl3EQfzwvz/content/tmp_files/load_file.txt b/UNE3T4oBgHgl3EQfzwvz/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dbb977c9f6a1617d1cd04cad9f20cf2dcc1464f0
--- /dev/null
+++ b/UNE3T4oBgHgl3EQfzwvz/content/tmp_files/load_file.txt
@@ -0,0 +1,1098 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf,len=1097
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='04732v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='QA]  11 Jan 2023  SEMI-INFINITE CONSTRUCTION FOR THE DOUBLE YANGIAN OF  TYPE A(1)  1  MARIJANA BUTORAC, NAIHUAN JING, SLAVEN KOˇZI´C AND FAN YANG  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We consider certain infinite dimensional modules of level 1 for the double  Yangian DY(gl2) which are based on the Iohara–Kohno realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We show that they  possess topological bases of Feigin–Stoyanovsky-type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' the bases expressed in terms  of semi-infinite monomials of certain integrable operators which stabilize and satisfy  the difference two condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Finally, we give some applications of these bases to the  representation theory of the corresponding quantum affine vertex algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Introduction  The integrable highest weight modules present one of the most fundamental notions  in the representation theory of affine Kac–Moody Lie algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=', the book by  Kac [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The problem of constructing different types of bases for such modules and  their various substructures, especially those which establish connection with Rogers–  Ramanujan-type identities via character formulae, has been extensively studied since the  pioneering paper of Lepowsky and Milne [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Our paper is motivated by the well known  Feigin–Stoyanovsky construction [6] of semi-infinite monomial bases for certain integrable  highest weight modules for the affine Lie algebra �sl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The construction relies on the fact  that these modules can be obtained from their principal subspaces using the Weyl trans-  lation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' At the level 1, the resulting bases consist of semi-infinite monomials  xα(r1)xα(r2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' in coefficients of the vertex operator xα(z) = �  r∈Z xα(r)z−r−1 associ-  ated with the positive simple root α of sl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Their degrees r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' satisfy the difference  two condition rj+1 ⩾ rj + 2 for all j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , which comes from the integrability  relation xα(z)2 = 0 of Lepowsky and Primc [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Moreover, these monomials stabilize,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' for a sufficiently large n all degrees rn, rn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' are consecutive odd or even integers,  depending on the choice of the highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Later on, the semi-infinite construction  was generalized to the case of quantum affine algebra Uq(�sl2) by Ding and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Feigin [1]  using the realization of its integrable highest weight modules found by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Frenkel and the  second author in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  The goal of this paper is to give a semi-infinite construction for certain infinite di-  mensional modules of level 1 for the centrally extended double Yangian DY(gl2) defined  over the commutative ring C[[h]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Their bosonic realization, which resembles the famous  Frenkel–Kac–Segal construction [5, 20] for affine Lie algebras, was given by Iohara and  Kohno in [11] for DY(gl2) and then generalized to the higher rank case by Iohara [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  We slightly modify the Iohara–Kohno realization as the action of the original translation  operator does not appear to be in tune with the semi-infinite construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' However, the  action of the double Yangian is still given on the same C[[h]]-module, which we denote by  Fi, i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In contrast with the aforementioned setting of affine Lie algebras and quan-  tum affine algebras, the general theory of integrable representations for double Yangians  has not yet been sufficiently developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Thus, in our construction, we often need to use  1  different and more technical arguments which rely on the explicit formulae for the action  of the double Yangian generators on Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Motivated by Ding–Feigin’s approach [1], we start by defining an auxiliary commutative  operator X(z) on Fi, i = 0, 1, which can be regarded as a Yangian counterpart of the  level 1 affine vertex operator xα(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In particular, it satisfies the h-adic integrability  relation X(z)X(z ± h) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We use its coefficients in parallel with [6, 9] to introduce  the notion of principal submodule Wi ⊂ Fi and, furthermore, to obtain the topological  basis for Wi which provides an interpretation of the sum-sides of Rogers–Ramanujan  identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Next, we employ the action of translation operator on Wi to recover irreducible  modules Li(sl2) ⊂ Fi for the double Yangian DY(sl2), such that their classical limits  are exactly the level 1 integrable highest weight �sl2-modules L(Λi) of highest weight Λi  with i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Finally, we construct the Feigin–Stoyanovsky-type semi-infinite monomial  bases for Li(sl2), which is the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In addition, we generalize this  construction to the corresponding modules Li(gl2) for the double Yangian DY(gl2) by  using the action of its Heisenberg subalgebra, which is generated by the coefficients of  the quantum determinant and commutes with DY(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  At the end of the paper, we obtain some applications of the semi-infinite construction  to the quantum vertex algebra theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In particular, by employing the Iohara–Kohno  isomorphism [11] between two realizations of the double Yangian, we show that Li(gl2)  are naturally equipped with the structure of irreducible modules for the corresponding  Etingof–Kazhdan quantum affine vertex algebra of level 1 from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Preliminaries  In this section, we recall the double Yangian for the general linear Lie algebra gl2 and  the Iohara–Kohno bosonic realization of its level 1 modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Double Yangian for gl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We follow the paper of Iohara and Kohno [11] to intro-  duce the centrally extended double Yangians for the Lie algebras gl2 and sl2 and recover  some of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Let I be the identity and P the permutation operator on C2⊗C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Consider the (normalized) Yang R-matrix over the commutative ring C[[h]],  R(u) =  1  1 + h/u  �  I + h  uP  �  ∈ End C2 ⊗ End C2[[h/u]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  The double Yangian DY(gl2) is defined as the associative algebra over the ring C[[h]]  generated by the central element C and the elements t(r)  ij , where i, j = 1, 2 and r ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Its  defining relations are given by  R12(u − v)T ±  13(u)T ±  23(v) = T ±  23(v)T ±  13(u)R12(u − v),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1)  R12(u − v − hC/2)T +  13(u)T −  23(v) = T −  23(v)T +  13(u)R12(u − v + hC/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2)  The generator matrices T ±(u) are defined by  T ±(u) =  �  i,j=1,2  eij ⊗ t±  ij(u),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3)  where eij ∈ End C2 denote the matrix units and the power series t±  ij(u) are given by  t+  ij(u) = δij − h  �  r⩾0  t(r)  ij u−r−1  and  t−  ij(u) = δij + h  �  r⩾1  t(−r)  ij  ur−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4)  2  In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2) we use the subscripts to indicate the tensor factors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' we have  T ±  13(u) =  �  i,j=1,2  eij ⊗ 1 ⊗ t±  ij(u)  and  T ±  23(u) =  �  i,j=1,2  1 ⊗ eij ⊗ t±  ij(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Let us discuss the classical limit of double Yangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Consider the affine Lie algebra  �gl2 = gl2 ⊗ C[t±1] ⊕ CK, where K is the central element and the Lie brackets are  [eij(r), ekl(s)] = δkj eil(r + s) − δilekj(r + s) + rδr+s0K (δkj δil − δij δkl)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5)  for eij(r) = eij ⊗ tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Introduce the ascending filtration over DY(gl2) by setting  deg t(r)  ij = r  for  i, j = 1, 2, r ∈ Z  and  deg C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6)  The images ¯t(r)  ij and ¯C of the double Yangian generators t(r)  ij and C in the corresponding  graded algebra grDY(gl2) satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Thus, the assignments eij(r) �→ ¯t(r)  ij and K �→ ¯C  define the algebra homomorphism  U(�gl2) ⊗ C[[h]] → grDY(gl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7)  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7) is an algebra isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The surjectivity of the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' On the other hand, the injectivity is a  consequence of the Poincar´e–Birkhoff–Witt theorem [12, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2], which states that the  suitably ordered monomials in the double Yangian generators form its basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' see also [13,  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1] and [19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' More specifically, although we define the double Yangian  using the normalization of the Yang R-matrix which differs from [12], the arguments from  the corresponding part of the proof of [12, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2] can be still carried out analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  It is worth noting that they rely on the Iohara–Kohno realization [11], which provides  level 1 representations of the double Yangian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The correspondence similar to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7), which employs the universal envelop-  ing algebra over C, can be also established by taking the classical limit DY(gl2)/hDY(gl2)  of the double Yangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Indeed, by extracting the coefficients of the matrix entries in its  defining relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2), one observes that the resulting top degree terms with  respect to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6) coincide with the terms which contain the lowest power of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  From now on, we shall assume that the double Yangian for gl2 is h-adically completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Using the series (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4) one obtains its Drinfeld generators [2] as follows:  k±  1 (u) = t±  11(u),  k±  2 (u) = t±  22(u) − t±  21(u)t±  11(u)−1t±  12(u),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8)  X+(u) = t+  11(u − hC/4)−1t+  12(u − hC/4) − t−  11(u + hC/4)−1t−  12(u + hC/4),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='9)  X−(u) = t+  21(u + hC/4)t+  11(u + hC/4)−1 − t−  21(u − hC/4)t−  11(u − hC/4)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10)  The commutation relations for these generators can be found in [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  The double Yangian DY(gl2) can be decomposed into two subalgebras: the double  Yangian DY(sl2), which is generated by the central element C1 := C and all coefficients  of the power series  E(u) = 1  hX+(u + h/2),  F(u) = 1  hX−(u + h/2),  H±(u) = k±  2 (u + h/2)k±  1 (u + h/2)−1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='11)  3  and the Heisenberg subalgebra H, which is generated by the coefficients of the series  K±(u) = k±  1 (u − h/2)k±  2 (u + h/2)  and the central element C2 = −2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The generators of the Heisenberg subalgebra commute  with all elements of DY(sl2) and satisfy the relations from [11, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2],  �  K±(u), K±(v)  �  = 0,  u − v − h + hC2/4  u − v + h + hC2/4K+(u)K−(v) = K−(v)K+(u)u − v − h − hC2/4  u − v + h − hC2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  The generator series of H are of the form K±(u) = 1 ∓ hκ±(u), where  κ+(u) =  �  r⩾0  κ(r)u−r−1  and  κ−(u) =  �  r⩾1  κ(−r)ur−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Consider the Heisenberg Lie subalgebra �h = h ⊗ C[t±1] ⊗ CK2 ⊂ �gl2, where K2 = −2K  is the central element and h = CI is the one-dimensional commutative subalgebra of gl2  spanned by the identity matrix I = e11 + e22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5), its Lie brackets are given by  [I(r), I(s)] = rδr+s0K2,  where I(r) = I ⊗tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Note that the restriction of the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7) produces the isomorphism  U(�h) ⊗ C[[h]] → grH,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='12)  where grH is the corresponding graded algebra of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' It is given by I(r) �→ ¯κ(r) for all  r ∈ Z and K2 �→ ¯C2, where ¯κ(r) and ¯C2 stand for the images of Heisenberg subalgebra  generators κ(r) and C2 in the corresponding component of grH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Iohara–Kohno realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We follow [11] to introduce a certain realization of  level 1 modules for the double Yangian DY(gl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Let s = Ce11 ⊕ Ce22 be the Cartan  subalgebra of gl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Denote by εj, j = 1, 2, the linear maps s → C given by εj(ekk) = δjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  The dual s∗ is equipped with the standard bilinear form (·, ·) determined by (εj, εk) = δjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Let Q = Zα with α = ε1 − ε2 be the root lattice of the simple Lie algebra sl2 and  λi = Λi − Λ0 the classical part of the i-th fundamental affine weight Λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Consider the Lie algebra �s generated by the central element C and the elements aj(r),  where j = 1, 2 and r ∈ Z, r ̸= 0, subject to the relations  [aj(r), ak(s)] = rδjkδr+s0C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='13)  Let �s− be its (commutative) subalgebra generated by all aj(r) with j = 1, 2 and r ∈ Z<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Denote by C [Q] the group algebra of the root lattice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' For any i = 0, 1 and s ∈ C set  Fis = U(�s−)[[h]] ⊗ C[Q][[h]]eλi+ s  2(ε1+ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14)  The tensor product in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14) is understood in the h-adically completed sense, so that the  C[[h]]-module Fis is topologically free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Define the action of C, aj(r), ∂εj and eεj, where  j = 1, 2 and r ̸= 0, on Fis so that for any f ⊗ eµ ∈ Fis we have  C · f ⊗ eµ = f ⊗ eµ,  aj(r) · f ⊗ eµ =  �  aj(r)f ⊗ eµ,  if r < 0,  [aj(r), f] ⊗ eµ,  if r > 0,  ∂εj · f ⊗ eµ = (εj, µ)f ⊗ eµ,  eεj · f ⊗ eµ = f ⊗ eεj+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  4  Let us write X±α(u) = h−1X±(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Introduce the following operator series on Fis:  E−(u) = exp  ��  r>0  �  −a1(−r)  r  �  u − 3  4h  �r  + a2(−r)  r  �  u + h  4  �r��  ,  E+(u) = exp  ��  r>0  �  a1(r) − a2(r)  r  �  u + h  4  �−r��  ,  E0(u) = eα  �  u + h  4  �∂α  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Finally, we recall the Iohara–Kohno realization [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The following assignments define a structure of DY(gl2)-module on Fis:  Xα(u) �→ E−(u)E+(u)E0(u),  X−α(u) �→ E−(u + h/2)−1E+(u − h/2)−1E0(u − h/2)−1,  k+  j (u) �→ exp  �  −  �  r>0  aj(r)  r  ��  u + h  2  �−r  −  �  u − h  2  �−r�� �  1 −  h  u + h  2  �∂εj  , j = 1, 2,  k−  j (u) �→ exp  ��  r>0  �δ1j a2(−r)  r  ((u + h)r − ur) + δ2j a1(−r)  r  (ur − (u − h)r)  ��  , j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Semi-infinite construction  Throughout the rest of the paper, we assume that s = 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14) and we write Fi = Fi0  for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Furthermore, from now on, we assume that the double Yangian DY(sl2) and  the Heisenberg subalgebra H, as given in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1, are h-adically completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Our  goal is to construct the topological bases for certain DY(gl2)-modules in parallel with the  semi-infinite construction [6] for the affine Kac–Moody Lie algebra �sl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Normalized Iohara–Kohno realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We now introduce another structure of  level 1 DY(gl2)-module over Fi by modifying the original formulae from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The following assignments define a structure of DY(gl2)-module on Fi:  Xα(u) �→ E−(0)−1E−(u)E+(u)E0(u)u∂α/2(u + h)∂α/2  (u + h/4)∂α  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  X−α(u) �→ E−(0)E−(u + h/2)−1E+(u − h/2)−1E0(u − h/2)−1  (u − h/4)∂α  (u − h/2)∂α/2(u + h/2)∂α/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  k+  1 (u) �→ exp  �  −  �  r>0  a1(r)  r  ��  u + h  2  �−r  −  �  u − h  2  �−r�� � u + h/4  u + 5h/4  �∂α/2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  k+  2 (u) �→ exp  �  −  �  r>0  a2(r)  r  ��  u + h  2  �−r  −  �  u − h  2  �−r�� � u + h/4  u − 3h/4  �∂α/2  and the action of k−  1 (u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' k−  2 (u) is given by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As with the original Iohara–Kohno realization from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3, one verifies by  a direct calculation that the given operators satisfy the defining relations for double  Yangian, which are established in [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As a demonstration, we shall prove that  the operators satisfy the level 1 defining relation  [Xα(u), X−α(v)] = 1  h  �  δ(u − v − h/2)k+  2 (u − h/4)k+  1 (u − h/4)−1  5  −δ(u − v + h/2)k−  2 (v − h/4)k−  1 (v − h/4)−1�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1)  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1], where δ(u − v) = �  r∈Z u−r−1vr is the δ-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Throughout the  proof, we denote by X′  ±α(u) and k  ′±  j (u) the corresponding operators from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3  in order to distinguish them from those given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  First, we rewrite the left hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1) as  [Xα(u), X−α(v)] =  �  X′  α(u), X′  −α(v)  �  f(u, v),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2)  where  f(u, v) = u∂α/2(u + h)∂α/2  (u + h/4)∂α  (v − h/4)∂α  (v − h/2)∂α/2(v + h/2)∂α/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Next, consider the right hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Using the properties of δ-function, we find  δ(u − v + h/2) = f(u, u + h/2)δ(u − v + h/2) = f(u, v)δ(u − v + h/2)  since f(u, u + h/2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Furthermore, we have k  ′−  j (u) = k−  j (u) for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Therefore, the  second summand in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1) can be written as  − δ(u − v + h/2)k−  2 (v − h/4)k−  1 (v − h/4)−1  = − f(u, v)δ(u − v + h/2)k  ′−  2 (v − h/4)k  ′−  1 (v − h/4)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3)  Finally, we consider the first summand on the right hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Observe that the  operators k+  j (u) and k  ′+  j (u) are connected by the identities  k+  1 (u) = k  ′+  1 (u)  �u + h/2  u − h/2  �∂ε1 � u + h/4  u + 5h/4  �∂α/2  ,  k+  2 (u) = k  ′+  2 (u)  �u + h/2  u − h/2  �∂ε2 � u + h/4  u − 3h/4  �∂α/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Thus, we have the equality  δ(u − v − h/2)k+  2 (u − h/4)k+  1 (u − h/4)−1  = δ(u − v − h/2)  �u − 3h/4  u + h/4  �∂α �u + h  u − h  �∂α/2  k  ′+  2 (u − h/4)k  ′+  1 (u − h/4)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4)  However, the properties of δ-function imply  δ(u − v − h/2)  �u − 3h/4  u + h/4  �∂α �u + h  u − h  �∂α/2  = f(u, u − h/2)δ(u − v − h/2) = f(u, v)δ(u − v − h/2),  so we conclude by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4) that the first summand on the right hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1) equals  f(u, v)δ(u − v − h/2)k  ′+  2 (u − h/4)k  ′+  1 (u − h/4)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5)  As the operators from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1), the commutation relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1) now  follows by comparing the expressions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  The formula for the action of Xα(u) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1 implies  Xα(u)1 ⊗ 1 ∈ 1 ⊗ eα + uF0[[u]]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6)  on F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' On the other hand, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6) does not hold for the action of Xα(u) given by Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As this property is required for the semi-infinite construction in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5, in the  6  rest of the paper we only consider the DY(gl2)-action from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1 and, furthermore,  we write X±α(u) and k±  j (u) for the corresponding vertex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Commutative operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In this subsection, we associate with the operators  Xα(u) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1 certain commutative operators X(u) which satisfy an h-adic  version of the integrability relations which go back to Lepowsky and Primc [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As with  the construction of commutative operators of Ding and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Feigin [1], which are associ-  ated with the Frenkel–Jing realization [4] of certain Uq(�sl2)-modules, we modify the term  E+(u) consisting of annihilation operators aj(r) with r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' However, in contrast with  the quantum affine algebra case, the term E0(u) needs to be renormalized as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Consider the operator series on Fi, i = 0, 1, defined by  X(u) = Xα(u)E  +(u)E  0(u),  where  E  0(u) = (1 − h/u)∂α/2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7)  the action of Xα(u) is given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1 and the term E  +(u) is  E  +(u) = exp  ��  r>0  �  a1(r)  r  �  u − 7  4h  �−r  + a2(r)  r  �  u + 1  4h  �−r��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The following identities hold for operators on Fi, i = 0, 1:  X(u)X(v) = X(v)X(u),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8)  X(u)X(u − h) = X(u)X(u + h) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='9)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' By using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='13) one easily verifies the following relations:  E+(u)E−(v) = (u − v)(u − v + h)  (u + h/4)2  E−(v)E+(u),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10)  E0(u)E0(v) = (u + h/4)2  (v + h/4)2 E0(v)E0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='11)  The commutativity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8) can be verified using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='11) and the identities  E  +(u)E−(v) = u − v − h  u − v  u + h/4  u − 7h/4E−(v)E  +(u),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='12)  E  0(u)E0(v) = (1 − h/u) E0(v)E  0(u),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='13)  which also follow from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As for the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='9), it is sufficient to observe that,  due to relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='11), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='13), we have the decomposition  X(u)X(v) = f(u, v)X(u, v),  where  f(u, v) = (u − v − h)(u − v + h),  X(u, v) = E−(0)−2E−(u)E−(v)E+(u)E  +(u)E+(v)E  +(v)e2α(u2 − h2)∂α/2 (v2 − h2)∂α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  By applying X(u, v) to an arbitrary element of Fi we get only finitely many negative  powers of the variables u and v modulo hn for any n ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Hence the formal limit v → u±h  of X(u, v) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Therefore, the limit v → u ± h of the product X(u)X(v) is  well-defined as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Furthermore, since f(u, u ± h) = 0, it equals zero, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Principal submodule W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Let us write the operator series Xα(u) and X(u) as  Xα(u) =  �  r∈Z  x(r)u−r−1  and  X(u) =  �  r∈Z  x(r)u−r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Let 1 = 1 ⊗ 1 ∈ F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Motivated by the notion of principal subspaces of affine Lie algebra  modules from [6,9], we define the principal submodule W0 of F0 as the h-adic completion  of the C[[h]]-module which is spanned by all  x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1,  where  n ⩾ 0  and  r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , rn ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14)  In this subsection, we use the monomials (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14) to construct a topological basis for W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Note that W0 coincides with the h-adically completed C[[h]]-span of all  x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1,  where  n ⩾ 0  and  r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , rn ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='15)  Indeed, the terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='15) are the coefficients of u−rn−1  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' u−r1−1  1  in X(un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' X(u1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  However, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='13), along with E  +(u) 1 = E  0(u) 1 = 1, we find  X(un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' X(u1) 1 = F · Xα(un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(u1) 1 for F =  �  1⩽r<s⩽n  �  1 −  h  us − ur  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='16)  As F is invertible in C((un)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' ((u1))[[h]] and both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='16) belong to the h-adic  completion of F0((un)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' ((u1)), we conclude that each element in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14) can be expressed  in terms of elements of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='15) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Moreover, by rewriting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='16) as  h−1 �  X(un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' X(u1) 1 −Xα(un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(u1) 1  �  = h−1(F − 1)Xα(un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(u1) 1 = h−1(1 − F −1)X(un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' X(u1) 1,  we conclude that the expressions of the form h−1 (x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1 −x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1) can  be also written as C[[h]]-linear combinations of elements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14) or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Observe that  such linear combinations can be infinite, but they are well-defined as they are finite  modulo hm for all m ⩾ 1 and the C[[h]]-module F0 is h-adically complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  In the following lemma, we consider the monomials of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In comparison  with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14), they are easier to handle since their factors x(r) commute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Let  BW0 be the set of all elements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='15) with n ⩾ 0 such that their indices satisfy  r1 ⩽ −1  and  rj+1 ⩽ rj − 2  for all  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='17)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The h-adic completion of the C[[h]]-span of BW0 coincides with W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' First, note that X(u) 1 possesses only nonnegative powers of the variable u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  we have x(r) 1 = 0 for r ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Thus, by commutativity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8), the terms  x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1  such that  r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , rn ∈ Z<0  and  rn ⩽ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' ⩽ r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='18)  span an h-adically dense C[[h]]-submodule of W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Next, we use the h-adic integrability  relation to establish the difference two condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='17) for the terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Although  this is just an adjustment of the well-known argument, originating from the (integrable)  representation theory of �sl2, to the h-adic setting, we give some details for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Extracting the coefficients of u−2r−3 and u−2r−2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='9) and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8) we find  x(r)x(r + 1) = −  �  l⩾1  x(r − l)x(r + l + 1)  mod h,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='19)  8  x(r)x(r) = −2  �  l⩾1  x(r − l)x(r + l)  mod h,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='20)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Suppose that we have two adjacent operators x(rj+1)x(rj) in the element  x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1 of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='18) which do not satisfy the difference two condition rj+1 ⩽ rj − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Then, they can be replaced modulo h by a sum of those which satisfy this condition  using the identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='19) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='20)) if rj+1 = rj − 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' rj+1 = rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='9),  the difference of the left and the right hand-side in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='19) and in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='20) belongs to hW0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  In other words, it is a C[[h]]-linear combination of monomials as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='18) such that its  coefficients belong to hC[[h]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Thus, for any m ⩾ 1, by repeating such procedure sufficient  number of times one can express the original element x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1 as a C[[h]]-linear  combination of elements which satisfy the difference two condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='17) modulo hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As  W0 is h-adically complete, this implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  In the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7) of the series X(u), we multiplied Xα(u) by the term E  +(u), thus  changing the classical limit of the resulting operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Therefore, we can not instantly  establish the linear independence of the spanning set from the previous lemma via classical  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Instead, we now turn to the monomials of operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Let BW0 be the set of  all elements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14) with n ⩾ 0 which satisfy the conditions imposed by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The set BW0 is linearly independent over C[[h]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The classical limit h → 0 of the operators X±α(u) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1 with i = 0  produces the action of �sl2 on its level 1 integrable highest weight module L(Λ0) of the  highest weight Λ0, as given by the Frenkel–Kac–Segal realization [5,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Thus, the classical  limit of BW0 produces the well-known quasi-particle basis for the principal subspace of  L(Λ0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' see [6,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Hence, in particular, BW0 is linearly independent over C[[h]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  Finally, we combine Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5 to obtain a basis of W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The set BW0 forms a topological basis of W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5, it is sufficient to check that the C[[h]]-span of BW0 is h-adically  dense in W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Choose any w ∈ W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4, it can be expressed as w = �  j ajbj,  where aj ∈ C[[h]] and bj ∈ BW0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Next, by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3, we have w = �  j ajbj mod h,  where bj ∈ BW0 denote the elements corresponding to bj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=', more precisely, if bj =  x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1, then we set bj = x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In addition, the remark implies that  h−1(w − �  j ajbj) is again a C[[h]]-linear combination of the elements of BW0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Hence, for  any m ⩾ 1, we can repeat this procedure until we express w as a C[[h]]-linear combination  of the elements of BW0 modulo hm, so the theorem now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' DY(sl2)-modules Li(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Consider the set  BL0(sl2) =  �  ekαb : k ∈ Z, b ∈ BW0  �  ⊂ F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Let L0(sl2) be the h-adically completed C[[h]]-span of BL0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In this subsection, we show  that L0(sl2) is closed with respect to the DY(sl2)-action from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' For any integer m we have  E+(u)mL0(sl2) ⊂ L0(sl2)[[u−1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='21)  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Suppose m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' It is sufficient to show that E+(u)ekαb belongs to L0(sl2)[[u−1]]  for all ekαb ∈ BL0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Using the formulae from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1 and E+(u) 1 = 1 we find  E+(u)ekαXα(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(v1) 1 = F · ekαXα(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(v1) 1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='22)  where, due to relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10), the factor F ∈ C[u−1][[vn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , v1, h]] is given by  F =  n  �  r=1  �  1 − vr  u  � �  1 −  vr  u + h  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  For any integers s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , sn, by extracting the coefficients of vs1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' vsn  n from the right hand-  side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='22) one obtains a power series in the variable u−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Clearly, its coefficients are  C[[h]]-linear combinations of the elements of the form ekαx(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Moreover, by  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6, every x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1 can be expressed in terms of elements of BW0, so  that every ekαx(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' x(r1) 1 belongs to L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' On the other hand, by extracting the  coefficients of vs1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' vsn  n from the left hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='22) for a suitable choice of s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , sn,  we obtain E+(u)ekαb with ekαb ∈ BL0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Thus, we conclude that all E+(u)ekαb such that  ekαb ∈ BL0(sl2) belong to L0(sl2)[[u−1]], as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  If m = −1, we can use E+(u)−1 1 = 1 and verify the inclusion in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='21) analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Finally, if m is an arbitrary integer, one easily proves by induction over |m| that we have  E+(u1)±1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' E+(u|m|)±1L0(sl2) ⊂ L0(sl2)[[u−1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , u−1  |m|]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='23)  Hence the statement of the lemma follows by setting u1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' = u|m| = u in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7 relies on the fact that the coefficients of E+(u) are C[[h]]-linear  combinations of monomials of annihilation operators aj(r) with r > 0, which implies  E+(u)± 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As k+  j (u)±1 1 = 1 for j = 1, 2, one can prove the next lemma analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' For any integer m and j = 1, 2 we have  k+  j (u)mL0(sl2) ⊂ L0(sl2)[[u−1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  By combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8 and the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='11) we get  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' H+(u)L0(sl2) ⊂ L0(sl2)[[u−1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Let us consider the remaining operators from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1 whose action employs the  creation operators aj(r) with r < 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' First, observe that  Xα(u)ekα = u2k(u+h)2kekαXα(u)  implies  Xα(u)L0(sl2) ⊂ L0(sl2)[[u±1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='24)  Next, write E−(u) = E−(0)−1E−(u), so that, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1, we have  Xα(u) = E−(u)E+(u)eαu∂α/2(u + h)∂α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='25)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' For any integer m we have  E−(u)mL0(sl2) ⊂ L0(sl2)[[u]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='26)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Let m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We can use the formulae from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10) to prove  E−(u)ekαXα(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(v1) 1 = F · ekαXα(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(v1)E−(u) 1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='27)  where the factor F ∈ C[v−1  n , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , v−1  1 ][[u, h]] is given by  F =  n  �  r=1  �  1 − u  vr  �−1 �  1 −  u  vr + h  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  10  Since E+(u)eαu∂α/2(u + h)∂α/2 1 = eα 1, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='25) the rightmost term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='27) equals  E−(u) 1 = e−αE−(u)E+(u)eαu∂α/2(u + h)∂α/2 1 = e−αXα(u) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Therefore, the right hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='27) is equal to  F · ekαXα(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(v1)e−αXα(u) 1 = G · e(k−1)αXα(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(v1)Xα(u) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='28)  The right hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='28) was found by moving the operator e−α all the way to the  left, which produced the term G ∈ C[v−1  n , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , v−1  1 ][[u, h]] given by  G = F ·  n  �  r=1  v−1  r  (vr + h)−1 =  n  �  r=1  (vr − u)−1 (vr − u + h)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Thus, we proved that the left hand-side in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='27) coincides with the right hand-side in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' However, it is clear that all coefficients of the right hand-side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='28) belong  to L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Hence, the inclusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='26) for m = 1 follows as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Furthermore, the general case m > 1 can be again verified by induction over m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Let m = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Note that E−(u) can be written as E−(u) = 1 − e−(u), where e−(u)  belongs to uU(�s−)[[h, u]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Hence E−(u) is invertible and its inverse takes the form  E−(u)−1 =  �  1 − e−(u)  �−1 =  �  l⩾0  e−(u)l =  �  l⩾0  �  1 − E−(u)  �l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='29)  Each summand in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='29) is given in terms of nonnegative powers of E−(u), which satisfy  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='26), so we conclude that E−(u)−1L0(sl2) ⊂ L0(sl2)[[u]], as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Finally, as before,  this is generalized to any negative integer m by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  By using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='11) and the action of k−  j (u) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1 one easily verifies the identity  H−(u − h/4) = E−(u)E−(u + h)−1 for operators on F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10 we have  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' H−(u)L0(sl2) ⊂ L0(sl2)[[u]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Let us turn our attention to the operator X−α(u), which takes the form  X−α(u) = E−(u + h/2)−1E+(u − h/2)−1e−α(u − h/2)−∂α/2(u + h/2)−∂α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='30)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' X−α(u)L0(sl2) ⊂ L0(sl2)[[u±1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The lemma follows by an argument which goes in parallel with the proof of Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' More specifically, it relies on the identity  X−α(u)ekαXα(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(v1) 1 = F · E−(u + h/2)−1e(k−1)αXα(vn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(v1) 1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='31)  where the series F ∈ C[u−1][[vn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , v1, h]] is given by  F = (u − h/2)−k (u + h/2)−k  n  �  r=1  (u − vr − h/2)−1 (u − vr + h/2)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  In addition, it employs Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10, which implies that all coefficients of the right hand-  side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='31) belong to L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As for the equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='31), it is verified using the  expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='30) for the operators X±α(u) and the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  Finally, Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='11, the inclusion in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='24) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='12 imply  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The C[[h]]-module L0(sl2) is a module for the double Yangian DY(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Consider the C[[h]]-module L1(sl2) =  �  eλ1v : v ∈ L0(sl2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The next lemma is verified  by a direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  11  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The following commutation relations hold:  Xα(u)eλ1 = u1/2(u + h)1/2eλ1 Xα(u),  X−α(u)eλ1 = (u − h/2)−1/2(u + h/2)−1/2eλ1X−α(u),  H+(u)eλ1 = (u + 7h/4)1/2(u − h/4)−1/2eλ1 H+(u),  H−(u)eλ1 = eλ1 H−(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  By using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='14, one easily checks that L1(sl2) is closed under the action of the  double Yangian DY(sl2) as well, so that we have  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The C[[h]]-module L1(sl2) is a module for the double Yangian DY(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Semi-infinite monomial bases for Li(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In this subsection, we construct topo-  logical bases for L0(sl2) and L1(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Let us start with L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Introduce the operator  �  X(u) =  �  r∈Z  �x(r)u−r−1 = Xα(u) (1 + h/u)−∂α/2 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='32)  where the action of Xα(u) is given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' It satisfies the identity  �  X(un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' �X(u1) 1 = F · Xα(un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Xα(u1) 1  for  F =  �  r=2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=',n  (1 + h/ur)1−r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='33)  Let �BW0 be the set of all elements �x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' �x(r1) 1, where n ⩾ 0 and the integers r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , rn  satisfy the conditions given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The set �BW0 forms a topological basis of W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The given set is linearly independent as its classical limit coincides with the classical  limit of the basis BW0 from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The fact that the C[[h]]-span of �BW0 is h-adically  dense in W0 is established by arguing as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' However, while the  argument therein relies on the identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='16), here one uses (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='33) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  For any integer m let �BL0(sl2),m be the set of all elements emαb such that b ∈ �BW0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Clearly, the union ∪m∈Z �BL0(sl2),m spans an h-adically dense C[[h]]-submodule of L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  We now employ the Feigin–Stoyanovsky-type construction [6] to reduce it to a linearly  independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='32) we have  �x(−1) 1 = eα 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='34)  Furthermore, by extracting the coefficients in the relation �  X(u)e−α = e−α �  X(u)u−2, which  can be easily verified by a direct calculation, we find  �x(r)e−α = e−α �x(r − 2)  for all  r ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='35)  Let L0(sl2)m be the h-adically completed C[[h]]-span of �BL0(sl2),m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Observe that the union  ∪m∈ZL0(sl2)m coincides with L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Also, we have L0(sl2)m ⊂ L0(sl2)m−1 for all integers  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Indeed, this is an immediate consequence of the inclusion �BL0(sl2),m ⊂ �BL0(sl2),m−1,  which can be proved by using the identities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='34) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='35) as follows:  emα �x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' �x(r1) 1 = emα �x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' �x(r1)e−αeα 1 = emα �x(rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' �x(r1)e−α �x(−1) 1  = e(m−1)α �x(rn − 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' �x(r1 − 2)�x(−1) 1 ∈ �BL0(sl2),m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Finally, we obtain the topological basis for L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  12  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The direct limit  �BL0(sl2) = lim  −→  �BL0(sl2),m  forms a topological basis of L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' It is clear that the elements of �BL0(sl2) span the h-adically dense C[[h]]-submodule  of L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' On the other hand, their classical limits produce the semi-infinite basis for the  integrable highest weight �sl2-module L(Λ0) of the highest weight Λ0, established by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Feigin and Stoyanovsky [6], so that they are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  Consider the DY(sl2)-module L1(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' For any integer m let �BL1(sl2),m be the set of  all elements eλ1+mαb with b ∈ �BW0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The union ∪m∈Z �BL1(sl2),m spans an h-adically dense  C[[h]]-submodule of L1(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The semi-infinite basis for L1(sl2) is established in parallel  with the case of L0(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Naturally, in this case, its linear independence follows from  the observation that its classical limit produces the semi-infinite basis for the integrable  highest weight �sl2-module L(Λ1) of the highest weight Λ1 from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Hence, we have  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The direct limit  �BL1(sl2) = lim  −→  �BL1(sl2),m  forms a topological basis of L1(sl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In this remark, we discuss an interpretation of the bases �BLi(sl2) in terms  of semi-infinite monomials [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' First, let us write vi,m = eλi+mα 1 for m ∈ Z and i = 0, 1  so that, in particular, we have vi,0 = eλi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We can express the elements vi,0 = eλi 1 as  vi,0 = eλi 1 = eλi−αeα 1 = eλi−α�x(−1) 1 = �x(1 − i)eλi−α 1 = �x(1 − i)vi,−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Next, by a similar calculation, which starts with vi,−1, we get  vi,−1 = eλi−α 1 = eλi−2αeα 1 = eλi−2α�x(−1) 1 = �x(3 − i)eλi−2α 1 = �x(3 − i)vi,−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Combining the above equalities we find that vi,0 = �x(1 − i)�x(3 − i)vi,−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' By repeating  such calculations we find  vi,0 = �x(1 − i)�x(3 − i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' �x(2m − 1 − i)vi,−m  for any  m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='36)  Note that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='36) the direct limits in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='17 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='18 correspond with taking  the limit m → ∞ of the elements of the form  emαbvi,0 = emαb�x(1 − i)�x(3 − i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' �x(2m − 1 − i)vi,−m,  where  b ∈ �BW0  Hence, arguing as above to move the term emα to the right, we see that the elements of  the bases �BLi(sl2) can be represented as the semi-infinite monomials  �x(r1)�x(r2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' vi,−∞  such that  (1) Their degrees r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' ∈ Z satisfy the difference two condition  rj ⩽ rj+1 − 2  for all  j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' ,  (2) For each monomial there exists an index n such that all degrees rn, rn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' are  consecutive odd (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' even) integers if i = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' i = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Semi-infinite monomial bases for Li(gl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Consider the DY(gl2)-modules  L0(gl2) = DY(gl2) · 1  and  L1(gl2) = DY(gl2) · eλ1,  where, as before, the action of the double Yangian is given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Recall that  the Heisenberg subalgebra H commutes with all elements of DY(sl2), so that we have  L0(gl2) = H·DY(sl2)·1 = DY(sl2)·H·1  and  L1(gl2) = H·DY(sl2)·eλ1 = DY(sl2)·H·eλ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Consider the subalgebra H− ⊂ H generated by 1 and all elements κ(−r) with r ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  The algebra H− possesses the Poincar´e–Birkhoff–Witt basis  BH− =  �  κ(−rn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' κ(−r1) : n ⩾ 0, rn ⩾ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' ⩾ r1 ⩾ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='37)  Indeed, the elements of BH− span H− as all κ(−r) with r ⩾ 1 commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' On the other hand,  their linear independence is established by the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='12) and the Poincar´e–Birkhoff–  Witt theorem for U(�h−), where �h− stands for the Lie subalgebra of �h generated by all  I(−r) = I ⊗ t−r with r ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' For i = 0, 1 introduce the sets  �BLi(gl2) =  �  hb : h ∈ BH−, b ∈ �BLi(sl2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Let i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The set �BLi(gl2) forms a topological basis for Li(gl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='17 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='18, along with the discussion preceding this theorem, imply  that the set �BLi(gl2) spans an h-adically dense C[[h]]-submodule of Li(gl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' On the other  hand, its linear independence follows from the observation that the classical limit of  Li(gl2) is equal to U(�h−) ⊗ L(Λi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Modules for the quantum vertex algebra Vc(gl2)  In this section, we study the underlying (quantum) vertex algebraic framework of the  semi-infinite construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In particular, we obtain examples of irreducible modules for  the quantized universal affine vertex algebra of gl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Quantum affine vertex algebra Vc(gl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The dual Yangian Y+(gl2) for gl2 is  defined as the associative algebra over the ring C[[h]] generated by the elements t(−r)  ij  ,  where i, j = 1, 2 and r = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Its defining relations are given by  R12(u − v)T −  13(u)T −  23(v) = T −  23(v)T −  13(u)R12(u − v),  where the notation is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We use the same symbols for the generators  of the dual Yangian and the corresponding generators of the double Yangian as Y+(gl2)  can be naturally regarded as a subalgebra of DY(gl2) due to the Poincar´e–Birkhoff–Witt  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Moreover, for any c ∈ C the h-adic completion of the dual Yangian is naturally  equipped with the DY(gl2)-action so that the central element C acts as the scalar multipli-  cation by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We shall denote this DY(gl2)-module by Vc(gl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Furthermore, Vc(gl2) can be  equipped with the quantum vertex algebra structure via Etingof–Kazhdan’s construction  as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' see [3, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' There exists a unique quantum vertex algebra structure on Vc(gl2) so  that the unit 1 is the vacuum vector and the vertex operator map satisfies  Y (T −  1n(u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T −  n−1n(un−1), z) = T −  1n(z + u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T −  n−1n(z + un−1)  × T +  n−1n(z + un−1 − hc/2)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T +  1n(z + u1 − hc/2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  14  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In comparison with the original result [3, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3], we use a different normaliza-  tion of the Yang R-matrix, which governs the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2) between the operators T ±(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  However, the theorem can be again proved by the analogous arguments, which can be  also recovered from the proofs of [9, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8] and [12, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The map Y (·, z) can be expressed modulo h as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Write  t±(u) = ±h−1(I − T ±(u))  and  t(u) = t−(u) + t+(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  One can prove by induction over n the identity  Y (t−  1n(u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' t−  n−1n(un−1), z) = :t1n(u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' tn−1n(un−1):  mod h,  where the normal-ordered product of operators is defined in a usual way:  :t(u1)t(u2): = t−(u1)t(u2) + t(u2)t+(u1)  and  :t(u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' t(ur): = :t(u1) :t(u2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' t(ur)::.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Thus, it is easy to see that the classical limit h → 0 of the quantum vertex algebra Vc(gl2)  coincides with the universal affine vertex algebra V c(gl2) which goes back to the papers  of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Frenkel and Zhu [7] and Lian [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' recall Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Let c2 = −2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Denote by Vc2(h) the level c2 module for the Heisenberg subalgebra  H ⊂ DY(gl2) defined over the h-adic completion of H−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Clearly, we have Vc2(h) ⊂ Vc(gl2)  and the topological basis of Vc2(h) is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The generator series K±(u) of H  can be expressed as the quantum determinants of the matrices T ±(u) from the RTT-  realization of DY(gl2), so that we have K±(u) = qdetT ±(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' see [10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Using  this observation, one obtains the following simple consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The C[[h]]-module Vc2(h) is a quantum vertex subalgebra of Vc(gl2) for  c = −c2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Its vacuum vector is 1 ∈ Vc(gl2) and its vertex operator map satisfies  Y (K−(u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' K−(un), z) = K−(z + u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' K−(z + un)  × K+(z + un + hc2/4)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' K+(z + u1 + hc2/4)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The results of this subsection are presented for gl2 in order to better fit the  setting of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' However, their generalization to glN with N > 2 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  This is also true for the notion of restricted module and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6 which we give below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Restricted modules for the double Yangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' A DY(gl2)-module V is said to  be restricted if it is a topologically free C[[h]]-module such that for any v ∈ V and n ⩾ 1  the expression T +(u)v possesses only finitely many negative powers of u modulo hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The  last requirement can be equivalently expressed as  T +(u)v ∈ End C2 ⊗ V [u−1]h  for all  v ∈ V,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1)  where V [u−1]h stands for the h-adic completion of the C[[h]]-module V [u−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Extending  this notation, we shall also write V ((u))h for the h-adic completion of V ((u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The DY(gl2)-modules Fis and Fi established by Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1  respectively are restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' It is clear that the underlying C[[h]]-module structure is topologically free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' More-  over, by examining the explicit formulae in the aforementioned theorems we find  X±(u) ∈ Hom(F, F((u))h),  k−  j (u) ∈ Hom(F, F[[u]]),  k+  j (u) ∈ Hom(F, F[u−1]h),  15  where F = Fis, Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' We now employ the identities in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10) to prove the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  First, we note that by the first equality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8) we have t+  11(u) ∈ Hom(F, F[u−1]h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Next,  from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='10) we obtain  t+  21(u) = X−(u − hC/4)t+  11(u) + t−  21(u − hC/2)t−  11(u − hC/2)−1t+  11(u),  which implies t+  21(u) ∈ Hom(F, F[u−1]h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As for t+  12(u), we rewrite the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='9) as  t+  11(u)−1t+  12(u) = X+(u + hC/4) + t−  11(u + hC/2)−1t−  12(u + hC/2)  to conclude that the product t+  11(u)−1t+  12(u) belongs to Hom(F, F[u−1]h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Multiplying this  product by t+  11(u) ∈ Hom(F, F[u−1]h), we obtain t+  12(u) ∈ Hom(F, F[u−1]h), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Finally, the second equality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8) implies  t+  22(u) = k+  2 (u) + t+  21(u)t+  11(u)−1t+  12(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  As t+  11(u)−1 ∈ Hom(F, F[u−1]h), we conclude that the remaining matrix entry t+  22(u) of  T +(u) belongs to Hom(F, F[u−1]h), thus verifying the requirement imposed by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  The next theorem was our main motivation for introducing the notion of restricted  module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Its proof relies on the RTT-realization of the double Yangian and goes in parallel  with the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Let V be a restricted DY(gl2)-module of level c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Then there exists a  unique structure of Vc(gl2)-module over V such that for all n ⩾ 1 we have  YV (T −  1n(u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T −  n−1n(un−1), z) = T −  1n(z + u1)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T −  n−1n(z + un−1)V  × T +  n−1n(z + un−1 − hc/2)−1  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T +  1n(z + u1 − hc/2)−1  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  The next corollary is an immediate consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' There exists a unique structure of V1(gl2)-module over Li = Li(gl2) with  i = 0, 1 such that for all n ⩾ 1 we have  YLi(T −  1n(u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T −  n−1n(un−1), z) = T −  1n(z + u1)Li .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T −  n−1n(z + un−1)Li  × T +  n−1n(z + un−1 − h/2)−1  Li .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T +  1n(z + u1 − h/2)−1  Li .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Consider the standard structure of bosonic Fock �h-module M(−2) of level −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' In  particular, recall that M(−2) is an irreducible �h-module which, as a vector space, coincides  with U(�h−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The tensor product Li = M(−2) ⊗ L(Λi) for i = 0, 1, where L(Λi) is the  integrable highest weight �sl2-module of highest weight Λi, is naturally equipped with the  structure of irreducible module for �gl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Thus, Li possesses the structure of irreducible  module for the universal affine vertex algebra V 1(gl2) of level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' On the other hand, the  classical limit of the V1(gl2)-module Li(gl2) coincides with the V 1(gl2)-module Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  In the next corollary, we use the notion of irreducible module in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  The topologically free C[[h]]-module V is said to be irreducible with respect to the action  of associative algebra (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' quantum vertex algebra) if it does not possess any nontrivial  topologically free C[[h]]-submodule W which is invariant under the corresponding action  of associative algebra (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' quantum vertex algebra) and satisfies that hnv ∈ W for  n ⩾ 1 and v ∈ V implies v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The C[[h]]-modules Li(gl2) are irreducible both as modules for the quan-  tum vertex algebra V1(gl2) and as level 1 modules for the double Yangian DY(gl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  16  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' The irreducibility with respect to the action of V1(gl2) follows by the discussion  preceding the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As for the action of the double Yangian, if the C[[h]]-submodule  W ⊂ Li(gl2) is DY(gl2)-invariant, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7 we have  YLi(T −  1n(u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T −  n−1n(un−1), z)W ⊂ (End CN)⊗(n−1) ⊗ W[[z±1, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' , un−1]]  for all n ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' As the coefficients of matrix entries of T −  1n(u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' T −  n−1n(un−1) with n ⩾ 1  span an h-adically dense C[[h]]-submodule of V1(gl2), it is clear that W is V1(gl2)-invariant  as well, so that the second assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Ding–Feigin’s construction of semi-infinite monomial bases [1] in the case of  quantum affine algebra Uq(�sl2) suggests that the suitable analogues of Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='7 and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='8 can be established for Etingof–Kazhdan’s quantum vertex algebra [3] associated with  the trigonometric R-matrix of type A(1)  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' On the other hand, by using different approach,  the h-adic quantum vertex algebra structure over certain wide class of irreducible modules  for untwisted quantum affinization algebras was recently obtained by Kong [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  Acknowledgement  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' would like to thank Mirko Primc for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' This work  has been supported in part by Chinese National Natural Science Foundation grant nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  12101261 and 12171303 and by Croatian Science Foundation under the project UIP-2019-  04-8488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Ding, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Feigin, Commutative quantum current operators, semi-infinite construction and functional  models, Represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Theory 4 (2000), 330–341;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' arXiv:q-alg/9612009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Drinfeld, A new realization of Yangians and quantized affine algebras, Soviet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' 36  (1988), 212–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Etingof, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' Kazhdan, Quantization of Lie bialgebras, V, Selecta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE3T4oBgHgl3EQfzwvz/content/2301.04732v1.pdf'}
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+arXiv:2301.12034v1  [cond-mat.supr-con]  28 Jan 2023
+Surface-termination-dependent electronic states in kagome superconductors AV3Sb5
+(A = K, Rb, Cs) studied by micro-ARPES
+Takemi Kato,1, ∗ Yongkai Li,2, 3, 4, ∗ Min Liu,2, 3, ∗ Kosuke Nakayama,1, 5, † Zhiwei Wang,2, 3, 4, † Seigo Souma,6, 7 Miho
+Kitamura,8 Koji Horiba,8, 9 Hiroshi Kumigashira,10 Takashi Takahashi,1 Yugui Yao,2, 3 and Takafumi Sato1, 6, 7, 11, †
+1Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan
+2Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE),
+School of Physics, Beijing Institute of Technology, Beijing 100081, P. R. China
+3Beijing Key Lab of Nanophotonics and Ultrafine Optoelectronic Systems,
+Beijing Institute of Technology, Beijing 100081, P. R. China
+4Material Science Center, Yangtze Delta Region Academy of Beijing Institute of Technology, Jiaxing, 314011, P. R. China
+5Precursory Research for Embryonic Science and Technology (PRESTO),
+Japan Science and Technology Agency (JST), Tokyo, 102-0076, Japan
+6Center for Science and Innovation in Spintronics (CSIS), Tohoku University, Sendai 980-8577, Japan
+7Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan
+8Institute of Materials Structure Science, High Energy Accelerator
+Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan
+9National Institutes for Quantum Science and Technology (QST), Sendai 980-8579, Japan
+10Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University, Sendai 980-8577, Japan
+11International Center for Synchrotron Radiation Innovation Smart (SRIS), Tohoku University, Sendai 980-8577, Japan
+(Dated: January 31, 2023)
+Recently discovered kagome superconductors AV3Sb5 (A = K, Rb, Cs) exhibit exotic bulk and
+surface physical properties such as charge-density wave (CDW) and chirality, whereas their origins
+remain unresolved. By using micro-focused angle-resolved photoemission spectroscopy, we discov-
+ered that AV3Sb5 commonly exhibits two distinct polar surfaces depending on the termination;
+electron- and hole-doped ones for the A- and Sb-termination, respectively. We observed that the
+kagome-derived band shows a clear splitting in the A-terminated surface while it is absent in the
+Sb-terminated counterpart, indicative of the polarity-dependent CDW at the surface. Close com-
+parison of the band-dependent splitting reveals that the three-dimensional CDW structure of the
+K-terminated surface is different from that of the Rb- or Cs-terminated surface, suggesting the di-
+versity of the CDW ground state. These results provide important insight into the origin of CDW
+in kagome superconductors AV3Sb5.
+I.
+INTRODUCTION
+Kagome lattices, consisting of a two-dimensional net-
+work of corner-sharing triangles, offer a fertile platform
+to explore a variety of quantum phases, such as magnetic
+Weyl semimetals, unconventional density waves, charge
+fractionalization, and superconductivity [1–16], owing to
+the unique band structure due to the symmetry of the
+kagome lattice; flat band, Dirac cone, and saddle point
+forming a van Hove singularity. Recent discovery of a
+new family of kagome metals AV3Sb5 (A = K, Rb, Cs)
+which have a saddle point in the vicinity of the Fermi
+level (EF) provides an excellent opportunity to investi-
+gate the charge-density wave (CDW; TCDW = 78–103
+K) in the kagome lattice [17–20], which is accompanied
+by the three-dimensional (3D) 2 × 2 × 2 or 2 × 2 × 4
+charge order [21–23]. Besides CDW, AV3Sb5 commonly
+exhibits superconductivity (Tc = 0.92–2.5 K) in the CDW
+phase [17–20]. The complex electronic phase diagram as
+∗ These authors contributed equally to this work.
+† Corresponding authors:
+k.nakayama@arpes.phys.tohoku.ac.jp
+zhiweiwang@bit.edu.cn
+t-sato@arpes.phys.tohoku.ac.jp
+a function of pressure and chemical doping indicates the
+intertwined nature of CDW and superconductivity [24–
+37]. Emergence of other exotic properties, such as broken
+time reversal symmetry [38–41] and electronic nematic-
+ity [42–44], has been also reported in the CDW phase.
+Thus, elucidating the origin of CDW and its interplay
+with other physical properties is urgently required to un-
+derstand the exotic properties of AV3Sb5.
+Besides unique bulk properties, AV3Sb5 shows several
+intriguing surface properties distinct from those of bulk,
+such as the unidirectional 4 × 1 stripe charge order and
+the surface-dependent vortex core state [21, 40–42, 45–
+47]. These surface properties are strongly coupled to the
+crystal termination at the surface. Namely, since AV3Sb5
+consists of two building blocks, A layer and V3Sb5 layer
+[Fig. 1(a)], cleavage of crystal produces either A- or Sb-
+terminated surface [Fig. 1(b)]. As demonstrated by scan-
+ning tunneling microscopy (STM) [21, 40–42, 45–47], the
+4 × 1 charge order and the vortex core states show a
+remarkable surface-termination dependence. It was also
+uncovered by a recent angle-resolved photoemission spec-
+troscopy (ARPES) with micro-focused photon beam that
+CsV3Sb5 shows a termination-dependent surface polarity
+that leads to the electron- and hole-rich characters for the
+Cs- and Sb-terminated surfaces, respectively [48]. The
+ARPES study also revealed the electronic reconstruc-
+
+2
+tion due to the 3D CDW at the Cs-terminated surface
+and its suppression at the Sb-terminated surface [48], in-
+dicating that the crystal termination modifies not only
+surface properties but also bulk-originated properties at
+the surface.
+Therefore, high-spatial-resolution ARPES
+is needed to elucidate the bulk and surface band struc-
+tures and their relationship with the physical properties
+in AV3Sb5. However, such a study is scarce especially re-
+garding the A-element dependence, despite the fact that
+the bulk CDW and surface stripe charge order show a
+strong A-element dependence [21, 38, 40, 41, 45–47, 49–
+51].
+In this paper, we report a spatially-resolved ARPES
+study of KV3Sb5, RbV3Sb5, and CsV3Sb5. We system-
+atically investigated the band structure of the A- and
+Sb-terminated surfaces of these materials, and found the
+universality of termination-dependent self-doping effect
+indicative of surface polarity. We also found the univer-
+sal and A-element dependent characteristics of 3D CDW.
+We discuss implications of the present results in relation
+to the nature of exotic CDW and surface properties of
+AV3Sb5.
+II.
+EXPERIMENTS
+A.
+Crystal growth and ARPES measurements
+High-quality AV3Sb5 single crystals (A = K, Rb, Cs)
+were synthesized by the self-flux method [41]. ARPES
+measurements were performed using Scienta-Omicron
+DA30 and SES2002 spectrometers at BL-28A and BL-2A
+in Photon Factory, KEK, with the micro-focused beam
+(spot size of 10 × 12 µm2) achieved by Kirkpatrick-Baez
+mirror optics [52]. We used energy tunable photons of
+hν = 85–350 eV. ARPES data were mainly obtained
+by circularly polarized 114-, 111-, and 106-eV photons
+for KV3Sb5, RbV3Sb5, and CsV3Sb5, respectively (cor-
+responding to the out-of-plane wave vector kz ∼ 0; see
+Appendix A). The energy resolution was set to be 25
+meV. Samples were quickly cooled down from room tem-
+perature to T = 8 K, and then cleaved and measured.
+ARPES measurements were also carried out using soft
+x ray with Scienta SES2002 spectrometer at BL-2A in
+Photon Factory, KEK.
+B.
+Band calculations
+The first-principles calculations were performed based
+on the density functional theory (DFT), as implemented
+in the Vienna Ab initio Simulation Package (VASP) [53,
+54]. The projected augmented wave (PAW) is adopted
+as the pseudopotentials [55]. The generalized gradient
+approximation with the Perdew-Burke-Ernzerhof (PBE)
+realization was used for the exchange-correlation func-
+tional [56]. We used an energy cutoff of 500 eV and a
+12 × 12 × 7 Γ-centered k-points for the self-consistent
+calculations. The interlayer van der Waals interactions
+are taken into account using the DFT-D3 correction [57].
+The lattice constants and internal coordinates are op-
+timized until the atomic forces become less than 10−4
+eV/˚A. We used the WANNIERTOOLS package [58] to
+perform the slab band calculations, in which we construct
+a 3-unit-cell slab structure with the Sb (A) termination
+at the top (bottom) surface.
+III.
+RESULTS AND DISCUSSION
+First, to investigate the surface termination, we mea-
+sured spatially-resolved core level spectra of KV3Sb5
+[Figs.
+1(c)–1(f)], RbV3Sb5 [Figs.
+1(g)–1(j)], and
+CsV3Sb5 [Figs. 1(k)–1(n)] by sweeping the micro-focused
+beam in the area of 500 × 500 µm2 of cleaved sur-
+faces. According to a previous study on CsV3Sb5 [48],
+the intensity ratio between the surface and bulk Cs-
+4d core peaks due to Cs atoms at the topmost surface
+and beneath the V3Sb5 layer, respectively, is a good
+measure for determining the surface termination.
+As
+seen from representative Cs-4d core-level spectra in Fig.
+1(k), each spin-orbit satellite (Cs 4d5/2 or 4d3/2) con-
+sists of surface and bulk components located at higher
+and lower binding energies (EB’s), respectively. The Cs-
+termination-dominant surface (red curve) is character-
+ized by a stronger surface-derived peak than the bulk
+one, whereas the Sb-termination-dominant surface (blue
+curve) is characterized by a stronger bulk-derived peak
+[48]. As displayed in Fig. 1(m), the spatial dependence of
+the surface/bulk Cs-4d core-peak intensity-ratio clearly
+maps out the Cs- and Sb-termination-dominant domains
+(red and blue regions, respectively). Hereafter, we simply
+call them Cs- and Sb-terminated surfaces, respectively.
+An intriguing property in relation to the difference in the
+surface termination is polarity [48]. This is seen in the
+Sb-4d core-level spectra in Fig. 1(l), in which both the
+Sb-4d5/2 and 4d3/2 peaks of the Cs-terminated surface
+(red curve) shift to higher EB by ∼130 meV compared
+to those of the Sb-terminated one (blue curve) due to
+the polarity-induced self-electron-doping. Polar domains
+visualized by the spatial map at the EB value of the Sb-
+4d5/2 core-level peak [Fig. 1(n)] show almost the same
+spatial distribution as that in Fig.
+1(m), indicating a
+close link between the surface polarity and the termina-
+tion. In the same way, we have mapped out the surface
+termination and polarity for KV3Sb5 and RbV3Sb5, and
+found an overall similarity to those observed for CsV3Sb5.
+As seen from Figs. 1(c) and 1(g), the K-3p and Rb-3d
+core levels consist of the bulk and surface components as
+in the case of CsV3Sb5 (note that the surface K-3p3/2
+and 3p1/2 peaks appear as a single broad peak due to
+their small energy separation). The stronger intensity of
+the surface component compared to the bulk one in the
+red curve signifies the A termination, whereas the weaker
+intensity of the surface component in the blue curve sug-
+gests the Sb termination. In the Sb-4d core level, the
+
+3
+Sb 4d5/2
+Sb 4d5/2
+Sb 4d5/2
+A1+
+(V3Sb5)1-
+A1+
+(V3Sb5)1-
+A1+
+(V3Sb5)1-
+A1+
+(V3Sb5)1-
+AV3Sb5 (A = K, Rb, Cs)
+a
+b
+c
+A
+Sb2
+V
+Sb1
+(a)
+Intensity (arb. units)
+80
+70
+Binding Energy (eV)
+4d3/2
+4d5/2
+33
+32
+Binding Energy (eV)
+112
+110
+Binding Energy (eV)
+33
+32
+Binding Energy (eV)
+19
+18
+Binding Energy (eV)
+33
+32
+Binding Energy (eV)
+(c)
+(d)
+(g)
+(h)
+(k)
+(l)
+3p1/2
+3p3/2
+3d3/2
+3d5/2
+4d3/2
+4d5/2
+4d3/2
+4d5/2
+4d3/2
+4d5/2
+y (μm)
+400
+200
+0
+0
+200
+400
+x (μm)
+(e)
+(f)
+(i)
+(j)
+(m)
+(n)
+Cs 4d5/2
+Rb 3d5/2
+K 3p3/2
+4
+2
+4
+3
+2
+1
+0
+200
+400
+x (μm)
+0
+200
+400
+x (μm)
+0
+200
+400
+x (μm)
+0
+200
+400
+x (μm)
+0
+200
+400
+x (μm)
+(b)
+K term.
+Sb term.
+K term.
+Sb term.
+Rb term.
+Sb term.
+Rb term.
+Sb term.
+Cs term.
+Sb term.
+Cs term.
+Sb term.
+114
+31.78
+31.73
+31.7 31.8
+31.82
+31.77
+Bulk
+Surf.
+Bulk
+Surf.
+Bulk
+Surf.
+K 3p
+Sb 4d
+Sb 4d
+Sb 4d
+Rb 3d
+Cs 4d
+B. E. (eV)
+Intensity ratio
+B. E. (eV)
+Intensity ratio
+B. E. (eV)
+Intensity ratio
+FIG. 1. (a) Crystal structure of AV3Sb5 (A = K, Rb, Cs). (b) Side view of crystal consisting of A1+ and (V3Sb5)1− layer.
+The cleaved surface is terminated by Sb or A atoms. (c) and (d) EDCs in the K-3p and Sb-4d core-level regions, respectively,
+for the K-(red) and Sb-(blue) terminated surfaces of KV3Sb5. (e) and (f) Spatial map of the intensity ratio of the surface/bulk
+K-3p3/2 core-level peaks and the EB position of Sb-4d5/2 core level, respectively, measured at T = 8 K in a surface area
+of 500 × 500 µm2 for KV3Sb5. (g) and (h) EDCs in the Rb-3d and Sb-4d core-level regions, respectively, for the Rb-(red)
+and Sb-(blue) terminated surfaces of RbV3Sb5. (i) and (j) Same as (e) and (f) but for the Rb-3d5/2 and Sb-4d5/2 core-level
+peaks of RbV3Sb5, respectively. (k) and (l) EDCs in the Cs-4d and Sb-4d core-level regions, respectively, for the Cs-(red) and
+Sb-(blue) terminated surfaces of CsV3Sb5. (m) and (n) Same as (e) and (f) but for the Cs-4d5/2 and Sb-4d5/2 core-level peaks
+of CsV3Sb5, respectively. The core-level spectra were measured with hν = 114, 150, and 106 eV for KV3Sb5, RbV3Sb5, and
+CsV3Sb5, respectively.
+peak position for the A termination [red curves in Figs.
+1(d) and 1(h)] shifts to higher EB by 100–150 meV com-
+pared to that for the Sb termination (blue curves), in-
+dicating the polar-surface formation. Figures 1(e) and
+1(i), and Figs. 1(f) and 1(j) show the spatial distribu-
+tion of the surface termination and polarity, respectively.
+All the images include the red and blue regions which
+correspond to the electron-doped K/Rb- and hole-doped
+Sb-rich surfaces, respectively. In addition, the red- and
+blue-colored patterns are almost identical between Figs.
+1(e) and 1(f), and Figs. 1(i) and 1(j), showing the uni-
+versality of the termination-dependent surface polarity in
+AV3Sb5 (note that the domain size appeared to be differ-
+ent at each cleavage and position of surface, suggesting
+that the A-dependent variation in the domain size seen
+in Fig. 1 may not be an essential feature).
+The universal feature among AV3Sb5 is also seen near
+EF.
+Figure 2 shows the Fermi-surface (FS) mapping
+measured at T
+= 8 K (well below bulk TCDW) for
+KV3Sb5 [Figs. 2(a)–2(d)], RbV3Sb5 [Figs. 2(e)–2(h)],
+and CsV3Sb5 [Figs. 2(i)–2(l)]. We have selected kz ∼ 0
+plane for the mapping by sweeping photon energies and
+thereby tracing the kz dispersion (see Appendix A for
+details). In the Sb-terminated surface [Figs. 2(a), 2(e),
+and 2(i)], the FS of all AV3Sb5 consists of a circu-
+lar pocket centered at the Γ point, a slightly distorted
+hexagon near the Brillouin-zone boundary, and a trian-
+gular pocket centered at the K point, in qualitative agree-
+ment with first-principles bulk-band calculations in the
+normal state [38, 46, 49, 59–69].
+The circular pocket
+is attributed to the 5pz band of Sb atoms embedded in
+the kagome-lattice plane [Sb1 in Fig.
+1(a)], while the
+hexagonal and triangular FSs are mainly due to the V
+kagome-derived 3dxz/dyz and 3dxy/dx2−y2 bands, respec-
+tively. In the A-terminated surface [Figs. 2(c), 2(g), and
+2(k)], one can find that each of the circular and hexago-
+nal FSs consists of two sheets [see two concentric circles
+centered at the Γ point in Figs. 2(c), 2(g), and 2(k) and
+two parallel lines indicated by arrows in Figs. 2(d), 2(h),
+and 2(l)], in contrast to their single-sheet nature in the
+Sb-terminated surface [Figs. 2(a), 2(b), 2(e), 2(f), 2(i),
+and 2(j)]. In addition to the increase in the number of
+FS, one can find an expansion of the circular pockets in
+the A-terminated surface compared to that in the Sb-
+terminated surface (see Appendix B for the comparison
+of FS volume). These results demonstrate that AV3Sb5
+commonly shows termination-dependent FSs.
+To understand the termination-dependent electronic
+states in more detail, we next measured the band dis-
+persions near EF.
+We plot the ARPES intensity ob-
+tained for the Sb-terminated surface at T = 8 K along
+the KMK cut (kx = −π) in Figs. 3(a)–3(c) for KV3Sb5,
+RbV3Sb5, and CsV3Sb5, respectively.
+In all AV3Sb5,
+there are highly dispersive β and γ bands crossing EF
+midway between the Γ and K points, the δ and ε bands
+forming a Dirac cone at the K point, and the hole-like η
+
+M人人从人人Sb 4 ds/2
+Sb 4d5/2
+Sb 4ds/24
+(i)
+ky (Å-1)
+0.0
+-0.5
+0.5
+ky (Å-1)
+0.0
+-0.5
+0.5
+ky (Å-1)
+0.0
+-0.5
+0.5
+ky (Å-1)
+0.0
+-0.5
+0.5
+(k)
+(e)
+(g)
+ky (Å-1)
+0.0
+-0.5
+0.5
+ky (Å-1)
+0.0
+-0.5
+0.5
+(a)
+(c)
+High
+Low
+High
+Low
+High
+Low
+K
+ky (Å-1)
+0.6
+0.4
+K
+ky (Å-1)
+0.6
+0.4
+K
+ky (Å-1)
+0.6
+0.4
+K
+ky (Å-1)
+0.6
+0.4
+K
+ky (Å-1)
+0.6
+0.4
+K
+ky (Å-1)
+0.6
+0.4
+0.0
+-0.5
+0.5
+kx (Å-1)
+0.0
+-0.5
+0.5
+kx (Å-1)
+CsV3Sb5
+kx (Å-1)
+-0.6
+-0.4
+-0.8
+kx (Å-1)
+-0.6
+-0.4
+-0.8
+(j)
+(f)
+(b)
+(l)
+(h)
+(d)
+0.0
+-0.5
+0.5
+kx (Å-1)
+0.0
+-0.5
+0.5
+kx (Å-1)
+RbV3Sb5
+kx (Å-1)
+-0.6
+-0.4
+-0.8
+kx (Å-1)
+-0.6
+-0.4
+-0.8
+0.0
+-0.5
+0.5
+kx (Å-1)
+0.0
+-0.5
+0.5
+kx (Å-1)
+KV3Sb5
+kx (Å-1)
+-0.6
+-0.4
+-0.8
+kx (Å-1)
+-0.6
+-0.4
+-0.8
+K
+M
+Γ
+K
+M
+Γ
+K
+M
+Γ
+K
+M
+Γ
+K
+M
+Γ
+K
+M
+Γ
+Sb term.
+Sb term.
+Sb term.
+Sb term.
+Sb term.
+Sb term.
+K term.
+Rb term.
+Cs term.
+K term.
+Rb term.
+Cs term.
+FIG. 2.
+(a) and (b) Fermi-surface mapping for the Sb-
+terminated surface of KV3Sb5 measured at T = 8 K and
+an enlarged view in the k region enclosed by a dashed rect-
+angle in (a), respectively. (c) and (d) Same as (a) and (b),
+respectively, but for the K-terminated surface. (e)–(h) Same
+as (a)–(d) but for RbV3Sb5. (i)–(l) Same as (a)–(d) but for
+CsV3Sb5.
+band topped at the M point. In the A-terminated sur-
+face [Figs. 3(d)–3(f)], two key differences compared to
+the Sb-terminated surfaces are commonly observed re-
+gardless of the A element. Firstly, several bands in the
+A-terminated surface are shifted downward compared to
+those in the Sb-terminated one. For instance, the Dirac
+point in the A-terminated surface shifts downward by
+∼30–60 meV compared to that in the Sb-terminated one
+(compare black arrows; also see Appendix C for detailed
+comparison). Also, the top of the δ band shifts down-
+ward to approach EF, so that it shows a bending be-
+havior near EF. These band shifts are consistent with
+the polarity-induced doping. It is noted that the bend-
+ing behavior of the δ band is partly associated with
+the termination-dependent CDW [48], as discussed later.
+Secondly, several V-derived bands exhibit splitting in the
+A-terminated surface.
+For instance, the β band splits
+into two subbands, as corroborated by comparison of mo-
+mentum distribution curves (MDCs) near EF between
+the A- and Sb-terminated surfaces [Figs. 3(j), 3(l), and
+3(n); see the double-vs-single peaked feature in MDCs
+for the A- and Sb-termination, respectively]. This split-
+ting accounts for the presence of two hexagonal FSs in
+Figs. 2(d), 2(h), and 2(l). In addition, the δ and η bands
+split into two subbands in the A-terminated surface. The
+splitting of δ band is supported by two-peaked structure
+in the energy distribution curve (EDC) near the Fermi
+wave vector [Figs. 3(k), 3(m), and 3(o)] and the splitting
+of the η band is seen in the second-derivative intensity
+plot [Figs. 3(g), 3(h), and 3(i)]. The band dispersion
+near EF is strongly termination-dependent irrespective
+of the A element.
+Binding Energy (eV)
+EF
+0.5
+(f)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+Binding Energy (eV)
+EF
+0.5
+(c)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+Binding Energy (eV)
+EF
+0.5
+(i)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+Binding Energy (eV)
+EF
+0.5
+(e)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+Binding Energy (eV)
+EF
+0.5
+(b)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+Binding Energy (eV)
+EF
+0.5
+(h)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+Binding Energy (eV)
+EF
+0.5
+(d)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+Binding Energy (eV)
+EF
+0.5
+(g)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+High
+Low
+High
+Low
+High
+Low
+High
+Low
+High
+Low
+High
+Low
+High
+Low
+High
+Low
+High
+Low
+Binding Energy (eV)
+EF
+0.5
+(a)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+KV3Sb5
+RbV3Sb5
+CsV3Sb5
+Intensity (arb. units)
+Intensity (arb. units)
+Intensity (arb. units)
+(n)
+(l)
+(j)
+(o)
+(m)
+(k)
+0.62
+0.52
+ky (Å-1)
+B. E. (eV)
+EF
+0.1
+-0.64
+-0.54
+ky (Å-1)
+0.62
+0.52
+ky (Å-1)
+B. E. (eV)
+EF
+0.1
+B. E. (eV)
+EF
+0.1
+β
+β
+β
+δ
+δ
+δ
+K term.
+Sb term.
+Rb term.
+Sb term.
+Cs term.
+Sb term.
+Sb term.
+K term.
+K term.
+Sb term.
+Rb term.
+Rb term.
+Sb term.
+Cs term.
+Cs term.
+hν = 
+92 eV
+hν = 
+88 eV
+hν = 
+94 eV
+FIG. 3. (a)–(c) ARPES intensity plots at T = 8 K measured
+along ky at kx = −π (KMK cut) for Sb-terminated surface of
+KV3Sb5, RbV3Sb5, and CsV3Sb5, respectively. (d)–(f) Same
+as (a)–(c) but for K-, Rb-, and Cs-terminated surface, respec-
+tively. (g)–(i) Second-derivative intensity of (d)–(f), respec-
+tively. (j) Comparison of MDC of the β band at EF between
+the K- and Sb-terminated surfaces. (k) Comparison of EDC
+of the δ band near the kF point between the two surface ter-
+minations. (l) and (m) Same as (j) and (k) but for RbV3Sb5.
+(n) and (o) Same as (j) and (k) but for CsV3Sb5. Red arrows
+in (j)–(o) indicate the splitting of the β and δ bands in the
+K-, Rb-, and Cs-terminated surfaces.
+Besides the universality among AV3Sb5 demonstrated
+above, we found an A-element dependence of the elec-
+tronic states.
+Side-by-side comparison of the band
+dispersions in the Sb-terminated surface [Figs.
+3(a)–
+3(c)] shows that the bottom of the ε band at the M
+point monotonically shifts downward from KV3Sb5 to
+
+5
+CsV3Sb5, as indicated by yellow circles and black dashed
+lines in Figs.
+3(a)–3(c).
+The ε band also exhibits A-
+dependent dispersion in the A-terminated surface.
+As
+seen from Figs.
+3(g)–3(i), the ε band splits into two
+subbands around the M point in RbV3Sb5 and CsV3Sb5
+[Figs. 3(h) and 3(i)], whereas it is not clearly visible in
+KV3Sb5 [Fig. 3(g)], in sharp contrast to the β, γ, and η
+bands in which the splitting is commonly observed in all
+AV3Sb5.
+The A-element dependence of electronic states is also
+observed around the Γ point. Figures 4(a)–4(c) and 4(d)–
+4(f) show the ARPES intensity along the KΓK cut for
+the Sb-terminated surface and the corresponding second-
+derivative intensity, respectively. There are two electron
+pockets (α and α’ bands) which produce circular FSs as
+shown in Fig. 2. The inner α band is the bulk band
+because it shows a clear kz dispersion (see Appendix A
+in the kz-dependent ARPES), while the outer α’ band is
+possibly of surface origin [68]. One can find that, while
+the position of α’ band is not sensitive to the A element,
+the bottom of α band shows a clear A-dependent energy
+shift; EB = 0.60 eV for KV3Sb5, 0.57 eV for RbV3Sb5,
+and 0.53 eV for CsV3Sb5. The systematic energy shift of
+the α-band bottom is also observed in the Sb-terminated
+surface [see Figs. 4(g)–4(i)].
+Binding Energy (eV)
+EF
+0.5
+1.0
+(c)
+K
+K
+Γ
+0.0
+0.2
+-0.2
+ky (Å-1)
+0.4
+-0.4
+Binding Energy (eV)
+EF
+0.5
+1.0
+(b)
+K
+K
+Γ
+0.0
+0.2
+-0.2
+ky (Å-1)
+0.4
+-0.4
+Binding Energy (eV)
+EF
+0.5
+1.0
+(a)
+K
+K
+Γ
+0.0
+0.2
+-0.2
+ky (Å-1)
+0.4
+-0.4
+High
+Low
+High
+Low
+High
+Low
+KV3Sb5
+RbV3Sb5
+CsV3Sb5
+α
+α’
+α
+α’
+α
+α’
+(f)
+K
+K
+Γ
+0.0
+0.2
+-0.2
+ky (Å-1)
+0.4
+-0.4
+(e)
+K
+K
+Γ
+0.0
+0.2
+-0.2
+ky (Å-1)
+0.4
+-0.4
+(d)
+K
+K
+Γ
+0.0
+0.2
+-0.2
+ky (Å-1)
+0.4
+-0.4
+High
+Low
+High
+Low
+High
+Low
+Binding Energy (eV)
+EF
+0.5
+1.0
+Binding Energy (eV)
+EF
+0.5
+1.0
+Binding Energy (eV)
+EF
+0.5
+1.0
+α
+α’
+α
+α’
+α
+α’
+(i)
+K
+K
+Γ
+0.0
+0.2
+-0.2
+ky (Å-1)
+0.4
+-0.4
+(h)
+K
+K
+Γ
+0.0
+0.2
+-0.2
+ky (Å-1)
+0.4
+-0.4
+(g)
+K
+K
+Γ
+0.0
+0.2
+-0.2
+ky (Å-1)
+0.4
+-0.4
+High
+Low
+High
+Low
+High
+Low
+Binding Energy (eV)
+EF
+0.5
+1.0
+Binding Energy (eV)
+EF
+0.5
+1.0
+Binding Energy (eV)
+EF
+0.5
+1.0
+α
+α
+α
+FIG. 4.
+(a)–(c) ARPES intensity plots at T = 8 K mea-
+sured around the Γ point for K-, Rb-, and Cs-terminated
+surfaces of KV3Sb5, RbV3Sb5, and CsV3Sb5, respectively.
+(d)–(f) Second-derivative intensity plots of (a)–(c), respec-
+tively. Dashed lines and arrows indicate the bottom of α and
+α’ bands. (g)–(i) Same as (d)–(f) but for the Sb-terminated
+surface.
+To explain the observed surface-termination and A-
+element dependences of the band structure, we performed
+first-principles band structure calculations for a 3-unit-
+cell slab with the Sb and A terminations at the top and
+bottom surfaces, respectively [Fig. 5(a)]. Figures 5(b)–
+5(d) are the calculated band structure along the ΓKM
+cut, in which red, gray, and blue colorings represent the
+contribution from the Sb-terminated surface at the top,
+the bulk inside the slab, and the A-terminated surface at
+the bottom, respectively. One can notice that the α and
+δ bands of the Sb-terminated surface shift upward com-
+pared to the A-terminated counterparts. This reasonably
+agrees with the experimental observation, supporting the
+polar nature of the termination-dependent band struc-
+ture. It is noted that, although the atomic rearrangement
+often modifies the band dispersion at the surface, we as-
+sumed the undistorted 1 × 1 lattice in the present calcu-
+lations, so the main trigger of the termination-dependent
+electronic states is unlikely atomic rearrangement but a
+band-bending effect due to the electrostatic potential gra-
+dient, i.e., polarity. The modulation of band structure
+and carrier concentration at the surface also indicates
+a need for caution when discussing/comparing physical
+properties probed by surface- and bulk-sensitive probes
+in AV3Sb5. Besides the surface-termination dependence,
+some A-element dependences in the band structure are
+also captured by our calculations. As indicated by dashed
+lines and arrows in Fig.
+5(d), the bottom of α band
+gradually shifts downward in going from KV3Sb5, to
+RbV3Sb5 and CsV3Sb5.
+Also, the bottom of ε band
+at the M point shows a similar tendency in qualitative
+agreement with the experimental observation. The A-
+dependent band shift likely originates from a variation of
+the relative energy position and the width of band due
+to a difference in the lattice parameter.
+Now we discuss implications of the present results in
+relation to CDW. As demonstrated above, the slab cal-
+culation with the 1 × 1 structure shows a reasonable
+agreement with the experimental band structure in the
+Sb-terminated surface, whereas the calculation cannot
+reproduce the band splitting in the A-terminated sur-
+face. Since the splitting of the V-3d band is probably
+due to the out-of-plane unit-cell doubling and the resul-
+tant band folding along kz [48, 49, 68, 70], the present
+result strongly suggests the formation of 2 × 2 × 2 CDW
+in the A-terminated surface and its suppression in the
+Sb-terminated one.
+While such termination/polarity-
+dependent CDW was suggested in a previous ARPES on
+CsV3Sb5 [48], the present result has clearly established
+the universality among AV3Sb5.
+Present results also provide deeper insight into the na-
+ture of CDW. According to the model calculations for
+various types of 3D CDW [49, 70], the ε band splits
+when the kagome layers with the star-of-David (SoD)-
+type and tri-hexagonal (TrH)-type distortions are stacked
+alternately, whereas it does not split when the kagome
+layers have only SoD or TrH distortion. Therefore, the
+present observations of the absence of ε-band splitting
+in KV3Sb5 and its presence in RbV3Sb5 and CsV3Sb5
+suggest that the 2 × 2 × 2 CDW in KV3Sb5 originates
+
+6
+Energy (eV)
+-0.5
+0.0
+KV3Sb5
+RbV3Sb5
+CsV3Sb5
+K
+K
+Γ
+M
+K
+Γ
+K
+K
+Γ
+M
+K
+Γ
+K
+K
+Γ
+M
+K
+Γ
+(b)
+(c)
+(d)
+(a)
+A
+Sb
+V
+α
+β
+γ
+δ
+ε
+η
+α
+β
+γ
+δ
+ε
+η
+α
+β
+γ
+δ
+ε
+η
+FIG. 5. (a) Slab structure for which first-principles band calculations were performed. (b)–(d) Calculated band structures
+along the ΓKM cut for KV3Sb5, RbV3Sb5, and CsV3Sb5 slabs, respectively. Dashed lines and arrows indicate the bottom of α
+band. Red and blue dots represent the contribution from the top and bottom surfaces, as indicated by gradual shading in (a).
+TABLE I. Summary of A-element and surface-termination dependences of band structure and 3D CDW. Ebottom (eV) indicates
+the binding energy of the band bottom.
+KV3Sb5
+RbV3Sb5
+CsV3Sb5
+Surface termination
+K rich
+Sb rich
+Rb rich
+Sb rich
+Cs rich
+Sb rich
+Ebottom of α band
+0.60
+0.50
+0.57
+0.49
+0.53
+0.46
+Ebottom of α’ band
+0.89
+—
+0.88
+—
+0.88
+—
+Doped carrier
+electron
+hole
+electron
+hole
+electron
+hole
+Split of β, γ, δ bands
+yes
+no
+yes
+no
+yes
+no
+Split of ε band
+no
+no
+yes
+no
+yes
+no
+3D CDW
+Staggered
+SoD or TrH
+no
+Alternate stacking
+of SoD and TrH
+no
+Alternate stacking
+of SoD and TrH
+no
+from only SoD or TrH distortion with a π phase shift
+along the c axis (staggered SoD or TrH CDW), while the
+2 × 2 × 2 CDW in RbV3Sb5 and CsV3Sb5 is an alternate
+stacking of the SoD and TrH distortions [49, 70] (see Ta-
+ble 1 for summary of the present key results). In this
+regard, it is noted that, while we observed the ε-band
+splitting in RbV3Sb5, it was absent in a recent study
+by another group [49]. This difference implies that the
+type of 3D CDW at the surface is not robust even in the
+same material. Possible key factors determining the type
+of 3D CDW are carrier/chemical doping, cooling speed,
+etc. [48, 49, 71].
+Another exotic property of AV3Sb5 is the unidirec-
+tional 4 × 1 charge order that appears only in the Sb-
+terminated surface. According to STM studies, the 4 × 1
+charge order is present in RbV3Sb5 and CsV3Sb5 but ab-
+sent in KV3Sb5 [21, 38, 40, 41, 45–47, 72, 73]. Despite
+such A-element dependence of the presence/absence of
+the 4 × 1 charge order, the present result shows that
+the FS topology and size in the Sb-terminated surface
+are similar among AV3Sb5 and a good FS nesting condi-
+tion with the 4 × 1 periodicity is not observed in any of
+AV3Sb5, suggesting that the mechanism of the unidirec-
+tional order may be beyond the framework with simple
+FS nesting.
+In conclusion, we reported a systematic study on the
+spatially-resolved band structure of AV3Sb5 by micro-
+ARPES in combination with first-principles band struc-
+ture calculations.
+We have established the universal-
+ity of the polar nature of cleaved surfaces characterized
+by the opposite band energy shift between the A- and
+Sb-terminated surfaces.
+We also found that the pres-
+ence/absence and the microscopic structure of 3D CDW
+depend on the surface polarity (termination) and the
+A element, pointing to an exceptional sensitivity of 3D
+CDW to the underlying electronic and chemical environ-
+ments. The present results provide foundation for under-
+standing and manipulating exotic physical properties of
+AV3Sb5.
+ACKNOWLEDGMENTS
+This work was supported by JST-CREST (No.
+JP-
+MJCR18T1),
+JST-PRESTO
+(No.
+JPMJPR18L7),
+Grant-in-Aid for Scientific Research (JSPS KAKENHI
+Grant Numbers JP21H04435 and JP20H01847), KEK-
+PF (Proposal number: 2021S2-001), and the Sasakawa
+Scientific Research Grant from the Japan Science Soci-
+ety.
+The work at Beijing Institute of Technology was
+supported by the National Key R&D Program of China
+(Grant
+Nos.
+2020YFA0308800,
+2022YFA1403400),
+
+三7
+the Natural Science Foundation of China (Grant No.
+92065109), and the Beijing Natural Science Foundation
+(Grant Nos.
+Z210006, Z190006).
+T.K. acknowledges
+support from GP-Spin at Tohoku University and JST-
+SPRING (No. JPMJSP2114). Z.W. thanks the Analysis
+& Testing Center at BIT for assistance in facility support.
+APPENDIX A: NORMAL EMISSION
+MEASUREMENT
+Figure 6 shows the second-derivative ARPES inten-
+sity measured at the normal-emission set up in the Rb-
+terminated surface of RbV3Sb5. The energy position of
+the inner electron-band (α) bottom shows a periodic dis-
+persion as a function of kz (see red dashed curve; note
+that the intensity of the outer electron band is weak),
+suggesting the bulk-band nature. From the observed pe-
+riodicity, we have estimated the inner-potential value to
+be V0 = 10.0 eV, which indicates that the kz ∼ 0 plane
+is probed by 111-eV photons in RbV3Sb5. In KV3Sb5
+and CsV3Sb5, the kz ∼ 0 plane is probed by hν = 114
+eV and 106 eV, respectively, according to our previous
+works [59, 69].
+Binding Energy (eV)
+EF
+0.5
+1.0
+A
+A
+Γ
+Γ
+High
+Low
+5.5
+6.0
+5.0
+kz (Å-1)
+FIG. 6.
+Second-derivative intensity of the ARPES spectra
+for RbV3Sb5 plotted as a function of kz and binding energy
+obtained with photon energies of 90–150 eV.
+APPENDIX B: SURFACE-TERMINATION
+DEPENDENCE OF THE FERMI-SURFACE SIZE
+Figure 7 shows side-by-side comparison of the ARPES
+intensity plots at EF obtained in the Sb- and A-
+terminated surfaces of KV3Sb5 [Figs.
+7(a) and (b)],
+RbV3Sb5 [Figs. 7(c) and 7(d)], and CsV3Sb5 [Figs. 7(e)
+and 7(f)] (note that the data are identical to those in Fig.
+2). One can see that both the two circular FSs centered
+at the Γ point in the A-terminated surface are larger than
+the one in the Sb-terminated surface, in agreement with
+the electron-doped character of the A-terminated surface.
+ky (Å-1)
+0.0
+-0.5
+0.5
+(b)
+ky (Å-1)
+0.0
+-0.5
+0.5
+(a)
+0.0
+-0.5
+0.5
+kx (Å-1)
+0.0
+-0.5
+0.5
+kx (Å-1)
+K
+M
+Γ
+K
+M
+Γ
+Sb term.
+K term.
+ky (Å-1)
+0.0
+-0.5
+0.5
+(d)
+ky (Å-1)
+0.0
+-0.5
+0.5
+(c)
+0.0
+-0.5
+0.5
+kx (Å-1)
+0.0
+-0.5
+0.5
+kx (Å-1)
+K
+M
+Γ
+K
+M
+Γ
+Sb term.
+Rb term.
+ky (Å-1)
+0.0
+-0.5
+0.5
+(f)
+ky (Å-1)
+0.0
+-0.5
+0.5
+(e)
+0.0
+-0.5
+0.5
+kx (Å-1)
+0.0
+-0.5
+0.5
+kx (Å-1)
+K
+M
+Γ
+K
+M
+Γ
+Sb term.
+Cs term.
+KV3Sb5
+RbV3Sb5
+CsV3Sb5
+High
+Low
+High
+Low
+High
+Low
+FIG. 7. (a) and (b) Fermi-surface mapping in the Sb- and
+A-terminated surfaces, respectively, in KV3Sb5. (c) and (d)
+Same as (a) and (b) but in RbV3Sb5. (e) and (f) Same as (a)
+and (b) but measured in CsV3Sb5.
+APPENDIX C: COMPARISON OF THE BAND
+DISPERSIONS BETWEEN SB- AND
+A-TERMINATED SURFACES
+Figure 8 shows comparison of the band dispersions
+along the KMK cut between the Sb- and A-terminated
+surfaces of KV3Sb5 [Figs. 8(a) and 8(b)], RbV3Sb5 [Figs.
+8(c) and 8(d)], and CsV3Sb5 [Figs. 8(e) and 8(f)]. The
+data are identical to those in Fig. 3, but the energy shift
+of the Dirac point is better recognized by side-by-side
+comparison. One can also see that the δ band near EF in
+the A-terminated surface is shifted downward compared
+
+8
+to that in the Sb-terminated surface (dotted line).
+Binding Energy (eV)
+EF
+0.5
+(b)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+High
+Low
+K term.
+Binding Energy (eV)
+EF
+0.5
+(c)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+High
+Low
+Sb term.
+Binding Energy (eV)
+EF
+0.5
+(e)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+High
+Low
+Sb term.
+Binding Energy (eV)
+EF
+0.5
+(d)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+High
+Low
+Rb term.
+KV3Sb5
+High
+Low
+Binding Energy (eV)
+EF
+0.5
+(a)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+Sb term.
+RbV3Sb5
+CsV3Sb5
+Binding Energy (eV)
+EF
+0.5
+(f)
+K
+M
+K
+Γ
+Γ
+β
+γ
+δ
+ε
+η
+0.0
+0.5
+-0.5
+ky (Å-1)
+High
+Low
+Cs term.
+FIG. 8. (a) and (b) ARPES intensity plots at T = 8 K mea-
+sured along ky at kx = −π (KMK cut) for the Sb- and A-
+terminated surfaces, respectively, in KV3Sb5.
+(c) and (d)
+Same as (a) and (b), respectively, but measured in RbV3Sb5.
+(e) and (f) Same as (a) and (b), respectively, but measured
+in CsV3Sb5. Black arrows indicate the energy position of the
+Dirac point at the K point. Dotted lines are the δ-band dis-
+persion determined in the Sb-terminated surface.
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+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' †  1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Sendai 980-8578,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Japan  2Centre for Quantum Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Beijing Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Beijing 100081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' China  3Beijing Key Lab of Nanophotonics and Ultrafine Optoelectronic Systems,  Beijing Institute of Technology, Beijing 100081, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' China  4Material Science Center, Yangtze Delta Region Academy of Beijing Institute of Technology, Jiaxing, 314011, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' China  5Precursory Research for Embryonic Science and Technology (PRESTO),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Japan Science and Technology Agency (JST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 102-0076,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Japan  6Center for Science and Innovation in Spintronics (CSIS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Sendai 980-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Japan  7Advanced Institute for Materials Research (WPI-AIMR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Sendai 980-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Japan  8Institute of Materials Structure Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' High Energy Accelerator  Research Organization (KEK),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Ibaraki 305-0801,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Japan  9National Institutes for Quantum Science and Technology (QST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Sendai 980-8579,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Japan  10Institute of Multidisciplinary Research for Advanced Materials (IMRAM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Sendai 980-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Japan  11International Center for Synchrotron Radiation Innovation Smart (SRIS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Sendai 980-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Japan  (Dated: January 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2023)  Recently discovered kagome superconductors AV3Sb5 (A = K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Rb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Cs) exhibit exotic bulk and  surface physical properties such as charge-density wave (CDW) and chirality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' whereas their origins  remain unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' By using micro-focused angle-resolved photoemission spectroscopy, we discov-  ered that AV3Sb5 commonly exhibits two distinct polar surfaces depending on the termination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  electron- and hole-doped ones for the A- and Sb-termination, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' We observed that the  kagome-derived band shows a clear splitting in the A-terminated surface while it is absent in the  Sb-terminated counterpart, indicative of the polarity-dependent CDW at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Close com-  parison of the band-dependent splitting reveals that the three-dimensional CDW structure of the  K-terminated surface is different from that of the Rb- or Cs-terminated surface, suggesting the di-  versity of the CDW ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' These results provide important insight into the origin of CDW  in kagome superconductors AV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  INTRODUCTION  Kagome lattices, consisting of a two-dimensional net-  work of corner-sharing triangles, offer a fertile platform  to explore a variety of quantum phases, such as magnetic  Weyl semimetals, unconventional density waves, charge  fractionalization, and superconductivity [1–16], owing to  the unique band structure due to the symmetry of the  kagome lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' flat band, Dirac cone, and saddle point  forming a van Hove singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Recent discovery of a  new family of kagome metals AV3Sb5 (A = K, Rb, Cs)  which have a saddle point in the vicinity of the Fermi  level (EF) provides an excellent opportunity to investi-  gate the charge-density wave (CDW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' TCDW = 78–103  K) in the kagome lattice [17–20], which is accompanied  by the three-dimensional (3D) 2 × 2 × 2 or 2 × 2 × 4  charge order [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Besides CDW, AV3Sb5 commonly  exhibits superconductivity (Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='92–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5 K) in the CDW  phase [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The complex electronic phase diagram as  ∗ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  † Corresponding authors:  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='nakayama@arpes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='tohoku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='jp  zhiweiwang@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='cn  t-sato@arpes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='tohoku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='jp  a function of pressure and chemical doping indicates the  intertwined nature of CDW and superconductivity [24–  37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Emergence of other exotic properties, such as broken  time reversal symmetry [38–41] and electronic nematic-  ity [42–44], has been also reported in the CDW phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Thus, elucidating the origin of CDW and its interplay  with other physical properties is urgently required to un-  derstand the exotic properties of AV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Besides unique bulk properties, AV3Sb5 shows several  intriguing surface properties distinct from those of bulk,  such as the unidirectional 4 × 1 stripe charge order and  the surface-dependent vortex core state [21, 40–42, 45–  47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' These surface properties are strongly coupled to the  crystal termination at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Namely, since AV3Sb5  consists of two building blocks, A layer and V3Sb5 layer  [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1(a)], cleavage of crystal produces either A- or Sb-  terminated surface [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' As demonstrated by scan-  ning tunneling microscopy (STM) [21, 40–42, 45–47], the  4 × 1 charge order and the vortex core states show a  remarkable surface-termination dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' It was also  uncovered by a recent angle-resolved photoemission spec-  troscopy (ARPES) with micro-focused photon beam that  CsV3Sb5 shows a termination-dependent surface polarity  that leads to the electron- and hole-rich characters for the  Cs- and Sb-terminated surfaces, respectively [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The  ARPES study also revealed the electronic reconstruc-  2  tion due to the 3D CDW at the Cs-terminated surface  and its suppression at the Sb-terminated surface [48], in-  dicating that the crystal termination modifies not only  surface properties but also bulk-originated properties at  the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Therefore, high-spatial-resolution ARPES  is needed to elucidate the bulk and surface band struc-  tures and their relationship with the physical properties  in AV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' However, such a study is scarce especially re-  garding the A-element dependence, despite the fact that  the bulk CDW and surface stripe charge order show a  strong A-element dependence [21, 38, 40, 41, 45–47, 49–  51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  In this paper, we report a spatially-resolved ARPES  study of KV3Sb5, RbV3Sb5, and CsV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' We system-  atically investigated the band structure of the A- and  Sb-terminated surfaces of these materials, and found the  universality of termination-dependent self-doping effect  indicative of surface polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' We also found the univer-  sal and A-element dependent characteristics of 3D CDW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  We discuss implications of the present results in relation  to the nature of exotic CDW and surface properties of  AV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  EXPERIMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Crystal growth and ARPES measurements  High-quality AV3Sb5 single crystals (A = K, Rb, Cs)  were synthesized by the self-flux method [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' ARPES  measurements were performed using Scienta-Omicron  DA30 and SES2002 spectrometers at BL-28A and BL-2A  in Photon Factory, KEK, with the micro-focused beam  (spot size of 10 × 12 µm2) achieved by Kirkpatrick-Baez  mirror optics [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' We used energy tunable photons of  hν = 85–350 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' ARPES data were mainly obtained  by circularly polarized 114-, 111-, and 106-eV photons  for KV3Sb5, RbV3Sb5, and CsV3Sb5, respectively (cor-  responding to the out-of-plane wave vector kz ∼ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' see  Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The energy resolution was set to be 25  meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Samples were quickly cooled down from room tem-  perature to T = 8 K, and then cleaved and measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  ARPES measurements were also carried out using soft  x ray with Scienta SES2002 spectrometer at BL-2A in  Photon Factory, KEK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Band calculations  The first-principles calculations were performed based  on the density functional theory (DFT), as implemented  in the Vienna Ab initio Simulation Package (VASP) [53,  54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The projected augmented wave (PAW) is adopted  as the pseudopotentials [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The generalized gradient  approximation with the Perdew-Burke-Ernzerhof (PBE)  realization was used for the exchange-correlation func-  tional [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' We used an energy cutoff of 500 eV and a  12 × 12 × 7 Γ-centered k-points for the self-consistent  calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The interlayer van der Waals interactions  are taken into account using the DFT-D3 correction [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  The lattice constants and internal coordinates are op-  timized until the atomic forces become less than 10−4  eV/˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' We used the WANNIERTOOLS package [58] to  perform the slab band calculations, in which we construct  a 3-unit-cell slab structure with the Sb (A) termination  at the top (bottom) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  First, to investigate the surface termination, we mea-  sured spatially-resolved core level spectra of KV3Sb5  [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  1(c)–1(f)], RbV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  1(g)–1(j)], and  CsV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1(k)–1(n)] by sweeping the micro-focused  beam in the area of 500 × 500 µm2 of cleaved sur-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' According to a previous study on CsV3Sb5 [48],  the intensity ratio between the surface and bulk Cs-  4d core peaks due to Cs atoms at the topmost surface  and beneath the V3Sb5 layer, respectively, is a good  measure for determining the surface termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  As  seen from representative Cs-4d core-level spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  1(k), each spin-orbit satellite (Cs 4d5/2 or 4d3/2) con-  sists of surface and bulk components located at higher  and lower binding energies (EB’s), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The Cs-  termination-dominant surface (red curve) is character-  ized by a stronger surface-derived peak than the bulk  one, whereas the Sb-termination-dominant surface (blue  curve) is characterized by a stronger bulk-derived peak  [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' As displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1(m), the spatial dependence of  the surface/bulk Cs-4d core-peak intensity-ratio clearly  maps out the Cs- and Sb-termination-dominant domains  (red and blue regions, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Hereafter, we simply  call them Cs- and Sb-terminated surfaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  An intriguing property in relation to the difference in the  surface termination is polarity [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' This is seen in the  Sb-4d core-level spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1(l), in which both the  Sb-4d5/2 and 4d3/2 peaks of the Cs-terminated surface  (red curve) shift to higher EB by ∼130 meV compared  to those of the Sb-terminated one (blue curve) due to  the polarity-induced self-electron-doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Polar domains  visualized by the spatial map at the EB value of the Sb-  4d5/2 core-level peak [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1(n)] show almost the same  spatial distribution as that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  1(m), indicating a  close link between the surface polarity and the termina-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In the same way, we have mapped out the surface  termination and polarity for KV3Sb5 and RbV3Sb5, and  found an overall similarity to those observed for CsV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  As seen from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1(c) and 1(g), the K-3p and Rb-3d  core levels consist of the bulk and surface components as  in the case of CsV3Sb5 (note that the surface K-3p3/2  and 3p1/2 peaks appear as a single broad peak due to  their small energy separation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The stronger intensity of  the surface component compared to the bulk one in the  red curve signifies the A termination, whereas the weaker  intensity of the surface component in the blue curve sug-  gests the Sb termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In the Sb-4d core level, the  3  Sb 4d5/2  Sb 4d5/2  Sb 4d5/2  A1+  (V3Sb5)1-  A1+  (V3Sb5)1-  A1+  (V3Sb5)1-  A1+  (V3Sb5)1-  AV3Sb5 (A = K, Rb, Cs)  a  b  c  A  Sb2  V  Sb1  (a)  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' units)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='Binding Energy (eV)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4d3/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Rb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Rb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Cs term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Cs term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  114  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='78  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='73  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='7 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='8  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='82  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='77  Bulk  Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Bulk  Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Bulk  Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  K 3p  Sb 4d  Sb 4d  Sb 4d  Rb 3d  Cs 4d  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (eV)  Intensity ratio  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (eV)  Intensity ratio  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (eV)  Intensity ratio  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (a) Crystal structure of AV3Sb5 (A = K, Rb, Cs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (b) Side view of crystal consisting of A1+ and (V3Sb5)1− layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  The cleaved surface is terminated by Sb or A atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (c) and (d) EDCs in the K-3p and Sb-4d core-level regions, respectively,  for the K-(red) and Sb-(blue) terminated surfaces of KV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (e) and (f) Spatial map of the intensity ratio of the surface/bulk  K-3p3/2 core-level peaks and the EB position of Sb-4d5/2 core level, respectively, measured at T = 8 K in a surface area  of 500 × 500 µm2 for KV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (g) and (h) EDCs in the Rb-3d and Sb-4d core-level regions, respectively, for the Rb-(red)  and Sb-(blue) terminated surfaces of RbV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (i) and (j) Same as (e) and (f) but for the Rb-3d5/2 and Sb-4d5/2 core-level  peaks of RbV3Sb5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (k) and (l) EDCs in the Cs-4d and Sb-4d core-level regions, respectively, for the Cs-(red) and  Sb-(blue) terminated surfaces of CsV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (m) and (n) Same as (e) and (f) but for the Cs-4d5/2 and Sb-4d5/2 core-level peaks  of CsV3Sb5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The core-level spectra were measured with hν = 114, 150, and 106 eV for KV3Sb5, RbV3Sb5, and  CsV3Sb5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  peak position for the A termination [red curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  1(d) and 1(h)] shifts to higher EB by 100–150 meV com-  pared to that for the Sb termination (blue curves), in-  dicating the polar-surface formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Figures 1(e) and  1(i), and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1(f) and 1(j) show the spatial distribu-  tion of the surface termination and polarity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  All the images include the red and blue regions which  correspond to the electron-doped K/Rb- and hole-doped  Sb-rich surfaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In addition, the red- and  blue-colored patterns are almost identical between Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  1(e) and 1(f), and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1(i) and 1(j), showing the uni-  versality of the termination-dependent surface polarity in  AV3Sb5 (note that the domain size appeared to be differ-  ent at each cleavage and position of surface, suggesting  that the A-dependent variation in the domain size seen  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 1 may not be an essential feature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  The universal feature among AV3Sb5 is also seen near  EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Figure 2 shows the Fermi-surface (FS) mapping  measured at T  = 8 K (well below bulk TCDW) for  KV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2(a)–2(d)], RbV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2(e)–2(h)],  and CsV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2(i)–2(l)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' We have selected kz ∼ 0  plane for the mapping by sweeping photon energies and  thereby tracing the kz dispersion (see Appendix A for  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In the Sb-terminated surface [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2(a), 2(e),  and 2(i)], the FS of all AV3Sb5 consists of a circu-  lar pocket centered at the Γ point, a slightly distorted  hexagon near the Brillouin-zone boundary, and a trian-  gular pocket centered at the K point, in qualitative agree-  ment with first-principles bulk-band calculations in the  normal state [38, 46, 49, 59–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  The circular pocket  is attributed to the 5pz band of Sb atoms embedded in  the kagome-lattice plane [Sb1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  1(a)], while the  hexagonal and triangular FSs are mainly due to the V  kagome-derived 3dxz/dyz and 3dxy/dx2−y2 bands, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In the A-terminated surface [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2(c), 2(g), and  2(k)], one can find that each of the circular and hexago-  nal FSs consists of two sheets [see two concentric circles  centered at the Γ point in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2(c), 2(g), and 2(k) and  two parallel lines indicated by arrows in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2(d), 2(h),  and 2(l)], in contrast to their single-sheet nature in the  Sb-terminated surface [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2(a), 2(b), 2(e), 2(f), 2(i),  and 2(j)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In addition to the increase in the number of  FS, one can find an expansion of the circular pockets in  the A-terminated surface compared to that in the Sb-  terminated surface (see Appendix B for the comparison  of FS volume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' These results demonstrate that AV3Sb5  commonly shows termination-dependent FSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  To understand the termination-dependent electronic  states in more detail, we next measured the band dis-  persions near EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  We plot the ARPES intensity ob-  tained for the Sb-terminated surface at T = 8 K along  the KMK cut (kx = −π) in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 3(a)–3(c) for KV3Sb5,  RbV3Sb5, and CsV3Sb5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  In all AV3Sb5,  there are highly dispersive β and γ bands crossing EF  midway between the Γ and K points, the δ and ε bands  forming a Dirac cone at the K point, and the hole-like η  M人人从人人Sb 4 ds/2  Sb 4d5/2  Sb 4ds/24  (i)  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (k)  (e)  (g)  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (a)  (c)  High  Low  High  Low  High  Low  K  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  K  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  K  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  K  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  K  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  K  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  kx (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  kx (Å-1)  CsV3Sb5  kx (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='8  kx (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='5  kx (Å-1)  KV3Sb5  kx (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='8  kx (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='8  K  M  Γ  K  M  Γ  K  M  Γ  K  M  Γ  K  M  Γ  K  M  Γ  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  K term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Rb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Cs term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  K term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Rb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Cs term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  (a) and (b) Fermi-surface mapping for the Sb-  terminated surface of KV3Sb5 measured at T = 8 K and  an enlarged view in the k region enclosed by a dashed rect-  angle in (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (c) and (d) Same as (a) and (b),  respectively, but for the K-terminated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (e)–(h) Same  as (a)–(d) but for RbV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (i)–(l) Same as (a)–(d) but for  CsV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  band topped at the M point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In the A-terminated sur-  face [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 3(d)–3(f)], two key differences compared to  the Sb-terminated surfaces are commonly observed re-  gardless of the A element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Firstly, several bands in the  A-terminated surface are shifted downward compared to  those in the Sb-terminated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' For instance, the Dirac  point in the A-terminated surface shifts downward by  ∼30–60 meV compared to that in the Sb-terminated one  (compare black arrows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' also see Appendix C for detailed  comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Also, the top of the δ band shifts down-  ward to approach EF, so that it shows a bending be-  havior near EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' These band shifts are consistent with  the polarity-induced doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' It is noted that the bend-  ing behavior of the δ band is partly associated with  the termination-dependent CDW [48], as discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Secondly, several V-derived bands exhibit splitting in the  A-terminated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  For instance, the β band splits  into two subbands, as corroborated by comparison of mo-  mentum distribution curves (MDCs) near EF between  the A- and Sb-terminated surfaces [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 3(j), 3(l), and  3(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' see the double-vs-single peaked feature in MDCs  for the A- and Sb-termination, respectively].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' This split-  ting accounts for the presence of two hexagonal FSs in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2(d), 2(h), and 2(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In addition, the δ and η bands  split into two subbands in the A-terminated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The  splitting of δ band is supported by two-peaked structure  in the energy distribution curve (EDC) near the Fermi  wave vector [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 3(k), 3(m), and 3(o)] and the splitting  of the η band is seen in the second-derivative intensity  plot [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 3(g), 3(h), and 3(i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The band dispersion  near EF is strongly termination-dependent irrespective  of the A element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (f)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='5  ky (Å-1)  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (c)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='5  (e)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='5  ky (Å-1)  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='5  ky (Å-1)  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (h)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='5  ky (Å-1)  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (d)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='5  ky (Å-1)  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (g)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='5  ky (Å-1)  High  Low  High  Low  High  Low  High  Low  High  Low  High  Low  High  Low  High  Low  High  Low  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (a)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='5  ky (Å-1)  KV3Sb5  RbV3Sb5  CsV3Sb5  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' units)  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' units)  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' units)  (n)  (l)  (j)  (o)  (m)  (k)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='52  ky (Å-1)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='54  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='52  ky (Å-1)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='1  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='1  β  β  β  δ  δ  δ  K term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Rb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Cs term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  K term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  K term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Rb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Rb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Cs term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Cs term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  hν =  92 eV  hν =  88 eV  hν =  94 eV  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (a)–(c) ARPES intensity plots at T = 8 K measured  along ky at kx = −π (KMK cut) for Sb-terminated surface of  KV3Sb5, RbV3Sb5, and CsV3Sb5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (d)–(f) Same  as (a)–(c) but for K-, Rb-, and Cs-terminated surface, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (g)–(i) Second-derivative intensity of (d)–(f), respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (j) Comparison of MDC of the β band at EF between  the K- and Sb-terminated surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (k) Comparison of EDC  of the δ band near the kF point between the two surface ter-  minations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (l) and (m) Same as (j) and (k) but for RbV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  (n) and (o) Same as (j) and (k) but for CsV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Red arrows  in (j)–(o) indicate the splitting of the β and δ bands in the  K-, Rb-, and Cs-terminated surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Besides the universality among AV3Sb5 demonstrated  above, we found an A-element dependence of the elec-  tronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Side-by-side comparison of the band  dispersions in the Sb-terminated surface [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  3(a)–  3(c)] shows that the bottom of the ε band at the M  point monotonically shifts downward from KV3Sb5 to  5  CsV3Sb5, as indicated by yellow circles and black dashed  lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  3(a)–3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  The ε band also exhibits A-  dependent dispersion in the A-terminated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  As  seen from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  3(g)–3(i), the ε band splits into two  subbands around the M point in RbV3Sb5 and CsV3Sb5  [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 3(h) and 3(i)], whereas it is not clearly visible in  KV3Sb5 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 3(g)], in sharp contrast to the β, γ, and η  bands in which the splitting is commonly observed in all  AV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  The A-element dependence of electronic states is also  observed around the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Figures 4(a)–4(c) and 4(d)–  4(f) show the ARPES intensity along the KΓK cut for  the Sb-terminated surface and the corresponding second-  derivative intensity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' There are two electron  pockets (α and α’ bands) which produce circular FSs as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The inner α band is the bulk band  because it shows a clear kz dispersion (see Appendix A  in the kz-dependent ARPES), while the outer α’ band is  possibly of surface origin [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' One can find that, while  the position of α’ band is not sensitive to the A element,  the bottom of α band shows a clear A-dependent energy  shift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' EB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='60 eV for KV3Sb5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='57 eV for RbV3Sb5,  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='53 eV for CsV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The systematic energy shift of  the α-band bottom is also observed in the Sb-terminated  surface [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 4(g)–4(i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  (c)  K  K  Γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  (b)  K  K  Γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  (a)  K  K  Γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  High  Low  High  Low  High  Low  KV3Sb5  RbV3Sb5  CsV3Sb5  α  α’  α  α’  α  α’  (f)  K  K  Γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  (e)  K  K  Γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  (d)  K  K  Γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  High  Low  High  Low  High  Low  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  α  α’  α  α’  α  α’  (i)  K  K  Γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  (h)  K  K  Γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  (g)  K  K  Γ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='2  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='4  High  Low  High  Low  High  Low  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  α  α  α  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  (a)–(c) ARPES intensity plots at T = 8 K mea-  sured around the Γ point for K-, Rb-, and Cs-terminated  surfaces of KV3Sb5, RbV3Sb5, and CsV3Sb5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  (d)–(f) Second-derivative intensity plots of (a)–(c), respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Dashed lines and arrows indicate the bottom of α and  α’ bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (g)–(i) Same as (d)–(f) but for the Sb-terminated  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  To explain the observed surface-termination and A-  element dependences of the band structure, we performed  first-principles band structure calculations for a 3-unit-  cell slab with the Sb and A terminations at the top and  bottom surfaces, respectively [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 5(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Figures 5(b)–  5(d) are the calculated band structure along the ΓKM  cut, in which red, gray, and blue colorings represent the  contribution from the Sb-terminated surface at the top,  the bulk inside the slab, and the A-terminated surface at  the bottom, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' One can notice that the α and  δ bands of the Sb-terminated surface shift upward com-  pared to the A-terminated counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' This reasonably  agrees with the experimental observation, supporting the  polar nature of the termination-dependent band struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' It is noted that, although the atomic rearrangement  often modifies the band dispersion at the surface, we as-  sumed the undistorted 1 × 1 lattice in the present calcu-  lations, so the main trigger of the termination-dependent  electronic states is unlikely atomic rearrangement but a  band-bending effect due to the electrostatic potential gra-  dient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=', polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The modulation of band structure  and carrier concentration at the surface also indicates  a need for caution when discussing/comparing physical  properties probed by surface- and bulk-sensitive probes  in AV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Besides the surface-termination dependence,  some A-element dependences in the band structure are  also captured by our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' As indicated by dashed  lines and arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  5(d), the bottom of α band  gradually shifts downward in going from KV3Sb5, to  RbV3Sb5 and CsV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Also, the bottom of ε band  at the M point shows a similar tendency in qualitative  agreement with the experimental observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The A-  dependent band shift likely originates from a variation of  the relative energy position and the width of band due  to a difference in the lattice parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Now we discuss implications of the present results in  relation to CDW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' As demonstrated above, the slab cal-  culation with the 1 × 1 structure shows a reasonable  agreement with the experimental band structure in the  Sb-terminated surface, whereas the calculation cannot  reproduce the band splitting in the A-terminated sur-  face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Since the splitting of the V-3d band is probably  due to the out-of-plane unit-cell doubling and the resul-  tant band folding along kz [48, 49, 68, 70], the present  result strongly suggests the formation of 2 × 2 × 2 CDW  in the A-terminated surface and its suppression in the  Sb-terminated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  While such termination/polarity-  dependent CDW was suggested in a previous ARPES on  CsV3Sb5 [48], the present result has clearly established  the universality among AV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Present results also provide deeper insight into the na-  ture of CDW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' According to the model calculations for  various types of 3D CDW [49, 70], the ε band splits  when the kagome layers with the star-of-David (SoD)-  type and tri-hexagonal (TrH)-type distortions are stacked  alternately, whereas it does not split when the kagome  layers have only SoD or TrH distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Therefore, the  present observations of the absence of ε-band splitting  in KV3Sb5 and its presence in RbV3Sb5 and CsV3Sb5  suggest that the 2 × 2 × 2 CDW in KV3Sb5 originates  6  Energy (eV)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  KV3Sb5  RbV3Sb5  CsV3Sb5  K  K  Γ  M  K  Γ  K  K  Γ  M  K  Γ  K  K  Γ  M  K  Γ  (b)  (c)  (d)  (a)  A  Sb  V  α  β  γ  δ  ε  η  α  β  γ  δ  ε  η  α  β  γ  δ  ε  η  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (a) Slab structure for which first-principles band calculations were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (b)–(d) Calculated band structures  along the ΓKM cut for KV3Sb5, RbV3Sb5, and CsV3Sb5 slabs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Dashed lines and arrows indicate the bottom of α  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Red and blue dots represent the contribution from the top and bottom surfaces, as indicated by gradual shading in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Summary of A-element and surface-termination dependences of band structure and 3D CDW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Ebottom (eV) indicates  the binding energy of the band bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  KV3Sb5  RbV3Sb5  CsV3Sb5  Surface termination  K rich  Sb rich  Rb rich  Sb rich  Cs rich  Sb rich  Ebottom of α band  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='46  Ebottom of α’ band  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='89  —  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='88  —  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='88  —  Doped carrier  electron  hole  electron  hole  electron  hole  Split of β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' δ bands  yes  no  yes  no  yes  no  Split of ε band  no  no  yes  no  yes  no  3D CDW  Staggered  SoD or TrH  no  Alternate stacking  of SoD and TrH  no  Alternate stacking  of SoD and TrH  no  from only SoD or TrH distortion with a π phase shift  along the c axis (staggered SoD or TrH CDW),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' while the  2 × 2 × 2 CDW in RbV3Sb5 and CsV3Sb5 is an alternate  stacking of the SoD and TrH distortions [49,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 70] (see Ta-  ble 1 for summary of the present key results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In this  regard, it is noted that, while we observed the ε-band  splitting in RbV3Sb5, it was absent in a recent study  by another group [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' This difference implies that the  type of 3D CDW at the surface is not robust even in the  same material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Possible key factors determining the type  of 3D CDW are carrier/chemical doping, cooling speed,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' [48, 49, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Another exotic property of AV3Sb5 is the unidirec-  tional 4 × 1 charge order that appears only in the Sb-  terminated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' According to STM studies, the 4 × 1  charge order is present in RbV3Sb5 and CsV3Sb5 but ab-  sent in KV3Sb5 [21, 38, 40, 41, 45–47, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Despite  such A-element dependence of the presence/absence of  the 4 × 1 charge order, the present result shows that  the FS topology and size in the Sb-terminated surface  are similar among AV3Sb5 and a good FS nesting condi-  tion with the 4 × 1 periodicity is not observed in any of  AV3Sb5, suggesting that the mechanism of the unidirec-  tional order may be beyond the framework with simple  FS nesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  In conclusion, we reported a systematic study on the  spatially-resolved band structure of AV3Sb5 by micro-  ARPES in combination with first-principles band struc-  ture calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  We have established the universal-  ity of the polar nature of cleaved surfaces characterized  by the opposite band energy shift between the A- and  Sb-terminated surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  We also found that the pres-  ence/absence and the microscopic structure of 3D CDW  depend on the surface polarity (termination) and the  A element, pointing to an exceptional sensitivity of 3D  CDW to the underlying electronic and chemical environ-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The present results provide foundation for under-  standing and manipulating exotic physical properties of  AV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported by JST-CREST (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  JP-  MJCR18T1),  JST-PRESTO  (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  JPMJPR18L7),  Grant-in-Aid for Scientific Research (JSPS KAKENHI  Grant Numbers JP21H04435 and JP20H01847), KEK-  PF (Proposal number: 2021S2-001), and the Sasakawa  Scientific Research Grant from the Japan Science Soci-  ety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  The work at Beijing Institute of Technology was  supported by the National Key R&D Program of China  (Grant  Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  2020YFA0308800,  2022YFA1403400),  三7  the Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  92065109), and the Beijing Natural Science Foundation  (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Z210006, Z190006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' acknowledges  support from GP-Spin at Tohoku University and JST-  SPRING (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' JPMJSP2114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' thanks the Analysis  & Testing Center at BIT for assistance in facility support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  APPENDIX A: NORMAL EMISSION  MEASUREMENT  Figure 6 shows the second-derivative ARPES inten-  sity measured at the normal-emission set up in the Rb-  terminated surface of RbV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The energy position of  the inner electron-band (α) bottom shows a periodic dis-  persion as a function of kz (see red dashed curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' note  that the intensity of the outer electron band is weak),  suggesting the bulk-band nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' From the observed pe-  riodicity, we have estimated the inner-potential value to  be V0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0 eV, which indicates that the kz ∼ 0 plane  is probed by 111-eV photons in RbV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' In KV3Sb5  and CsV3Sb5, the kz ∼ 0 plane is probed by hν = 114  eV and 106 eV, respectively, according to our previous  works [59, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  A  A  Γ  Γ  High  Low  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  kz (Å-1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Second-derivative intensity of the ARPES spectra  for RbV3Sb5 plotted as a function of kz and binding energy  obtained with photon energies of 90–150 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  APPENDIX B: SURFACE-TERMINATION  DEPENDENCE OF THE FERMI-SURFACE SIZE  Figure 7 shows side-by-side comparison of the ARPES  intensity plots at EF obtained in the Sb- and A-  terminated surfaces of KV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  7(a) and (b)],  RbV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 7(c) and 7(d)], and CsV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 7(e)  and 7(f)] (note that the data are identical to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' One can see that both the two circular FSs centered  at the Γ point in the A-terminated surface are larger than  the one in the Sb-terminated surface, in agreement with  the electron-doped character of the A-terminated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (b)  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  kx (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  kx (Å-1)  K  M  Γ  K  M  Γ  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  K term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (d)  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  kx (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  kx (Å-1)  K  M  Γ  K  M  Γ  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Rb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (f)  ky (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (e)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  kx (Å-1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  kx (Å-1)  K  M  Γ  K  M  Γ  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Cs term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  KV3Sb5  RbV3Sb5  CsV3Sb5  High  Low  High  Low  High  Low  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (a) and (b) Fermi-surface mapping in the Sb- and  A-terminated surfaces, respectively, in KV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (c) and (d)  Same as (a) and (b) but in RbV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (e) and (f) Same as (a)  and (b) but measured in CsV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  APPENDIX C: COMPARISON OF THE BAND  DISPERSIONS BETWEEN SB- AND  A-TERMINATED SURFACES  Figure 8 shows comparison of the band dispersions  along the KMK cut between the Sb- and A-terminated  surfaces of KV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 8(a) and 8(b)], RbV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  8(c) and 8(d)], and CsV3Sb5 [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 8(e) and 8(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' The  data are identical to those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 3, but the energy shift  of the Dirac point is better recognized by side-by-side  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' One can also see that the δ band near EF in  the A-terminated surface is shifted downward compared  8  to that in the Sb-terminated surface (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (b)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  High  Low  K term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (c)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  High  Low  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (e)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  High  Low  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (d)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  High  Low  Rb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  KV3Sb5  High  Low  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (a)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  Sb term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  RbV3Sb5  CsV3Sb5  Binding Energy (eV)  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  (f)  K  M  K  Γ  Γ  β  γ  δ  ε  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='5  ky (Å-1)  High  Low  Cs term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' (a) and (b) ARPES intensity plots at T = 8 K mea-  sured along ky at kx = −π (KMK cut) for the Sb- and A-  terminated surfaces, respectively, in KV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  (c) and (d)  Same as (a) and (b), respectively, but measured in RbV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  (e) and (f) Same as (a) and (b), respectively, but measured  in CsV3Sb5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Black arrows indicate the energy position of the  Dirac point at the K point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Dotted lines are the δ-band dis-  persion determined in the Sb-terminated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  106, 236802 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 106, 236803 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Nugroho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Akebi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Nakat-  suji, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' 16, 1090 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Nayak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Fischer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Sun, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Yin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Guguchia, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Lian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Jiang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Neupert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Sun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Wei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Antony, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Ehrlich, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Krieg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Song, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+page_content=' Yin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Gong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Tu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Lei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Cheng,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Hasan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Yan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Thomale, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Lee,  and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Miao, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' 13, 6348 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='  [73] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Zhao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Ortiz, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Oey, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Wilson,  and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content=' Zeljkovic, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
+page_content='15057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFLT4oBgHgl3EQfQy8p/content/2301.12034v1.pdf'}
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+Cluster Formation and Relaxation in Particle Rafts under Uniform Radial Expansion
+Kh´a-ˆI Tˆo
+The Department of Physics and the James Franck and Enrico Fermi Institute
+The University of Chicago IL 60637
+(Dated: January 10, 2023)
+Particle rafts floating at an air-liquid interface exhibit a variety of behaviors when interfacial
+flow is introduced. Motivated by previous experimental observations, the failure pattern of particle
+rafts under uniform radial expansion is reported in this paper. The expansion process is specifically
+designed to expand the system affinely in the radial direction and to keep the velocity gradient
+constant throughout. A strong resemblance to the results of particle rafts under uniform uniaxial
+expansion [1] is found. The size of the cluster emerging as the rafts are pulled apart scales inversely
+with the pulling velocity. This is a result of two competing velocities: the inter-particle separation
+speed provided by the flow and a size-dependent relaxation speed for clustering. A model, generalized
+from a one-dimensional linear (in)stability calculation, is in agreement with the failure morphology
+found for this radially expanded system. Nonlinear relaxation and particle rearrangement is observed
+after the initial clustering occurs.
+This is a feature unique to a two-dimensional system.
+With
+its easily accessible particle dynamics at the microscopic level, this system provides insights into
+the morphology controlled by two competing mechanisms in two or higher dimensions and across
+different scales.
+I.
+INTRODUCTION
+Sub-millimeter particles floating at an air-liquid inter-
+face can aggregate into particle rafts. The inter-particle
+attraction arises from the coupling between the parti-
+cles and the fluid interface [2–4]. The resulting particle
+rafts can be treated like two-dimensional solids [5]. For
+example, compressing the raft from two of its opposing
+ends at small strain can induce an instability in which
+the particles buckle out of the plane [6–8].
+Likewise,
+pulling the raft from its opposing ends quasi-statically
+can induce ductile deformation [9]. The underlying fluid
+plays a crucial role. Not only, does it produce the cap-
+illary forces that create the inter-particle attractions, it
+also provides substantial expansion or compression forces
+when the fluid surface area is varied sufficiently rapidly.
+Thus the fluid interface is a valuable tool with which to
+control particle-raft dynamics when the flow is relevant.
+Different regimes of particle-raft behavior have been
+investigated. A previous study [1] explored the failure
+of particle rafts under expansion with a uniformly ex-
+panding metric in one dimension. This produced cracks
+throughout the material. By keeping the velocity gradi-
+ent uniform along the extensional axis, the failure was
+found to be homogeneous with no single crack domi-
+nating the resultant morphology. A smooth change in
+the distance between neighboring cracks was observed as
+the pulling speed was varied; the failure pattern evolved
+from an almost unperturbed, densely-packed raft at low
+shear rates to a uniformly distributed set of rifts sepa-
+rating small clusters of particles at high shear rates. In
+these experiments, the orientations of the particle clus-
+ters were found to be nearly isotropic; despite having a
+specific pulling direction, the clusters were oriented with
+no strong variation with respect to the extension axis.
+This suggests that the direction of the applied shear plays
+no significant role in determining the failure pattern of
+particle rafts.
+In other studies, the rafts were pulled apart radially so
+that the extension was in all directions simultaneously.
+In one case, this expansion was applied by adding surfac-
+tant near its center in order to impose a surface-tension
+gradient across the raft [10, 11]. In another study, ad-
+vection was generated by pumping water into a funnel
+so that the raft expanded as the water level rose [12].
+In both cases, as the systems evolved under the applied
+forces, wide cracks appeared preferentially near the cen-
+ter of the rafts. The morphology in these cases is thus
+significantly different from what was found in the one-
+dimensional experiments.
+This raises a puzzle as to what aspects of the expansion
+gives rise to the differences observed between the radial
+and linear pulling geometries.
+To address this puzzle,
+this paper examines the failure morphology of particle
+rafts pulled apart under radial expansion using a proto-
+col that more closely aligns with that used in the one-
+dimensional experiments. In particular, the experiments
+are designed to produce a nearly uniform velocity gradi-
+ent everywhere on the air-water interface. I verify the ve-
+locity field using Particle Image Velocimetry (PIV). The
+results of these experiments are more similar to the one-
+dimensional pulling studies suggesting the importance of
+minimizing velocity gradients in the expansion dynamics.
+In order to analyze these results, I generalize two di-
+mensions to an (in)stability calculation used to determine
+the cluster sizes, i.e., the distance between rifts, in the
+one-dimensional (1D) situation [1]. With no further as-
+sumptions on the elastic properties of the aggregate, the
+calculation showed that the morphology was determined
+by a cross-over phenomenon due to the competition be-
+tween the rate of separation (caused by the pulling ve-
+locity) and the rate of relaxation (caused by the inter-
+particle interactions) at different cluster sizes.
+In this
+model, the average cluster length scales inversely with
+arXiv:2301.03170v1  [cond-mat.soft]  9 Jan 2023
+
+2
+the velocity, in good agreement with the experimental
+results.
+In addition to exploring the failure pattern of parti-
+cle rafts under a different expansion metric, the present
+study is also inspired by the recognition that the study
+of structure formation under isotropic competing mech-
+anisms is relevant in many systems across different
+scales [13].
+At the largest scale, the competition be-
+tween gravity and cosmological expansion leads to the
+cluster formation in the universe [14]. In particle rafts,
+the capillary attraction is asymptotically inversely pro-
+portional to the distance between particles, which has
+the same form as two-dimensional gravity.
+Therefore,
+working with an radial-expansion metric provides a two-
+dimensional table-top experiment for studying this type
+of phenomenon.
+At much smaller scales, the cluster-
+ing of colloids [15, 16] and the microbial growth on liq-
+uid substrate [17] both have isotropic competing inter-
+action/flow. Two-dimensional particle aggregates under
+radial expansion can offer a macroscopic platform (with
+access to the microscopic entities involved) to study the
+morphology controlled by these competing processes.
+II.
+EXPERIMENTAL DETAILS
+The particle rafts consist of polyethylene spheres float-
+ing at an air-water interface.
+The polyethlene parti-
+cles have density 1080 kgm−3 and are a mixture of
+two diameter ranges:
+d = 550 ± 50µm (small) and
+d = 655 ± 55µm (large). Polydisperse packings are made
+by mixing roughly equal volumes of these two sizes. The
+particles naturally aggregate into a raft due to the lat-
+eral capillary attraction, also known as the “Cheerios”
+effect [2–4]. A monolayer of densely-packed particles is
+prepared by gathering the particles manually into the
+center of the apparatus as shown in the left panel of
+Fig. 1(A). The rafts are compacted by slight vibrations
+of the interface and have an initial packing fraction of
+0.72 ± 0.01. The particle raft floats on deionized water
+which has density ρw = 998 kgm−3; dynamic viscosity
+ηw = 0.95 mPas and surface tension, σw = 0.073 Nm−1
+at 22◦C.
+To create an isotropic flow, a two-dimensional radial
+expander is custom-built to create a coordinated move-
+ment of twelve nodes going radially outward simulta-
+neously. A schematic of the experimental apparatus is
+shown in Fig. 1A. The nodes are connected by twelve
+moving Mylar strips to form an expanding twelve-sided
+polygon. The Mylar strips are inserted vertically into the
+water as shown. Because Mylar is slightly hydrophilic,
+the particles rafts are enclosed inside the boundaries but
+do not make direct contact with the boundaries. This is
+shown in the middle panel of Fig. 1A. The rafts are thus
+pulled only by the flow of the fluid which rises incremen-
+tally and uniformly everywhere inside the boundaries as
+the Mylar boundaries are extended outward radially. The
+fluid interface can appropriately be considered as an ex-
+panding metric on which the particles float. As the nodes
+simultaneously move outward, as shown in “Top View”
+in Fig. 1A, an approximately isotropic flow is generated
+by the moving Mylar strips.
+(See Appendix A about
+how Mylar strips are attached and how the expander is
+driven.)
+The distance between the midpoint of each side to
+the center of the dodecagon is L0/2 and L0 ≈ 61 mm.
+Each side moves outwards away from the center at the
+same pulling speed V/2. The Particle Image Velocimetry
+(PIV) measurement at various V is shown in Appendix B.
+This allows the flow field u(x, y) to be described as an
+affine expansion:
+u(x, y) = V
+L (xˆx + yˆy)
+(1)
+where the origin is the center of the raft and x and y are
+the two orthogonal coordinates in the plane. The flow is
+radially outward away from the center.
+The pulling speed, V , is varied from 2.8 mm/s to
+260 mm/s. The raft diameter increases as L = L0 + V t
+and the stretch ratio is defined as λ = L/L0. The initial
+rate of stretch, ˙λ0 = V/L0, is varied from 5 × 10−2 s−1
+to 4.6 s−1. The experiments are stopped at a maximum
+stretch ratio λmax ∼ 1.75.
+I use the method described in [1] to measure the clus-
+ter length, ℓ, inside each raft.
+This requires counting
+the number of pixels between rifts on each line of the
+raster-scanned image. The only complication is due to
+the difference in pixelization at angles different from the
+rectangular coordinates of the images produced by the
+camera.
+III.
+EXPERIMENTAL RESULTS
+As the twelve-sided boundary moves outward so that
+the diameter increases at velocity V , the raft of particles
+is stretched by the underlying flow so that micro-cracks,
+or rifts, begin to form as shown in Fig. 1B. At interme-
+diate to high V , the rifts emerge soon after the onset
+of expansion.
+They are distributed diffusely through-
+out the entire raft and grow larger with time. The ex-
+periments are stopped at a stretch ratio (λmax ∼ 1.75).
+After their formation and initial growth, the same rifts
+remain up to a larger stretch ratio; once a rift is formed,
+it does not collapse at later times. This can be observed
+in the movies included in the Supplementary Informa-
+tion (SI). Likewise, the particles in between the initially-
+formed rifts remain connected. These clusters of particles
+maintain their configurations and are distributed homo-
+geneously throughout the entire system with a character-
+istic size. The average size of these clusters of particles,
+ℓ, is strongly dependent on the pulling speed V , as shown
+in Fig. 1B, C.
+When V is small enough, the raft shape is only very
+slightly perturbed at the initiation of expansion and re-
+mains unchanged afterwards. There is a small drift in the
+
+3
+Side View
+𝑉	 = 2.8	𝑚𝑚/𝑠
+𝑉	 = 26	𝑚𝑚/𝑠
+𝑉	 = 260	𝑚𝑚/𝑠
+B
+A
+Top View
+C
+Expander
+FIG. 1. Failure morphology at different pulling velocity V . (A) A schematic of the experimental apparatus. Left: The side
+view of the whole apparatus. A radial expander inserted into water creates an nearly-radial flow to stretch the particle raft
+enclosed inside. Middle: The side view of the boundary of the Mylar strip boundary. Right: The top view of the expander at
+early and late time. The gray arms are connected and move coherently to make the twelve orange inner nodes move outward.
+The inner nodes are connected by Mylar strips painted in blue, which set the boundaries of the system. The dotted black lines
+show the weight tied to the Mylar strips to keep them tout during the motion. (B, C) Snapshots and zoomed-in images of
+experiments at a fixed stretch ratio λ = 1.5 for different pulling velocities V .
+position due to secondary flows but the internal structure
+is undisturbed. In this low-velocity regime, no significant
+rifts can be observed and the cluster width, ℓ, remains
+close to the initial system size, L0.
+With increasing V , the number of rifts increases while
+the cluster size, ℓ, decreases. The cluster size, ℓ decreases
+until it reaches the size of a single particle at large veloc-
+ities, as shown in the right column of Fig. 1B. Figure 1C
+shows how the internal features change with increasing
+V : a single relatively closely-packed structure at low V ,
+evolves upon increasing V into separate clusters consist-
+ing of only a few (sometimes only one) particles.
+The average cluster length, ⟨ℓ⟩, is obtained by aver-
+aging the measurement at all different angles.
+This is
+because there is no specific tensile axis in this radial ex-
+pansion system. The percentage standard deviation of
+this average quantity is smaller than 3% for all veloci-
+ties, which is comparable to the fluctuation due to rotat-
+ing pixelated images. This indicates that the variation in
+cluster orientations is very low and below the uncertainty
+level of detection.
+Figure 2 shows the angular-averaged distribution of
+
+4
+FIG. 2. Angle-averaged distribution of cluster lengths, ℓ, for
+different velocities at fixed stretch ratio, λ = 1.5. The most
+dominant length, that is the probability of cluster length ℓ
+multiplied by ℓ normalized by ⟨ℓ⟩ = � ℓP(ℓ), is plotted versus
+ℓ with three pulling speeds, V : 2.8 mm/s (blue), 26 mm/s
+(green) and 260 mm/s (purple).
+cluster sizes at a liquid stretch ratio λ = 1.5 for different
+pulling velocities.
+The ordinate is the most dominant
+cluster length, ℓP(ℓ)/⟨ℓ⟩, where P(ℓ) is the probability of
+finding a cluster of length ℓ and the average cluster length
+⟨ℓ⟩ = � ℓP(ℓ).
+Since the statistics with small cluster
+lengths always outnumber those with larger ℓ, I plot the
+most dominant length, that is how much material has
+cluster length ℓ. This can help better identify the cluster
+distribution at each velocity, V .
+At small V , a large
+portion of the distribution remains close to the initial
+size of the raft, L0. The distribution shifts to smaller ℓ
+as V increases. At the highest pulling speed, V , most
+clusters have lengths between d and 2d.
+Figure 3A shows the average cluster length ⟨ℓ⟩ versus
+pulling speed V at a fixed value of λ = 1.5, for all the ex-
+periments. One can see that ⟨ℓ⟩ decreases monotonically
+with increasing V and levels off at high velocity. Because
+a cluster cannot be significantly larger than the initial
+system size, L0, or smaller than a particle diameter, d.
+The data can be interpolated between these two extremes
+using a similar form as was used in the one-dimensional
+pulling experiments [1]:
+⟨ℓ⟩ =
+1
+aV b + 1/(L0/
+√
+2 − dmax) + dmax.
+(2)
+This assumes a power-law dependence of the average
+cluster length away from the two extremes. The differ-
+ence from the one-dimensional fitting form is that initial
+raft size L0 is scaled by
+√
+2 because the cluster lengths
+are measured in a square box of size L0/
+√
+2 located in
+the center of the raft to avoid complexity caused by the
+round shape of the raft.
+The fit gives us a = (9 ± 2) × 103 and b = 1.20 ± 0.05.
+This result is similar to the results in uniaxial expansion.
+FIG. 3.
+Averaged cluster length, ⟨ℓ⟩ = � ℓP(ℓ), versus
+pulling speed, V at fixed stretch ratio λ = 1.5.
+(A) The
+blue points are measurements of ⟨ℓ⟩ versus V . The blue curve
+is a fit of Eq. 2. The red solid line indicates L0/
+√
+2, which
+is the size of the measuring box. The red dashed line shows
+the largest particle diameter dmax. The blue curve is a fit to
+Eq. 2. (B) ⟨ℓ⟩ is plotted versus V for both radial (blue dots)
+and uniaxial (green squares) experiments at similar expansion
+condition (λ = 1.5 and ε = 0.5). The black line shows the
+theoretical prediction for water from Ref. [1].
+The average cluster length, ⟨ℓ⟩, scales inversely with the
+pulling velocity V .
+IV.
+DISCUSSION
+Figure 3B compares the results from these radial ex-
+pansion experiments with those found in the 1D exper-
+iments. Both data sets are shown at a similar time in
+the expansion: the liquid strain ε = 0.5 for 1D experi-
+ment and λ = 1.5 for radial experiment both occur when
+the system size is 1.5 times larger than the original size
+of the raft. The experiments under radial, isotropic ex-
+pansion give very similar results to the 1D experiments.
+
+0.02
+V=2.8mm/s
+V=26mm/s
+V=260mm/s
+P()/([)
+0.01
+0.00
+0
+20
+40
+60
+[w] A
+10-2
+[mm]
+10-3
+10-2
+10-1
+V[ms-1]
+B
+Radial 入 = 1.5
+Linear = 0.5
+1D model
+[mm]
+10-3
+10-2
+10-1
+V[ms-1]5
+This suggests that the experiments with isotropic expan-
+sion can in fact be generalized from the analysis used in
+one-dimension.
+To understand this similarity, let us reexamine the flow
+field described in Eq. 1. The velocity field can be ex-
+panded around any arbitrary point, (x0, y0), within the
+boundaries:
+u(x, y) = u0 + V
+L ((x − x0)ˆx + (y − y0)ˆy)
+where u0 is the velocity at (x0, y0). Independent of the
+position, (x0, y0), the velocity field expands radially out-
+ward from that point; all points are equivalent and every
+particle feels the surrounding fluid retreating at the same
+constant velocity in all directions. Thus, the local mo-
+tion can be reduced to multiple one-dimensional chains
+as treated in the case of uniaxial expansion.
+The large cracks that were observed near the raft cen-
+ters in previous experiments where the rafts were pulled
+radially in two-dimensions [10–12], are not present in the
+data presented here. The difference in the way the pulling
+of the raft is accomplished is likely the reason for this
+difference. The present experiment was designed to min-
+imize higher-order gradients in the flow.
+When using
+the funnel method [12], I also observed the formation of
+large cracks which I attributed to gradients in the veloc-
+ity field produced by the influx of liquid near the center
+of the raft.
+To understand the two-dimensional effects intrinsic to
+this systems, I analyze both the uniaxial and radial ex-
+pansion datasets. As shown in Fig. 4A, the time evo-
+lution of the cluster sizes shows a similar trend in both
+cases. Here I focus on the time evolution in the uniax-
+ial expansion experiments because they have the greater
+range. Only the behavior for liquid strain ε ≥ 1.0 are
+examined because the cluster sizes at early times is com-
+plicated by limitations of image processing.
+A square
+observation window of size Lx0 located at the center of
+the rafts is also chosen to avoid dealing with the irregular
+edges of the rafts.
+At low V , the cluster size is nearly independent of ε. At
+higher pulling speed, however, the average cluster sizes
+consistently decrease with increasing ε.
+This suggests
+non-linear, late-time relaxation of the raft.
+One possibility for this is that, after the initial expan-
+sion, the particles which are further apart experience a
+smaller velocity gradient.
+However, the particles that
+are initially more compact continue to experience strong
+attraction from their neighbors; they rearrange so the
+particle clusters become more compact. Another possi-
+bility is that the clusters simply continue to break up as
+time goes on.
+To understand the cause of the decrease in cluster sizes,
+I calculate the particle ratio inside the clusters that have
+been identified. The particle ratio is defined as the area
+of particles inside the clusters divided by the total area
+of the clusters. This quantity indicates if the voids inside
+B
+A
+C
+FIG. 4.
+Time evolution of cluster sizes in both radial and
+uniaxial experiments. (A) Average cluster length, ⟨ℓ⟩, versus
+liquid stretch ratio λ or strain ε at all V . The results from ra-
+dial and uniaxial expansion experiments are shown in crosses
+and dots respectively.
+The color from dark purple to light
+orange represent results from fast to slow V , and span the
+entire pulling velocity range in both experiments.) (B) The
+ratio of particles occupied in the cluster versus strain ε for
+uniaxial experiments. The results from four fastest pulling
+speeds, V = 20 mm/s (orange) to 200 mm/s (purple), are
+presented. (C) The change in ⟨ℓ⟩ , divided by ⟨ℓ⟩ at ε = 1.0
+(solid circles), and the inverse of normalized number of clus-
+ters, nc, scaled by nc at ε = 1.0 (empty circles), are plotted
+versus ε.
+
+1.0 -
+1.0
+0.8
+0.8
+(0'T = 30)/(0)
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+Strain ε0.76
+0.74
+ratio
+0.72
+Particle
+0.70
+0.68
+0.66
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+Strain Strain
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+wj
+10~3
+1.50
+1.75
+2.00
+2.25
+2.50
+2.75
+3.00
+3.25
+Stretch ratio 入6
+the clusters are shrinking in time. As shown in Fig. 4B,
+for four higher pulling velocities, the particle ratios all
+increase in time.
+This suggests that rearrangement of
+neighboring particles inside a cluster is indeed taking
+place.
+On the other hand, Fig. 4C examines whether the clus-
+ters are breaking up with time. The solid dots represent
+the cluster lengths ⟨ℓ⟩ normalized by ⟨ℓε=1.0⟩, the mea-
+surement at ε = 1.0 versus strain.
+I also plot the in-
+verse of the normalized number of clusters, nc (scaled by
+nc,ε=1.0) as empty circles. nc is the number of clusters
+in the observation region normalized by the total area
+of particles. This quantity shows how many clusters are
+formed per unit area of materials. If the inverse of nc
+matches the change in cluster length, this implies that
+the change in the number of clusters is the main con-
+tributor to the change in cluster sizes, which is the case
+at V = 20 mm/s Moreover, the overall change in ⟨ℓ⟩ is
+small because the particles are already at a lower-energy
+configuration to start with. The slight increase in n−1
+c
+after ε = 1.7, that is the decrease in the number of clus-
+ters, suggests that a larger-scale coalescence has occurred
+as the global shear became very weak.
+At V = 200 mm/s, a significant difference between
+n−1
+c
+and ⟨ℓ⟩ can be observed. The average cluster length
+decreases by more than 20% while the inverse of nc
+only changed by 10%. A substantial rearrangement be-
+tween neighboring particles is therefore responsible for
+this change, as shown in Fig. 4B as well.
+V.
+CONCLUSIONS
+In this paper, the morphology of particle rafts float-
+ing at an air-water interface is studied as the underlying
+fluid is forced to flow radially outward at a uniform con-
+stant velocity gradient.
+The experiments are designed
+to minimize any high-order velocity gradients in the sys-
+tem. By examining the flow field of the radial expansion
+with particle image velocimetry (PIV), the local expan-
+sion metric is found to be uniform inside the apparatus.
+As was found in the case of uniaxial expansion described
+in Ref. [1], the failure morphology due to the outward
+flow varies smoothly as a function of the fluid expan-
+sion velocity. When the velocity is low, the rafts remain
+intact throughout the expansion without the formation
+of visible internal cracks.
+For larger expansion veloci-
+ties, rifts open up inside the rafts.
+The rifts are dis-
+tributed diffusely with a spacing between them that de-
+creases with increasing velocity. At the largest velocities
+used in the experiment, the separation decreases until
+the cluster size, that is the length between rifts, becomes
+only slightly larger than the size of a single particle. The
+measured average cluster sizes are very close to that of
+the one-dimensional expansion experiment and follow the
+same scaling behavior despite the fact that the geometry
+of the expansion metric is different. This indicates that
+the 1D (in)stability model that was used to describe the
+relaxation and cluster formation in [1] can be generalized
+to two-dimensions and is sufficient to explain the radial
+experiment.
+By measuring the time evolution of the clusters, a non-
+linear relaxation is found at a later time; after the initial
+formation of clusters, the clusters start to shrink in sizes.
+The decrease in particle ratio inside the clusters shows
+that the particles continue to rearrange and become more
+compact even while the overall outward expansion con-
+tinues. This can be understood because, once the initial
+expansion rate sets the dominant cluster length, the ve-
+locity gradient between clusters is no longer that used
+initially. Because the particles are farther apart (in the
+rifts), the rifts will increase in size faster than the spaces
+between the more densely packed particles. Moreover,
+the global velocity gradient is also decreasing monoton-
+ically as the system size becomes larger. The particles
+inside an individual cluster will rearrange according to
+the competition between the local relaxation rate and
+the local shear rate set by neighboring particles. This
+later-time aspect of the relaxation is not captured by the
+one-dimensional linear (in)stability analysis.
+This experiment provides a flexible platform for study-
+ing how morphology can form by the competition be-
+tween two isotropic mechanisms. There are many other
+systems that share this common attribute. As pointed
+out in [1], an obvious analog is cosmological expansion
+[14].
+In this table top experiment, the interaction be-
+tween particles has the same form as gravity in two di-
+mensions. Other examples include gel formation[15] and
+structure formation by colloids in polymer solutions[16].
+This two dimensional system gives us a direct access to
+the clustering morphology with minimal imaging com-
+plexity which would be troublesome in three dimensional
+systems. In addition, without tuning the details of the
+potentials chemically or at a small scale, one of the com-
+peting processes can be easily manipulated by changing
+the mechanical driving protocol, while the other can be
+controlled by changing specific particle-liquid or particle-
+particle interactions.
+VI.
+ACKNOWLEDGEMENT
+I thank T. A. Witten, V. Vitelli and M. M. Bandi
+for insightful discussion.
+I am deeply grateful to S.
+R. Nagel for his support and mentorship.
+This work
+was supported by the National Science Foundation (MR-
+SEC program NSF-DMR 2011854), by the Simons Foun-
+dation for the collaboration Cracking the Glass Prob-
+lem Award #348125 and by Government Scholarship to
+Study Abroad by the Ministry of Education in Taiwan.
+Appendix A: Details of the Radial Expander
+The design of the radial expander used to create the
+flow with minimal velocity gradients is shown in Fig. 5.
+
+7
+FIG. 5. Details of the radial expander design. (A) Schemat-
+ics of the top view of the radial expander. The inner nodes
+are shown as orange and the outer nodes are green. Two of
+the inner (orange) nodes are connected to the opposite sides
+of a belt which is driven by a motor. The movement of the
+belt drives those two nodes in opposite directions. This opens
+(closes) the entire structure when the belt is rotated clockwise
+(counter-clockwise). Because of the geometry connecting all
+the nodes, pulling on just two of the inner nodes is sufficient
+for the entire structure to open smoothly. (B) A side view of
+the green and orange nodes. The green nodes are placed to
+ensure that the entire structure is stable and does not wobble
+vertically during the opening of the expander. The bottom
+of a green node is attached to a ball bearing so that it slides
+smoothly on a clear acrylic platform. Each orange node is
+a screw with a 3D-printed attachment at its tip.
+This at-
+tachment is composed of two closely-spaced columns that are
+threaded by two adjacent Mylar ribbons. These ribbons slide
+smoothly around the two columns as the expander is moving.
+Each ribbon is held in place by the attachments at two ad-
+jacent orange nodes. To keep the Mylar ribbon taut during
+movement of the expander, a weight is attached that pulls
+the ribbon outwards on the opposite side from where the par-
+ticles are located. The Mylar ribbons are partly submerged
+below the liquid surface. It is these ribbons which create the
+boundary for the particles as shown in Fig. 1A. (C) Top: A
+photograph of the angled top view of the expander. Bottom:
+The side view of the chips and Mylar strips before inserting
+into water.
+The twenty-four arms are connected by binding posts
+(green nodes), which allow them to rotate freely, as shown
+in the gray pieces in the Fig. 5A. Half of them are on the
+top (light gray) and half of them are at the bottom (dark
+gray).
+Two of the opposite inner nodes are each con-
+nected to the opposite side of a belt. As the belt is driven
+by a motor, a motion of the arms to move the twelve in-
+ner (orange) nodes outward is created. Ball bearings are
+glued underneath the binding posts (shown in Fig. 5B)
+to hold the expander and make it glide smoothly on the
+platform. The inner nodes are long screws, that go be-
+low the platform and a 3D-printed chip is attached to the
+tip. This chip, as shown in Fig. 5B, has two threaded
+cylinders with a 1.22 mm slit in between them. Each of
+the cylinder serves as a column for one side of the Mylar
+strip. Each Mylar strip forms a triangle and is attached
+to a weight at the corner against the side connecting ad-
+jacent nodes. This provides sufficient tension to the strip
+during the motion to create an outward-going boundary.
+Appendix B: Particle Image Velocimetry
+FIG. 6. The Particle Image Velocimetry results of the radial
+expander across whole velocity range. The magnitude of the
+velocity u normalized by the pulling speed V is plotted versus
+the position relative to the center, normalized by the radius
+of the raft, L0/2.
+A linear relation can be observed in all
+V , indicating an radially affine flow field can be created by
+the expander.
+The measurements near the boundaries are
+excluded due to the difficulty of image processing.
+To measure the expansion of the liquid surface due to
+the extension of the pullers, the Particle Image Velocime-
+try (PIV) measurement on water at V = 2.8, 26 and 260
+mm/s. The result is shown in Fig. 6. A layer of light,
+non-interacting floating particles are spread sparsely on
+the water surface prior to expansion. By tracking the mo-
+tion of these particles and computing the correlation be-
+tween adjacent frames, the underlying fluid flow is shown
+to lead to an uniform radial expansion of the surface,
+as described in 1; the spacing of the particles increases
+proportional to their relative position. Fig. 6 shows the
+gradient of the velocity is constant radially.
+
+V=2.8mm/s
+0.6
+V=26mm/s
+V=260mm/s
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+r/(Lo/2)8
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+
diff --git a/VtE1T4oBgHgl3EQfbQQ6/content/tmp_files/load_file.txt b/VtE1T4oBgHgl3EQfbQQ6/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,515 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf,len=514
+page_content='Cluster Formation and Relaxation in Particle Rafts under Uniform Radial Expansion  Kh´a-ˆI Tˆo  The Department of Physics and the James Franck and Enrico Fermi Institute  The University of Chicago IL 60637  (Dated: January 10, 2023)  Particle rafts floating at an air-liquid interface exhibit a variety of behaviors when interfacial  flow is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Motivated by previous experimental observations, the failure pattern of particle  rafts under uniform radial expansion is reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The expansion process is specifically  designed to expand the system affinely in the radial direction and to keep the velocity gradient  constant throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A strong resemblance to the results of particle rafts under uniform uniaxial  expansion [1] is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The size of the cluster emerging as the rafts are pulled apart scales inversely  with the pulling velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This is a result of two competing velocities: the inter-particle separation  speed provided by the flow and a size-dependent relaxation speed for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A model, generalized  from a one-dimensional linear (in)stability calculation, is in agreement with the failure morphology  found for this radially expanded system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Nonlinear relaxation and particle rearrangement is observed  after the initial clustering occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This is a feature unique to a two-dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  With  its easily accessible particle dynamics at the microscopic level, this system provides insights into  the morphology controlled by two competing mechanisms in two or higher dimensions and across  different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  INTRODUCTION  Sub-millimeter particles floating at an air-liquid inter-  face can aggregate into particle rafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The inter-particle  attraction arises from the coupling between the parti-  cles and the fluid interface [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The resulting particle  rafts can be treated like two-dimensional solids [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' For  example, compressing the raft from two of its opposing  ends at small strain can induce an instability in which  the particles buckle out of the plane [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Likewise,  pulling the raft from its opposing ends quasi-statically  can induce ductile deformation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The underlying fluid  plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Not only, does it produce the cap-  illary forces that create the inter-particle attractions, it  also provides substantial expansion or compression forces  when the fluid surface area is varied sufficiently rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Thus the fluid interface is a valuable tool with which to  control particle-raft dynamics when the flow is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Different regimes of particle-raft behavior have been  investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A previous study [1] explored the failure  of particle rafts under expansion with a uniformly ex-  panding metric in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This produced cracks  throughout the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' By keeping the velocity gradi-  ent uniform along the extensional axis, the failure was  found to be homogeneous with no single crack domi-  nating the resultant morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A smooth change in  the distance between neighboring cracks was observed as  the pulling speed was varied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' the failure pattern evolved  from an almost unperturbed, densely-packed raft at low  shear rates to a uniformly distributed set of rifts sepa-  rating small clusters of particles at high shear rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' In  these experiments, the orientations of the particle clus-  ters were found to be nearly isotropic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' despite having a  specific pulling direction, the clusters were oriented with  no strong variation with respect to the extension axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This suggests that the direction of the applied shear plays  no significant role in determining the failure pattern of  particle rafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  In other studies, the rafts were pulled apart radially so  that the extension was in all directions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  In one case, this expansion was applied by adding surfac-  tant near its center in order to impose a surface-tension  gradient across the raft [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' In another study, ad-  vection was generated by pumping water into a funnel  so that the raft expanded as the water level rose [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  In both cases, as the systems evolved under the applied  forces, wide cracks appeared preferentially near the cen-  ter of the rafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The morphology in these cases is thus  significantly different from what was found in the one-  dimensional experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This raises a puzzle as to what aspects of the expansion  gives rise to the differences observed between the radial  and linear pulling geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  To address this puzzle,  this paper examines the failure morphology of particle  rafts pulled apart under radial expansion using a proto-  col that more closely aligns with that used in the one-  dimensional experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' In particular, the experiments  are designed to produce a nearly uniform velocity gradi-  ent everywhere on the air-water interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' I verify the ve-  locity field using Particle Image Velocimetry (PIV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The  results of these experiments are more similar to the one-  dimensional pulling studies suggesting the importance of  minimizing velocity gradients in the expansion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  In order to analyze these results, I generalize two di-  mensions to an (in)stability calculation used to determine  the cluster sizes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=', the distance between rifts, in the  one-dimensional (1D) situation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' With no further as-  sumptions on the elastic properties of the aggregate, the  calculation showed that the morphology was determined  by a cross-over phenomenon due to the competition be-  tween the rate of separation (caused by the pulling ve-  locity) and the rate of relaxation (caused by the inter-  particle interactions) at different cluster sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  In this  model, the average cluster length scales inversely with  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='03170v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='soft]  9 Jan 2023  2  the velocity, in good agreement with the experimental  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  In addition to exploring the failure pattern of parti-  cle rafts under a different expansion metric, the present  study is also inspired by the recognition that the study  of structure formation under isotropic competing mech-  anisms is relevant in many systems across different  scales [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  At the largest scale, the competition be-  tween gravity and cosmological expansion leads to the  cluster formation in the universe [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' In particle rafts,  the capillary attraction is asymptotically inversely pro-  portional to the distance between particles, which has  the same form as two-dimensional gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Therefore,  working with an radial-expansion metric provides a two-  dimensional table-top experiment for studying this type  of phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  At much smaller scales, the cluster-  ing of colloids [15, 16] and the microbial growth on liq-  uid substrate [17] both have isotropic competing inter-  action/flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Two-dimensional particle aggregates under  radial expansion can offer a macroscopic platform (with  access to the microscopic entities involved) to study the  morphology controlled by these competing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  EXPERIMENTAL DETAILS  The particle rafts consist of polyethylene spheres float-  ing at an air-water interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The polyethlene parti-  cles have density 1080 kgm−3 and are a mixture of  two diameter ranges:  d = 550 ± 50µm (small) and  d = 655 ± 55µm (large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Polydisperse packings are made  by mixing roughly equal volumes of these two sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The  particles naturally aggregate into a raft due to the lat-  eral capillary attraction, also known as the “Cheerios”  effect [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A monolayer of densely-packed particles is  prepared by gathering the particles manually into the  center of the apparatus as shown in the left panel of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The rafts are compacted by slight vibrations  of the interface and have an initial packing fraction of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The particle raft floats on deionized water  which has density ρw = 998 kgm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' dynamic viscosity  ηw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='95 mPas and surface tension, σw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='073 Nm−1  at 22◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  To create an isotropic flow, a two-dimensional radial  expander is custom-built to create a coordinated move-  ment of twelve nodes going radially outward simulta-  neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A schematic of the experimental apparatus is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The nodes are connected by twelve  moving Mylar strips to form an expanding twelve-sided  polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The Mylar strips are inserted vertically into the  water as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Because Mylar is slightly hydrophilic,  the particles rafts are enclosed inside the boundaries but  do not make direct contact with the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This is  shown in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The rafts are thus  pulled only by the flow of the fluid which rises incremen-  tally and uniformly everywhere inside the boundaries as  the Mylar boundaries are extended outward radially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The  fluid interface can appropriately be considered as an ex-  panding metric on which the particles float.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' As the nodes  simultaneously move outward, as shown in “Top View”  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1A, an approximately isotropic flow is generated  by the moving Mylar strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  (See Appendix A about  how Mylar strips are attached and how the expander is  driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=')  The distance between the midpoint of each side to  the center of the dodecagon is L0/2 and L0 ≈ 61 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Each side moves outwards away from the center at the  same pulling speed V/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The Particle Image Velocimetry  (PIV) measurement at various V is shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This allows the flow field u(x, y) to be described as an  affine expansion:  u(x, y) = V  L (xˆx + yˆy)  (1)  where the origin is the center of the raft and x and y are  the two orthogonal coordinates in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The flow is  radially outward away from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The pulling speed, V , is varied from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='8 mm/s to  260 mm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The raft diameter increases as L = L0 + V t  and the stretch ratio is defined as λ = L/L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The initial  rate of stretch, ˙λ0 = V/L0, is varied from 5 × 10−2 s−1  to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='6 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The experiments are stopped at a maximum  stretch ratio λmax ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  I use the method described in [1] to measure the clus-  ter length, ℓ, inside each raft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This requires counting  the number of pixels between rifts on each line of the  raster-scanned image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The only complication is due to  the difference in pixelization at angles different from the  rectangular coordinates of the images produced by the  camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  EXPERIMENTAL RESULTS  As the twelve-sided boundary moves outward so that  the diameter increases at velocity V , the raft of particles  is stretched by the underlying flow so that micro-cracks,  or rifts, begin to form as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' At interme-  diate to high V , the rifts emerge soon after the onset  of expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  They are distributed diffusely through-  out the entire raft and grow larger with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The ex-  periments are stopped at a stretch ratio (λmax ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  After their formation and initial growth, the same rifts  remain up to a larger stretch ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' once a rift is formed,  it does not collapse at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This can be observed  in the movies included in the Supplementary Informa-  tion (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Likewise, the particles in between the initially-  formed rifts remain connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' These clusters of particles  maintain their configurations and are distributed homo-  geneously throughout the entire system with a character-  istic size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The average size of these clusters of particles,  ℓ, is strongly dependent on the pulling speed V , as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  When V is small enough, the raft shape is only very  slightly perturbed at the initiation of expansion and re-  mains unchanged afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' There is a small drift in the  3  Side View  𝑉  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='8 𝑚𝑚/𝑠  𝑉  = 26 𝑚𝑚/𝑠  𝑉  = 260 𝑚𝑚/𝑠  B  A  Top View  C  Expander  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Failure morphology at different pulling velocity V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' (A) A schematic of the experimental apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Left: The side  view of the whole apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A radial expander inserted into water creates an nearly-radial flow to stretch the particle raft  enclosed inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Middle: The side view of the boundary of the Mylar strip boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Right: The top view of the expander at  early and late time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The gray arms are connected and move coherently to make the twelve orange inner nodes move outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The inner nodes are connected by Mylar strips painted in blue, which set the boundaries of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The dotted black lines  show the weight tied to the Mylar strips to keep them tout during the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' (B, C) Snapshots and zoomed-in images of  experiments at a fixed stretch ratio λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5 for different pulling velocities V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  position due to secondary flows but the internal structure  is undisturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' In this low-velocity regime, no significant  rifts can be observed and the cluster width, ℓ, remains  close to the initial system size, L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  With increasing V , the number of rifts increases while  the cluster size, ℓ, decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The cluster size, ℓ decreases  until it reaches the size of a single particle at large veloc-  ities, as shown in the right column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Figure 1C  shows how the internal features change with increasing  V : a single relatively closely-packed structure at low V ,  evolves upon increasing V into separate clusters consist-  ing of only a few (sometimes only one) particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The average cluster length, ⟨ℓ⟩, is obtained by aver-  aging the measurement at all different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This is  because there is no specific tensile axis in this radial ex-  pansion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The percentage standard deviation of  this average quantity is smaller than 3% for all veloci-  ties, which is comparable to the fluctuation due to rotat-  ing pixelated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This indicates that the variation in  cluster orientations is very low and below the uncertainty  level of detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Figure 2 shows the angular-averaged distribution of  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Angle-averaged distribution of cluster lengths, ℓ, for  different velocities at fixed stretch ratio, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The most  dominant length, that is the probability of cluster length ℓ  multiplied by ℓ normalized by ⟨ℓ⟩ = � ℓP(ℓ), is plotted versus  ℓ with three pulling speeds, V : 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='8 mm/s (blue), 26 mm/s  (green) and 260 mm/s (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  cluster sizes at a liquid stretch ratio λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5 for different  pulling velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The ordinate is the most dominant  cluster length, ℓP(ℓ)/⟨ℓ⟩, where P(ℓ) is the probability of  finding a cluster of length ℓ and the average cluster length  ⟨ℓ⟩ = � ℓP(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Since the statistics with small cluster  lengths always outnumber those with larger ℓ, I plot the  most dominant length, that is how much material has  cluster length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This can help better identify the cluster  distribution at each velocity, V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  At small V , a large  portion of the distribution remains close to the initial  size of the raft, L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The distribution shifts to smaller ℓ  as V increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' At the highest pulling speed, V , most  clusters have lengths between d and 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Figure 3A shows the average cluster length ⟨ℓ⟩ versus  pulling speed V at a fixed value of λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5, for all the ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' One can see that ⟨ℓ⟩ decreases monotonically  with increasing V and levels off at high velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Because  a cluster cannot be significantly larger than the initial  system size, L0, or smaller than a particle diameter, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The data can be interpolated between these two extremes  using a similar form as was used in the one-dimensional  pulling experiments [1]:  ⟨ℓ⟩ =  1  aV b + 1/(L0/  √  2 − dmax) + dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  (2)  This assumes a power-law dependence of the average  cluster length away from the two extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The differ-  ence from the one-dimensional fitting form is that initial  raft size L0 is scaled by  √  2 because the cluster lengths  are measured in a square box of size L0/  √  2 located in  the center of the raft to avoid complexity caused by the  round shape of the raft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The fit gives us a = (9 ± 2) × 103 and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This result is similar to the results in uniaxial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Averaged cluster length, ⟨ℓ⟩ = � ℓP(ℓ), versus  pulling speed, V at fixed stretch ratio λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  (A) The  blue points are measurements of ⟨ℓ⟩ versus V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The blue curve  is a fit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The red solid line indicates L0/  √  2, which  is the size of the measuring box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The red dashed line shows  the largest particle diameter dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The blue curve is a fit to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' (B) ⟨ℓ⟩ is plotted versus V for both radial (blue dots)  and uniaxial (green squares) experiments at similar expansion  condition (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5 and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The black line shows the  theoretical prediction for water from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The average cluster length, ⟨ℓ⟩, scales inversely with the  pulling velocity V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  DISCUSSION  Figure 3B compares the results from these radial ex-  pansion experiments with those found in the 1D exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Both data sets are shown at a similar time in  the expansion: the liquid strain ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5 for 1D experi-  ment and λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5 for radial experiment both occur when  the system size is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5 times larger than the original size  of the raft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The experiments under radial, isotropic ex-  pansion give very similar results to the 1D experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='02  V=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='8mm/s  V=26mm/s  V=260mm/s  P()/([)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='00  0  20  40  60  [w] A  10-2  [mm]  10-3  10-2  10-1  V[ms-1]  B  Radial 入 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5  Linear = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5  1D model  [mm]  10-3  10-2  10-1  V[ms-1]5  This suggests that the experiments with isotropic expan-  sion can in fact be generalized from the analysis used in  one-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  To understand this similarity, let us reexamine the flow  field described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The velocity field can be ex-  panded around any arbitrary point, (x0, y0), within the  boundaries:  u(x, y) = u0 + V  L ((x − x0)ˆx + (y − y0)ˆy)  where u0 is the velocity at (x0, y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Independent of the  position, (x0, y0), the velocity field expands radially out-  ward from that point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' all points are equivalent and every  particle feels the surrounding fluid retreating at the same  constant velocity in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Thus, the local mo-  tion can be reduced to multiple one-dimensional chains  as treated in the case of uniaxial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The large cracks that were observed near the raft cen-  ters in previous experiments where the rafts were pulled  radially in two-dimensions [10–12], are not present in the  data presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The difference in the way the pulling  of the raft is accomplished is likely the reason for this  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The present experiment was designed to min-  imize higher-order gradients in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  When using  the funnel method [12], I also observed the formation of  large cracks which I attributed to gradients in the veloc-  ity field produced by the influx of liquid near the center  of the raft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  To understand the two-dimensional effects intrinsic to  this systems, I analyze both the uniaxial and radial ex-  pansion datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 4A, the time evo-  lution of the cluster sizes shows a similar trend in both  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Here I focus on the time evolution in the uniax-  ial expansion experiments because they have the greater  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Only the behavior for liquid strain ε ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0 are  examined because the cluster sizes at early times is com-  plicated by limitations of image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  A square  observation window of size Lx0 located at the center of  the rafts is also chosen to avoid dealing with the irregular  edges of the rafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  At low V , the cluster size is nearly independent of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' At  higher pulling speed, however, the average cluster sizes  consistently decrease with increasing ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This suggests  non-linear, late-time relaxation of the raft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  One possibility for this is that, after the initial expan-  sion, the particles which are further apart experience a  smaller velocity gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  However, the particles that  are initially more compact continue to experience strong  attraction from their neighbors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' they rearrange so the  particle clusters become more compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Another possi-  bility is that the clusters simply continue to break up as  time goes on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  To understand the cause of the decrease in cluster sizes,  I calculate the particle ratio inside the clusters that have  been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The particle ratio is defined as the area  of particles inside the clusters divided by the total area  of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This quantity indicates if the voids inside  B  A  C  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Time evolution of cluster sizes in both radial and  uniaxial experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' (A) Average cluster length, ⟨ℓ⟩, versus  liquid stretch ratio λ or strain ε at all V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The results from ra-  dial and uniaxial expansion experiments are shown in crosses  and dots respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The color from dark purple to light  orange represent results from fast to slow V , and span the  entire pulling velocity range in both experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=') (B) The  ratio of particles occupied in the cluster versus strain ε for  uniaxial experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The results from four fastest pulling  speeds, V = 20 mm/s (orange) to 200 mm/s (purple), are  presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' (C) The change in ⟨ℓ⟩ , divided by ⟨ℓ⟩ at ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0  (solid circles), and the inverse of normalized number of clus-  ters, nc, scaled by nc at ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0 (empty circles), are plotted  versus ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0 -  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content="8  (0'T = 30)/(0)  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
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+page_content='25  wj  10~3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='75  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='25  Stretch ratio 入6  the clusters are shrinking in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 4B,  for four higher pulling velocities, the particle ratios all  increase in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This suggests that rearrangement of  neighboring particles inside a cluster is indeed taking  place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  On the other hand, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 4C examines whether the clus-  ters are breaking up with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The solid dots represent  the cluster lengths ⟨ℓ⟩ normalized by ⟨ℓε=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0⟩, the mea-  surement at ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0 versus strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  I also plot the in-  verse of the normalized number of clusters, nc (scaled by  nc,ε=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0) as empty circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' nc is the number of clusters  in the observation region normalized by the total area  of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This quantity shows how many clusters are  formed per unit area of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' If the inverse of nc  matches the change in cluster length, this implies that  the change in the number of clusters is the main con-  tributor to the change in cluster sizes, which is the case  at V = 20 mm/s Moreover, the overall change in ⟨ℓ⟩ is  small because the particles are already at a lower-energy  configuration to start with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The slight increase in n−1  c  after ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='7, that is the decrease in the number of clus-  ters, suggests that a larger-scale coalescence has occurred  as the global shear became very weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  At V = 200 mm/s, a significant difference between  n−1  c  and ⟨ℓ⟩ can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The average cluster length  decreases by more than 20% while the inverse of nc  only changed by 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A substantial rearrangement be-  tween neighboring particles is therefore responsible for  this change, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 4B as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  CONCLUSIONS  In this paper, the morphology of particle rafts float-  ing at an air-water interface is studied as the underlying  fluid is forced to flow radially outward at a uniform con-  stant velocity gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The experiments are designed  to minimize any high-order velocity gradients in the sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' By examining the flow field of the radial expansion  with particle image velocimetry (PIV), the local expan-  sion metric is found to be uniform inside the apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  As was found in the case of uniaxial expansion described  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' [1], the failure morphology due to the outward  flow varies smoothly as a function of the fluid expan-  sion velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' When the velocity is low, the rafts remain  intact throughout the expansion without the formation  of visible internal cracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  For larger expansion veloci-  ties, rifts open up inside the rafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The rifts are dis-  tributed diffusely with a spacing between them that de-  creases with increasing velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' At the largest velocities  used in the experiment, the separation decreases until  the cluster size, that is the length between rifts, becomes  only slightly larger than the size of a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The  measured average cluster sizes are very close to that of  the one-dimensional expansion experiment and follow the  same scaling behavior despite the fact that the geometry  of the expansion metric is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This indicates that  the 1D (in)stability model that was used to describe the  relaxation and cluster formation in [1] can be generalized  to two-dimensions and is sufficient to explain the radial  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  By measuring the time evolution of the clusters, a non-  linear relaxation is found at a later time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' after the initial  formation of clusters, the clusters start to shrink in sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The decrease in particle ratio inside the clusters shows  that the particles continue to rearrange and become more  compact even while the overall outward expansion con-  tinues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This can be understood because, once the initial  expansion rate sets the dominant cluster length, the ve-  locity gradient between clusters is no longer that used  initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Because the particles are farther apart (in the  rifts), the rifts will increase in size faster than the spaces  between the more densely packed particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Moreover,  the global velocity gradient is also decreasing monoton-  ically as the system size becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The particles  inside an individual cluster will rearrange according to  the competition between the local relaxation rate and  the local shear rate set by neighboring particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This  later-time aspect of the relaxation is not captured by the  one-dimensional linear (in)stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This experiment provides a flexible platform for study-  ing how morphology can form by the competition be-  tween two isotropic mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' There are many other  systems that share this common attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' As pointed  out in [1], an obvious analog is cosmological expansion  [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  In this table top experiment, the interaction be-  tween particles has the same form as gravity in two di-  mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Other examples include gel formation[15] and  structure formation by colloids in polymer solutions[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This two dimensional system gives us a direct access to  the clustering morphology with minimal imaging com-  plexity which would be troublesome in three dimensional  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' In addition, without tuning the details of the  potentials chemically or at a small scale, one of the com-  peting processes can be easily manipulated by changing  the mechanical driving protocol, while the other can be  controlled by changing specific particle-liquid or particle-  particle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  I thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Witten, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Vitelli and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Bandi  for insightful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  I am deeply grateful to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Nagel for his support and mentorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This work  was supported by the National Science Foundation (MR-  SEC program NSF-DMR 2011854), by the Simons Foun-  dation for the collaboration Cracking the Glass Prob-  lem Award #348125 and by Government Scholarship to  Study Abroad by the Ministry of Education in Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Appendix A: Details of the Radial Expander  The design of the radial expander used to create the  flow with minimal velocity gradients is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Details of the radial expander design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' (A) Schemat-  ics of the top view of the radial expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The inner nodes  are shown as orange and the outer nodes are green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Two of  the inner (orange) nodes are connected to the opposite sides  of a belt which is driven by a motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The movement of the  belt drives those two nodes in opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This opens  (closes) the entire structure when the belt is rotated clockwise  (counter-clockwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Because of the geometry connecting all  the nodes, pulling on just two of the inner nodes is sufficient  for the entire structure to open smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' (B) A side view of  the green and orange nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The green nodes are placed to  ensure that the entire structure is stable and does not wobble  vertically during the opening of the expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The bottom  of a green node is attached to a ball bearing so that it slides  smoothly on a clear acrylic platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Each orange node is  a screw with a 3D-printed attachment at its tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  This at-  tachment is composed of two closely-spaced columns that are  threaded by two adjacent Mylar ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' These ribbons slide  smoothly around the two columns as the expander is moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Each ribbon is held in place by the attachments at two ad-  jacent orange nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' To keep the Mylar ribbon taut during  movement of the expander, a weight is attached that pulls  the ribbon outwards on the opposite side from where the par-  ticles are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The Mylar ribbons are partly submerged  below the liquid surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' It is these ribbons which create the  boundary for the particles as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' (C) Top: A  photograph of the angled top view of the expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Bottom:  The side view of the chips and Mylar strips before inserting  into water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The twenty-four arms are connected by binding posts  (green nodes), which allow them to rotate freely, as shown  in the gray pieces in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 5A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Half of them are on the  top (light gray) and half of them are at the bottom (dark  gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Two of the opposite inner nodes are each con-  nected to the opposite side of a belt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' As the belt is driven  by a motor, a motion of the arms to move the twelve in-  ner (orange) nodes outward is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Ball bearings are  glued underneath the binding posts (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 5B)  to hold the expander and make it glide smoothly on the  platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The inner nodes are long screws, that go be-  low the platform and a 3D-printed chip is attached to the  tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This chip, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 5B, has two threaded  cylinders with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='22 mm slit in between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Each of  the cylinder serves as a column for one side of the Mylar  strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Each Mylar strip forms a triangle and is attached  to a weight at the corner against the side connecting ad-  jacent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' This provides sufficient tension to the strip  during the motion to create an outward-going boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Appendix B: Particle Image Velocimetry  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The Particle Image Velocimetry results of the radial  expander across whole velocity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The magnitude of the  velocity u normalized by the pulling speed V is plotted versus  the position relative to the center, normalized by the radius  of the raft, L0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  A linear relation can be observed in all  V , indicating an radially affine flow field can be created by  the expander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  The measurements near the boundaries are  excluded due to the difficulty of image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  To measure the expansion of the liquid surface due to  the extension of the pullers, the Particle Image Velocime-  try (PIV) measurement on water at V = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='8, 26 and 260  mm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' A layer of light,  non-interacting floating particles are spread sparsely on  the water surface prior to expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' By tracking the mo-  tion of these particles and computing the correlation be-  tween adjacent frames, the underlying fluid flow is shown  to lead to an uniform radial expansion of the surface,  as described in 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' the spacing of the particles increases  proportional to their relative position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 6 shows the  gradient of the velocity is constant radially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  V=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='8mm/s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='6  V=26mm/s  V=260mm/s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='6  r/(Lo/2)8  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' To and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Nagel, Rifts in rafts, Soft Matter ,  (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Paunov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Kralchevsky, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Denkov, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Na-  gayama, Lateral capillary forces between floating submil-  limeter particles, Journal of colloid and interface science  157, 100 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Vella and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Mahadevan, The “cheerios effect”, Amer-  ican journal of physics 73, 817 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Dalbe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Cosic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Berhanu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Kudrolli, Ag-  gregation of frictional particles due to capillary attrac-  tion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
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+page_content=' E 83, 051403 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
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+page_content=' Cicuta and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Vella, Granular character of particle  rafts, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' 102, 138302 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
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+page_content=' Varshney, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Sane, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Ghosh, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Bhattacharya,  Amorphous to amorphous transition in particle rafts,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' E 86, 031402 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Jambon-Puillet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Josserand, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Proti`ere, Wrin-  kles, folds, and plasticity in granular rafts, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Materials 1, 042601 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Razavi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Cao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Lin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
+page_content='  Tu, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE1T4oBgHgl3EQfbQQ6/content/2301.03170v1.pdf'}
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+arXiv:2301.04080v1  [math.AP]  10 Jan 2023
+NULL CONTROLLABILITY OF ONE-DIMENSIONAL LINEARIZED
+COMPRESSIBLE NAVIER-STOKES SYSTEM IN PERIODIC SETUP USING ONE
+BOUNDARY CONTROL
+JITEN KUMBHAKAR†
+Abstract. In this article, we study boundary null controllability properties of the linearized compress-
+ible Navier-Stokes equations in one dimension for both barotropic and non-barotropic fluids using only
+one boundary control. The control acts through periodic boundary conditions. We prove null control-
+lability of the linearized compressible Navier-Stokes system (for both barotropic and non-barotropic
+fluids) at large time T. We also prove that our result is sharp with respect to the regularity of the
+initial states. Finally, for both barotropic and non-barotropic fluids, we prove a lack of controllability
+at small time T when a control acts in density.
+Contents
+1.
+Introduction and Main Results
+1
+2.
+Controllability of Linearized Compressible Navier-Stokes System (Barotropic)
+6
+3.
+Controllability of Linearized Compressible Navier-Stokes System (Non-barotropic)
+15
+4.
+Further Comments and Conclusions
+27
+Appendix A.
+Proof of the Well-Posedness Results
+28
+References
+31
+1. Introduction and Main Results
+1.1. Linearized Compressible Navier-Stokes System in 1d. Let I = (0, +∞) be the time interval
+and Ω ⊂ R be a spatial domain. For a compressible, isentropic (barotropic) fluid, that is, when the
+pressure depends only on the density and the temperature is constant, the Navier-Stokes system in I ×Ω
+consists of the equation of continuity and the momentum equation
+ρt(t, x) + (ρu)x(t, x) = 0,
+ρ(t, x)[ut(t, x) + u(t, x)ux(t, x)] + px(t, x) − (λ + 2µ)uxx(t, x) = 0,
+where ρ denotes the density of the fluid, u is the velocity. The constants λ, µ are called the viscosity
+coefficients that satisfy µ > 0, λ + µ ≥ 0. The pressure p satisfies the following constitutive equation in
+I × Ω
+p(t, x) = aργ(t, x),
+(t, x) ∈ I × Ω,
+for some constants a > 0, γ ≥ 1. In the case of non-barotropic fluids, that is, when the pressure is a
+function of both density and temperature of the fluid, the Navier-Stokes system consists of the equation
+of continuity, the momentum equation, and an additional thermal energy equation
+cνρ(t, x)[θt(t, x) + u(t, x)θx(t, x)] + θ(t, x)pθ(t, x)ux(t, x) − κθxx(t, x) − (λ + 2µ)u2
+x(t, x) = 0,
+where θ is the temperature of the fluid, cν is the specific heat constant, and κ is the heat conductivity
+constant. For an ideal gas, Boyles law gives the pressure p(t, x) = Rρ(t, x)θ(t, x) in I × Ω with R as the
+universal gas constant. See [19, Chapter 1] for more about compressible flows.
+2010 Mathematics Subject Classification. 35M30, 35Q30, 76N25, 93B05, 93B07, 93C20.
+Key words and phrases. Linearized compressible Navier-Stokes system, null-controllability, observability, boundary
+control, Ingham-type inequalities.
+†Indian Institute of Science Education and Research Kolkata, Campus road, Mohanpur, West Bengal 741246, India;
+jk17ip021@iiserkol.ac.in.
+1
+
+1.2. The Barotropic Case. Let T > 0 be a finite time. We first consider the Navier-Stokes system
+for compressible, isentropic (barotropic) fluids linearized around some constant steady state (¯ρ, ¯u) with
+¯ρ > 0 and ¯u > 0
+(1.1)
+
+
+
+ρt(t, x) + ¯uρx(t, x) + ¯ρux(t, x) = 0,
+in (0, T ) × (0, 2π),
+ut(t, x) − µ
+¯ρ uxx(t, x) + ¯uux(t, x) + aγ¯ργ−2ρx(t, x) = 0,
+in (0, T ) × (0, 2π).
+The initial conditions are
+(1.2)
+ρ(0, x) = ρ0(x),
+u(0, x) = u0(x),
+x ∈ (0, 2π).
+We will consider two different problems, based on the act of control, by imposing any one of the following
+boundary conditions on the system (1.1).
+• Control in Density:
+(1.3)
+ρ(t, 0) = ρ(t, 2π) + p(t),
+u(t, 0) = u(t, 2π),
+ux(t, 0) = ux(t, 2π),
+t ∈ (0, T ).
+• Control in Velocity:
+(1.4)
+ρ(t, 0) = ρ(t, 2π),
+u(t, 0) = u(t, 2π) + q(t),
+ux(t, 0) = ux(t, 2π),
+t ∈ (0, T ),
+where p and q are boundary controls. Our main goal is to study null controllability of the system (1.1)
+at a given time T > 0 with the initial condition (1.2) and one of the boundary conditions (1.3) and (1.4).
+More precisely, given any initial state (ρ0, u0) in some suitable Hilbert space, we want to find a boundary
+control p (resp. q) such that the solution (ρ, u) to the system (1.1)-(1.2)-(1.3) (resp. (1.1)-(1.2)-(1.4))
+satisfies
+(ρ(T, x), u(T, x)) = (0, 0) in (0, 2π).
+Before stating our main results, we first introduce the Sobolev space for any s > 0
+Hs
+per(0, 2π) =
+�
+ϕ : ϕ =
+�
+n∈Z
+cneinx,
+�
+n∈Z
+|n|2s |cn|2 < ∞
+�
+,
+with the norm
+∥ϕ∥Hs
+per(0,2π) :=
+��
+n∈Z
+(1 + |n|2)s |cn|2
+� 1
+2
+.
+For s > 0, we denote H−s
+per(0, 2π) to be the dual of the Sobolev space Hs
+per(0, 2π) with respect to the
+pivot space L2(0, 2π). We also define the space
+˙L2(0, 2π) :=
+�
+ϕ ∈ L2(0, 2π) :
+� 2π
+0
+ϕ(x)dx = 0
+�
+and
+˙Hs
+per(0, 2π) :=
+�
+ϕ ∈ Hs
+per(0, 2π) :
+� 2π
+0
+ϕ(x)dx = 0
+�
+.
+If the system (1.1) is null controllable in time T by using a boundary control p, then integrating both
+equations in (1.1), we get a compatibility condition on the initial states
+aγ¯ργ−2
+� 2π
+0
+ρ0(x)dx = ¯u
+� 2π
+0
+u0(x)dx = −aγ¯ργ−2¯u
+� T
+0
+p(t)dt.
+If the system (1.1) is null controllable in time T by using a boundary control q, then also we will get
+a similar compatibility condition on the initial states. Since every initial state (ρ0, u0) in (L2(0, 2π))2
+will not satisfy this compatibility condition, we will work on the Hilbert space ( ˙L2(0, 2π))2 to avoid this
+difficulty.
+When a boundary control q acts in the velocity component, it is known in [7] that the system (1.1)-(1.2)-
+(1.4) is null controllable at time T > 2π
+¯u provided that the initial state is regular enough, in particular,
+lies in the space ˙Hs+1
+per (0, 2π) × ˙Hs
+per(0, 2π) for s > 9
+2. In the first part of our article, we generalize this
+result (with respect to the regularity of initial states). We also prove null controllability of the system
+(1.1) when there is a boundary control p acts in the density component. We write all the statements
+below.
+Theorem 1.1. For any given time T >
+2π
+¯u
+and initial state (ρ0, u0) ∈ ( ˙L2(0, 2π))2, there exists a
+boundary control p ∈ L2(0, T ) such that the system (1.1)-(1.2)-(1.3) is null controllable at time T .
+2
+
+For small time, we have a lack of null controllability result for the system (1.1)-(1.2)-(1.3).
+Proposition 1.2. For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2, the system (1.1)-(1.2)-(1.3) is not
+null controllable at small time 0 < T < 2π
+¯u by means of any boundary control p ∈ L2(0, T ).
+We have a similar null controllability result for the system (1.1) when a boundary control acts in the
+velocity component.
+Theorem 1.3. Let s ≥ 1. For any given time T > 2π
+¯u and initial state (ρ0, u0) ∈ ˙Hs
+per(0, 2π)× ˙L2(0, 2π),
+there exists a boundary control q ∈ L2(0, T ) such that the system (1.1)-(1.2)-(1.4) is null controllable at
+time T .
+The following result shows that the null controllability of the system (1.1)-(1.2)-(1.4) at time T > 2π
+¯u is
+optimal in the space ˙H1
+per(0, 2π) × ˙L2(0, 2π).
+Proposition 1.4. For any given initial state (ρ0, u0) ∈ Hs
+per(0, 2π) × L2(0, 2π) with 0 ≤ s < 1, the
+system (1.1)-(1.2)-(1.4) is not null controllable at any time T > 0.
+Remark 1.5. Following the proof of Proposition 1.2, the lack of null controllability of the system (1.1)-
+(1.2)-(1.4) cannot be obtained when the time is small, in particular, when 0 < T < 2π
+¯u . Lack of con-
+trollability at small time may be possible to obtain by constructing a Gaussian beam, as mentioned
+in [31, Theorem 1.5] for the interior control case.
+Remark 1.6. Null controllability of the system (1.1) at time T =
+2π
+¯u
+is inconclusive in both cases,
+whether there is a control act in density or velocity.
+1.3. The Non-Barotropic Case. We next consider the Navier-Stokes system for compressible non-
+barotropic fluids linearized around some constant steady state (¯ρ, ¯u, ¯θ) with ¯ρ, ¯u, ¯θ > 0
+
+
+
+
+
+
+
+
+
+
+
+
+
+ρt(t, x) + ¯uρx(t, x) + ¯ρux(t, x) = 0, in (0, T ) × (0, 2π),
+ut(t, x) − λ + 2µ
+¯ρ
+uxx(t, x) + R¯θ
+¯ρ ρx(t, x) + ¯uux(t, x) + Rθx(t, x) = 0, in (0, T ) × (0, 2π),
+θt(t, x) − κ
+¯ρcν
+θxx(t, x) + R¯θ
+cν
+ux(t, x) + ¯uθx(t, x) = 0, in (0, T ) × (0, 2π).
+(1.5)
+The initial conditions are
+(1.6)
+ρ(0, x) = ρ0(x),
+u(0, x) = u0(x),
+θ(0, x) = θ0(x),
+x ∈ (0, 2π).
+In this case, we will consider three different problems, based on the act of control, by imposing any one
+of the following boundary conditions on the system (1.5).
+• Control in Density:
+(1.7)
+ρ(t, 0) = ρ(t, 2π) + p(t), u(t, 0) = u(t, 2π), ux(t, 0) = ux(t, 2π), θ(t, 0) = θ(t, 2π), θx(t, 0) = θx(t, 2π).
+• Control in Velocity:
+(1.8)
+ρ(t, 0) = ρ(t, 2π), u(t, 0) = u(t, 2π) + q(t), ux(t, 0) = ux(t, 2π), θ(t, 0) = θ(t, 2π), θx(t, 0) = θx(t, 2π).
+• Control in Temperature:
+(1.9)
+ρ(t, 0) = ρ(t, 2π), u(t, 0) = u(t, 2π), ux(t, 0) = ux(t, 2π), θ(t, 0) = θ(t, 2π) + r(t), θx(t, 0) = θx(t, 2π).
+for t ∈ (0, T ), where p, q and r are boundary controls.
+In this case also, we want to prove null controllability of the system (1.5) at any given time T > 0
+depending on the act of the control. Similar to the barotropic case, we will work on the Hilbert space
+( ˙L2(0, 2π))3. We denote the positive constants
+λ0 := λ + 2µ
+¯ρ
+,
+κ0 :=
+κ
+¯ρcν
+,
+and define the set
+(1.10)
+S :=
+�
+(λ0, κ0) :
+�
+λ0
+κ0
+/∈ Q
+�
+.
+We prove the following results.
+3
+
+Theorem 1.7. For any given time T > 2π
+¯u , (λ0, κ0) ∈ S, and initial state (ρ0, u0, θ0) ∈ ( ˙L2(0, 2π))3,
+there exists a boundary control p ∈ L2(0, T ) such that the system (1.5)-(1.6)-(1.7) is null controllable at
+time T .
+The following proposition gives a lack of null controllability of the system (1.5)-(1.6)-(1.7) when the time
+is small enough, that is, 0 < T < 2π
+¯u .
+Proposition 1.8. For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3, the system (1.5)-(1.6)-(1.7) is
+not null controllable at a small time 0 < T < 2π
+¯u using any boundary control p ∈ L2(0, T ).
+Similar to the barotropic case, we have the null controllability result of the system (1.1) at time T (large
+enough) when there is a boundary control acts in the velocity component.
+Theorem 1.9. Let s ≥ 1. For any given time T > 2π
+¯u , (λ0, κ0) ∈ S, and initial state (ρ0, u0, θ0) ∈
+˙Hs
+per(0, 2π) × ( ˙L2(0, 2π))2, there exists a boundary control q ∈ L2(0, T ) such that the system (1.5)-(1.6)-
+(1.8) is null controllable at time T .
+The following result proves that the space ˙H1
+per(0, 2π) × ( ˙L2(0, 2π))2 is optimal for null controllability of
+the system (1.5)-(1.6)-(1.8).
+Proposition 1.10. For any given initial state (ρ0, u0, θ0) ∈ Hs
+per(0, 2π) × (L2(0, 2π))2 with 0 ≤ s < 1,
+the system (1.5)-(1.6)-(1.8) is not null controllable at any time T > 0.
+We have similar result when there is a control act in temperature.
+Theorem 1.11. Let s ≥ 1. For any given time T > 2π
+¯u , (λ0, κ0) ∈ S and initial state (ρ0, u0, θ0) ∈
+˙Hs
+per(0, 2π) × ( ˙L2(0, 2π))2, there exists a boundary control r ∈ L2(0, T ) such that the system (1.5)-(1.6)-
+(1.9) is null controllable at time T .
+Similar to the previous one (control acts in velocity), we have optimality of the space
+˙H1
+per(0, 2π) ×
+( ˙L2(0, 2π))2 in the sense below.
+Proposition 1.12. For any given initial state (ρ0, u0, θ0) ∈ Hs
+per(0, 2π) × (L2(0, 2π))2 with 0 ≤ s < 1,
+the system (1.5)-(1.6)-(1.9) is not null controllable at any time T > 0.
+Remark 1.13. Following the proof of Proposition 1.8, lack of null controllability of the systems (1.5)-
+(1.6)-(1.8) and (1.5)-(1.6)-(1.9) cannot be obtained when the time is small, in particular, when 0 <
+T < 2π
+¯u . However, this result may be possible to obtain by constructing a Gaussian beam, as mentioned
+in [31, Theorem 1.5] for the interior control case.
+Remark 1.14. Null controllability of the system (1.5) at time T = 2π
+¯u is inconclusive in both cases,
+whether there is a control act in density, velocity, or temperature.
+1.4. An Ingham-Type Inequality. To prove the null controllability results for both barotropic and
+non-barotropic systems, we need the following Ingham-type inequality.
+Lemma 1.15 ( [3, Proposition 1.6]). Let {νh
+n}n∈Z and {νp
+n}n∈Z be two sequences in C with the following
+properties: there is N ∈ N, such that
+(H1) for all n, l ∈ Z, νh
+n ̸= νh
+l unless n = l;
+(H2) νh
+n = β + τni + en for all |n| ≥ N;
+where τ > 0, β ∈ C and {en}|n|≥N ∈ ℓ2.
+Also, there exists constants A0 ≥ 0, B0 ≥ δ with δ > 0 and some ǫ > 0, r > 1 for which {νp
+n}n∈Z
+satisfies
+(P1) for all n, l ∈ Z, νp
+n ̸= νp
+l unless n = l;
+(P2)
+−ℜ(νp
+n)
+|ℑ(νp
+n)| ≥ �c for some �c > 0 and for all |n| ≥ N;
+(P3) |νp
+n − νp
+l | ≥ δ |nr − lr| for all n ̸= l with |n| , |l| ≥ N and
+(P4) ǫ(A0 + B0nr) ≤ |νp
+n| ≤ A0 + B0nr for all |n| ≥ N.
+We also assume that the families are disjoint, i.e.,
+�
+νh
+n, n ∈ Z
+�
+∩ {νp
+n, n ∈ Z} = ∅.
+Then, for any time T > 2π
+τ , there exists a positive constant C depending only on T such that
+� T
+0
+�����
+�
+n∈Z
+aneνp
+nt +
+�
+n∈Z
+bneνh
+nt
+�����
+2
+dt ≥ C
+��
+n∈Z
+|an|2e2ℜ(νp
+n)T +
+�
+n∈Z
+|bn|2
+�
+,
+(1.11)
+4
+
+for all sequences {an}n∈Z and {bn}n∈Z in ℓ2.
+Notations. For any vector v, we denote its transpose by v† (instead of vT ). Throughout the article,
+C > 0 denotes a generic constant that may depend on the time T .
+Proving null controllability of the systems (1.1) and (1.5) using a boundary control is equivalent to
+proving an observability inequality for the corresponding adjoint systems. Spectrum of the associated
+linearized operators (for the adjoint systems) and the above Ingham-type inequality (1.11) plays a crucial
+role to prove such observability inequalities. For the system (1.1) (barotropic fluids), spectrum of the
+associated adjoint operator consists of two branches of complex eigenvalues, namely, the hyperbolic and
+parabolic branches. The hyperbolic branch has eigenvalues with the real part converging to − b¯ρ
+µ0 , whereas
+real part of the parabolic branch diverges to −∞. We have obtained explicit expressions of eigenvalues
+and eigenfunctions in terms of a Riesz basis (See Lemma 2.6 for details). For the non-barotropic fluids
+(that is, system (1.5)), we get three branches of complex eigenvalues; one is of the hyperbolic type,
+and two are parabolic types.
+Similar to the barotropic case, the real part of the hyperbolic branch
+converges to − R¯θ
+λ0 and the real parts of both parabolic branches diverge to −∞. In this case, we have
+obtained explicit expressions of eigenfunctions and asymptotic behavior of the eigenvalues (Lemma 3.7).
+We also proved that the eigenfunctions form a Riesz basis in ( ˙L2(0, 2π))2 for the barotropic system
+and ( ˙L2(0, 2π))3 for the non-barotropic system. Then, by writing the solutions to the corresponding
+adjoint systems in terms of the eigenfunctions, the null controllability results have been proved using the
+combined parabolic-hyperbolic Ingham type inequality (1.11).
+A vast amount of literature is available on the controllability of Navier-Stokes equations for incom-
+pressible fluids. For instance, one can see the works of Coron [12], Coron and Fursikov [13], Fursikov
+and Imanuvilov [22, 23], Imanuvilov [26, 27], Fern´andez-Cara et al. [20, 21], Guerrero [25], Coron and
+Guerrero [14], Chapouly [4], Coron and Lissy [15], Badra, Ervedoza and Guerrero [1], Coron, Marbach
+and Sueur [16]. In comparison, for compressible fluids, less works are available on the Navier-Stokes
+system’s controllability. In this context, we first mention the work of Ervedoza et al. [17], where the
+authors established local exact controllability of one dimensional compressible Navier-Stokes system at
+a large time T in the space H3(0, L) × H3(0, L) using two boundary controls. This result has been
+improved in [18] where the null controllability is achieved in the space H1(0, L) × H1(0, L).
+It is known in [10] that, for barotropic fluids, the one-dimensional compressible Navier-Stokes system
+linearized around (¯ρ, 0) (with ¯ρ > 0) cannot be null controllable at any time T > 0 by using a boundary
+control or a localized distributed control. For the linearized system around (¯ρ, ¯u) (with ¯ρ, ¯u > 0), the
+authors in [8] proved null controllability of the Navier-Stokes equations (with homogeneous periodic
+boundary conditions) for viscous, compressible isothermal barotropic fluids at time T (large) in the
+space ˙H1
+per(0, 2π) × L2(0, 2π), when there is an interior control act only in the velocity equation. They
+also proved that the space ˙H1
+per(0, 2π) × L2(0, 2π) is optimal in the sense that if we choose the initial
+state from
+˙Hs
+per(0, 2π) × L2(0, 2π) with 0 ≤ s < 1, the linearized system cannot be null controllable
+at any time T > 0. In the case of linearization around (¯ρ, ¯u) with ¯ρ, ¯u > 0, the compressible Navier-
+Stokes system (1.1) is equivalent (in some sense) to the transformed system in [32]. Using a moving
+distributed control, the authors in [32] proved the null controllability of a one-dimensional structurally
+damped wave equation in the space Hs+2 × Hs for s >
+15
+2 . There is a generalization to this result
+in higher dimensions by Chaves-Silva, Rosier, and Zuazua [5]. Inspired by the work of Martin, Rosier
+and Rouchon [32], Chowdhury and Mitra in [7] proved the null controllability of the same compressible
+Navier-Stokes system linearized around (¯ρ, ¯u) at time T (large) by using a boundary control that acts
+on the velocity component through periodic conditions, provided the initial states are regular enough,
+more precisely, in the space
+˙H1+s
+per (0, 2π) × ˙Hs
+per(0, 2π) with s > 4.5.
+However, the question of null
+controllability at a large time T in the space ˙H1+s
+per (0, 2π)× ˙Hs
+per(0, 2π) with 0 < s ≤ 4.5 was unaddressed
+in [7], and up to the author’s knowledge, there has been no improvement in this result. In this article, we
+have answered this question (Theorem 1.3 and Proposition 1.4). We have proved null controllability of
+the linearized compressible Navier-Stokes system for barotropic fluids (1.1) at large time T in the space
+˙Hs
+per(0, 2π) × ˙L2(0, 2π) with s ≥ 1 by using one boundary control acting in the velocity component. We
+have also proved that our result is optimal in the sense that the system cannot be null controllable by a
+boundary control (acts in velocity) when the initial states belong to the space ˙Hs
+per(0, 2π)× ˙L2(0, 2π) with
+0 ≤ s < 1. When a control acts only in the density component through periodic boundary conditions,
+we have established null controllability of the linearized system (1.1)-(1.2)-(1.3) at large time T in the
+space ˙L2(0, 2π) × ˙L2(0, 2π) and that null controllability fails at small time T .
+5
+
+For the non-barotropic fluids, it is known in [31] that the compressible Navier-Stokes system linearized
+around (¯ρ, 0, ¯θ) (with ¯ρ, ¯θ > 0) is not null controllable at any time T > 0 by using a boundary control
+or a localized distributed control. For the linearization around (¯ρ, ¯u, ¯θ) with ¯ρ, ¯u, ¯θ > 0, it is only known
+that the system is not null controllable at small time by a localized interior control or a boundary control
+acting on the velocity component (see [31, Theorem 1.5] for instance). To the author’s knowledge, no
+controllability result is known for the linearized system around (¯ρ, ¯u, ¯θ), that is, the system (1.5), when
+the time is large, which is studied for the first time in this article.
+The main difficulty in the linearized compressible Navier-Stokes system is the presence of transport
+and parabolic coupling. The thermoelasticity system is also an example involving both transport and
+parabolic effects. Lebeau and Zuazua [30] have studied distributed Controllability for thermoelasticity
+systems. Following [30], Beauchard et al. in [2] proved null controllability for some coupled transport-
+parabolic systems when an interior control acts. They proved null controllability at large time T in
+the space L2(0, 2π) × ˙L2(0, 2π) by one interior control acts in the density equation and in the space
+˙H2(0, 2π) × H2(0, 2π) when only one interior control acts in the velocity equation; see also [28].
+The main contribution of this article is that we prove the null controllability of the one-dimensional
+linearized compressible Navier-Stokes system for both barotropic and non-barotropic fluids by using only
+one boundary control. We consider all the possible cases of the act of control for both systems (1.1)
+and (1.5). We get better regularity of the initial states for the controllability of barotropic system (1.1)
+compared to [7]. In the case of non-barotropic fluids, since the transport equation does not affect the
+temperature equation, it is pretty natural to obtain similar spaces of null controllability of the system
+(1.5). The combined parabolic-hyperbolic Ingham type inequality (Lemma 1.15) helps us obtain each
+case’s best possible results (with respect to the state space).
+Our results cannot be obtained as a
+consequence of interior control results by the extension method. When the boundary control acts in the
+density component, we prove that both systems (1.1) and (1.5) are not null controllable at small time.
+The proof is inspired from [2] and is independent of that in [31].
+The result stated in Theorem 1.1 is similar to the results in [2], showing that we can achieve the space
+( ˙L2(0, 2π))2 in the case of only one boundary control (acts in density) also. Likewise the case of interior
+control [8], we also obtain similar results for our boundary control case (acts in velocity) (Theorem 1.3
+and Proposition 1.4).
+The rest of the article is organized as follows:
+– In Section 2, we prove the null controllability of the linearized compressible Navier-Stokes system
+for barotropic fluids (1.1) at a large time T using a boundary control that acts either in density
+or velocity, that is, Theorem 1.1 and Theorem 1.3 respectively.
+The proofs of lack of null
+controllability at small time T (Proposition 1.2) and at any time T with less regular initial states
+(Proposition 1.4) are also included in this section.
+– In Section 3, we give all the null controllability results of linearized compressible Navier-Stokes
+system for non-barotropic fluids (1.5) based on the act of the control, namely the proofs of
+Theorem 1.7, Theorem 1.9 and Theorem 1.11. We have also included the proofs of lack of null
+controllability results at small time T (Proposition 1.8) and at any time T when the initial states
+are less regular (Proposition 1.10 and Proposition 1.12).
+– In section 4, we give few comments and discuss some open problems.
+– For the sake of completeness, we give the proof of well-posedness result (Lemma 3.1) for the
+non-barotropic system (1.5) in Appendix A.1.
+Acknowledgments. The author would like to thank his PhD supervisor Dr. Shirshendu Chowdhury
+for suggesting this problem and fruitful discussions. The author would also like to thank Dr. Rajib
+Dutta for careful reading and improvement of the manuscript.
+This work is supported by the Prime Minister’s Research Fellowship (ref. no. 41-1/2018-TS-1/PMRF),
+Government of India.
+2. Controllability of Linearized Compressible Navier-Stokes System (Barotropic)
+2.1. Functional Setting. We denote the positive constants
+µ0 := µ
+¯ρ ,
+b := aγ¯ργ−2.
+6
+
+We define the inner product in the space (L2(0, 2π))2 as follows
+��
+f1
+g1
+�
+,
+�
+f2
+g2
+��
+:= b
+� 2π
+0
+f1(x)f2(x)dx + ¯ρ
+� 2π
+0
+g1(x)g2(x)dx,
+for fi, gi ∈ L2(0, 2π), i = 1, 2. We write the system (1.1) in abstract differential equation
+(2.1)
+U ′(t) = AU(t),
+U(0) = U0,
+t ∈ (0, T ),
+where U := (ρ, u)†, U0 := (ρ0, u0)† and the operator A is given by
+A :=
+�
+−¯u∂x
+−¯ρ∂x
+−b∂x
+µ0∂xx − ¯u∂x
+�
+with the domain
+D(A) := H1
+per(0, 2π) × H2
+per(0, 2π).
+The adjoint of the operator A∗ is given by
+A∗ :=
+�
+¯u∂x
+¯ρ∂x
+b∂x
+µ0∂xx + ¯u∂x
+�
+with the same domain D(A∗) = D(A). The adjoint system is then given by
+(2.2)
+
+
+
+
+
+
+
+
+
+−σt(t, x) − ¯uσx(t, x) − ¯ρvx(t, x) = 0, in (0, T ) × (0, 2π),
+−vt(t, x) − µ0vxx(t, x) − ¯uvx(t, x) − bσx(t, x) = 0, in (0, T ) × (0, 2π),
+σ(t, 0) = σ(t, 2π),
+v(t, 0) = v(t, 2π),
+vx(t, 0) = vx(t, 2π), t ∈ (0, T ),
+σ(T, x) = σT (x),
+v(T, x) = vT (x),
+x ∈ (0, 2π).
+We now write the adjoint system with source terms f and g.
+(2.3)
+
+
+
+
+
+
+
+
+
+−σt(t, x) − ¯uσx(t, x) − ¯ρvx(t, x) = f, in (0, T ) × (0, 2π),
+−vt(t, x) − µ0vxx(t, x) − ¯uvx(t, x) − bσx(t, x) = g, in (0, T ) × (0, 2π),
+σ(t, 0) = σ(t, 2π),
+v(t, 0) = v(t, 2π),
+vx(t, 0) = vx(t, 2π), t ∈ (0, T ),
+σ(T, x) = σT (x),
+v(T, x) = vT (x),
+x ∈ (0, 2π).
+2.2. Well-Posedness of the System. This section devotes to the well-posedness of the system (1.1)
+under the boundary conditions (1.3), (1.4) and the initial conditions (1.2), and the adjoint system (2.3).
+When there is no control act on the system, we have the existence of solutions to the system (1.1) using
+semigroups.
+Lemma 2.1 ( [8, Lemma 2.1]). The operator A (resp. A∗) generates a C0-semigroup of contractions
+on (L2(0, 2π))2. Moreover, for every U0 ∈ (L2(0, 2π))2 the system (2.1) admits a unique solution U in
+C0([0, T ]; (L2(0, 2π))2) and
+∥U(t)∥(L2(0,2π))2 ≤ C ∥U0∥(L2(0,2π))2
+for all t ≥ 0.
+The following lemma shows the existence of a unique solution to the adjoint system (2.3).
+Lemma 2.2 ( [24]). For any given source term (f, g) ∈ L2(0, T ; (L2(0, 2π))2), the adjoint system (2.3)
+(with (σT , vT ) = (0, 0)) has a unique solution (σ, v) in the space
+C0([0, T ]; L2(0, 2π)) × [C0([0, T ]; L2(0, 2π)) ∩ L2(0, T ; H1
+per(0, 2π))].
+Once we have the existence results of the homogeneous system (without any boundary control) associated
+to the system (1.1), we can now guarantee the existence of a unique solution to the system (1.1) (in the
+sense of transposition) when there is a boundary control p (resp. q) act in density (resp. velocity) in the
+space L2(0, T ). Before writing the statements, let us first define the notion of a solution in the sense of
+transposition.
+7
+
+Definition 2.3.
+(1) For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2 and boundary control p ∈
+L2(0, T ), a function (ρ, u) ∈ L2(0, T ; (L2(0, 2π))2) is a solution to the system (1.1)-(1.2)-(1.3) if
+for any given (f, g) ∈ L2(0, T ; (L2(0, 2π))2), the following identity holds true:
+� T
+0
+� 2π
+0
+ρ(t, x)f(t, x)dxdt +
+� T
+0
+� 2π
+0
+u(t, x)g(t, x)dxdt
+= ⟨(ρ0(·), u0(·)), (σ(0, ·), v(0, ·))⟩L2×L2 +
+� T
+0
+�
+¯uσ(t, 2π) + ¯ρv(t, 2π)
+�
+p(t)dt,
+where (σ, v) is the unique weak solution to the adjoint system (2.3) with (σt, vT ) = (0, 0).
+(2) For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2 and boundary control q ∈ L2(0, T ), a function
+(ρ, u) ∈ L2(0, T ; (H1(0, 2π))′ × L2(0, 2π)) is a solution to the system (1.1)-(1.2)-(1.4) if for any
+given (f, g) ∈ L2(0, T ; H1(0, 2π) × L2(0, 2π)), the following identity holds true:
+� T
+0
+⟨ρ(t, ·), f(t, ·)⟩(H1)′,H1 dt +
+� T
+0
+� 2π
+0
+u(t, x)g(t, x)dxdt
+= ⟨(ρ0(·), u0(·)), (σ(0, ·), v(0, ·))⟩L2×L2 +
+� T
+0
+�
+bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)
+�
+q(t)dt,
+(σ, v) is the unique weak solution to the adjoint system (2.3) with (σT , vT ) = (0, 0).
+Proposition 2.4 ( [3, Theorem 2.4]). For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2 and boundary
+control p ∈ L2(0, T ), the system (1.1)-(1.2)-(1.3) admits a unique solution (ρ, u) in the space
+C0([0, T ]; L2(0, 2π)) × [C0([0, T ]; L2(0, 2π)) ∩ L2(0, T ; H1
+per(0, 2π))].
+Proposition 2.5 ( [9, Theorem 3.2]). For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2 and boundary
+control q ∈ L2(0, T ), the system (1.1)-(1.2)-(1.4) admits a unique solution (ρ, u) in the space
+L2(0, T ; (H1
+per(0, 2π))′) × L2(0, T ; L2(0, 2π)).
+Moreover, the operator q �→ (ρ, u) is linear and continuous from L2(0, T ) into L2(0, T ; (H1
+per(0, 2π))′) ×
+L2(0, T ; L2(0, 2π)).
+2.3. Spectral Analysis of A∗. We denote the spectrum of A∗ by σ(A∗). The following lemma gives
+behavior of the spectrum of the operator A∗.
+Lemma 2.6. The following statements holds.
+(1) ker(A∗) = span
+��
+1
+1
+�
+,
+�
+1
+−1
+��
+.
+(2) sup {ℜ(ν) : ν ∈ σ(A∗), ν ̸= 0} < 0.
+(3) The spectrum of A∗ consists of the eigenvalue 0 and pairs of complex eigenvalues {νh
+n, νp
+n}n∈Z∗
+given as
+νh
+n = −1
+2
+�
+µ0n2 − n
+�
+µ2
+0n2 − 4b¯ρ − 2¯uin
+�
+,
+(2.4)
+νp
+n = −1
+2
+�
+µ0n2 + n
+�
+µ2
+0n2 − 4b¯ρ − 2¯uin
+�
+,
+(2.5)
+for all n ∈ Z∗.
+(4) The eigenvalues satisfy the following properties
+
+
+
+lim|n|→∞ ℜ(νh
+n) = −ω0,
+lim|n|→∞
+ℜ(νp
+n)
+n2
+= −µ0
+lim|n|→∞
+ℑ(νh
+n)
+n
+= ¯u,
+lim|n|→∞
+ℑ(νp
+n)
+n
+= ¯u
+with ω0 = b¯ρ
+µ0 .
+(5) The eigenfunctions of A∗ corresponding to νh
+n and νp
+n are respectively
+(2.6)
+Φh
+n :=
+�
+ξh
+n
+ηh
+n
+�
+=
+�
+¯ρ
+νn
+2 − ¯u
+�
+einx,
+Φp
+n :=
+�
+ξp
+n
+ηp
+n
+�
+=
+�
+¯ρ
+νn
+1 −¯u
+1
+�
+einx,
+for n ∈ Z, where
+(2.7)
+νn
+1 := 1
+2
+�
+µ0in + 2¯u + i
+�
+µ2
+0n2 − 4b¯ρ
+�
+,
+νn
+2 := 1
+2
+�
+µ0in + 2¯u − i
+�
+µ2
+0n2 − 4b¯ρ
+�
+,
+n ∈ Z.
+8
+
+(6) The eigenfunctions {Φh
+n, Φp
+n : n ∈ Z∗} of A∗ forms a Riesz basis of ( ˙L2(0, 2π))2.
+Proof. We will prove only the parts (2), (3) and (5). Part (4) can be proved using part (2).
+Let Φ = (ξ, η)† be the eigenfunction of A∗ corresponding to the eigenvalue ν ̸= 0. Then, we have
+�
+A∗
+�
+ξ
+η
+�
+,
+�
+ξ
+η
+��
+=
+�
+ν
+�
+ξ
+η
+�
+,
+�
+ξ
+η
+��
+,
+that is,
+b¯u
+� 2π
+0
+ξ(x)ξx(x)dx + b¯ρ
+� 2π
+0
+ξ(x)ηx(x)dx + µ0¯ρ
+� 2π
+0
+η(x)ηxx(x)dx + ¯ρ¯u
+� 2π
+0
+η(x)ηx(x)dx
++b¯ρ
+� 2π
+0
+ξx(x)η(x)dx = ν
+� 2π
+0
+|ξ(x)|2 dx + ν
+� 2π
+0
+|η(x)|2 dx.
+An integration by parts yields
+ℜ(ν) = −
+∥ηx∥2
+L2(0,2π)
+|ξ|2
+L2(0,2π) + ∥η∥2
+L2(0,2π)
+< 0,
+which proves part (2).
+We denote
+ϕn(x) := einx,
+n ∈ Z.
+Then the set
+��
+ϕn
+0
+�
+,
+�
+0
+ϕn
+��
+forms an orthogonal basis of (L2(0, 2π))2. Let us define
+En :=
+�
+ϕn
+0
+0
+ϕn
+�
+,
+and Φn := (ξn, ηn)†,
+for all n ∈ Z. Then, we have the following relation
+(2.8)
+A∗EnΦn = inEnRnΦn,
+n ∈ Z,
+where the matrix Rn for n ∈ Z is given by
+(2.9)
+Rn :=
+�
+¯u
+¯ρ
+b
+µ0in + ¯u
+�
+,
+n ∈ Z.
+Thus, if (αn, νn) is an eigenpair of Rn, then (Enαn, inνn) will be an eigenpair of A∗. Therefore, it’s
+remains to find the eigenvalues and eigenvectors of the matrix Rn for n ∈ Z. The characteristics equation
+of Rn is
+(2.10)
+ν2 − (µ0in + 2¯u)ν + µ0¯uin + ¯u2 − b¯ρ = 0,
+for all n ∈ Z. Therefore, the eigenvalues of the matrix Rn are
+νn
+1 := 1
+2
+�
+µ0in + 2¯u + i
+�
+µ2
+0n2 − 4b¯ρ
+�
+,
+νn
+2 := 1
+2
+�
+µ0in + 2¯u − i
+�
+µ2
+0n2 − 4b¯ρ
+�
+,
+for all n ∈ Z. Note that, 0 and ¯u cannot be an eigenvalue of the matrix Rn for all n ∈ Z∗ and ¯u cannot
+be an eigenvalue of Rn for all n ∈ Z, because b, ¯ρ, µ0, ¯u > 0. To find the eigenvectors of the matrix Rn,
+we first consider the equation
+Rnαh
+n = νn
+2 αh
+n,
+n ∈ Z,
+where αh
+n := (αn
+1 , αn
+2 )†, that is,
+(¯u − νn
+2 )αn
+1 + ¯ραn
+2 = 0,
+(2.11)
+bαn
+1 + (µ0in + ¯u − νn
+2 )αn
+2 = 0,
+(2.12)
+for all n ∈ Z. One solution is given by
+(2.13)
+αh
+n =
+�
+αn
+1
+αn
+2
+�
+:=
+�
+¯ρ
+νn
+2 − ¯u
+�
+,
+n ∈ Z.
+We next consider the equation
+Rnαp
+n = νn
+1 αp
+n,
+n ∈ Z,
+9
+
+where αp
+n := (βn
+1 , βn
+2 )†, that is,
+(¯u − νn
+1 )βn
+1 + ¯ρβn
+2 = 0,
+(2.14)
+bβn
+1 + (µ0in + ¯u − νn
+1 )βn
+2 = 0,
+(2.15)
+for all n ∈ Z. One solution is given by
+(2.16)
+αp
+n =
+�
+βn
+1
+βn
+2
+�
+:=
+�
+¯ρ
+νn
+1 −¯u
+1
+�
+,
+n ∈ Z.
+Thus, the eigenvectors of Rn corresponding to the eigenvalues νn
+2 and νn
+1 are respectively
+αh
+n =
+�
+αn
+1
+αn
+2
+�
+=
+�
+¯ρ
+νn
+2 − ¯u
+�
+,
+αp
+n =
+�
+βn
+1
+βn
+2
+�
+=
+�
+¯ρ
+νn
+1 −¯u
+1
+�
+,
+n ∈ Z.
+Hence, the eigenvalues of the operator A∗ are
+νh
+n := inνn
+2 = 1
+2
+�
+−µ0n2 + n
+�
+µ2
+0n2 − 4b¯ρ + 2¯uin
+�
+, νp
+n := inνn
+1 = 1
+2
+�
+−µ0n2 − n
+�
+µ2
+0n2 − 4b¯ρ + 2¯uin
+�
+for n ∈ Z and the corresponding eigenfunctions are respectively
+Φh
+n :=
+�
+ξh
+n
+ηh
+n
+�
+= Enαh
+n = αh
+neinx,
+Φp
+n :=
+�
+ξp
+n
+ηp
+n
+�
+= Enαp
+n = αp
+neinx,
+for all n ∈ Z and x ∈ (0, 2π). This completes the proof.
+□
+Remark 2.7. Note that, for all |n| large, all the eigenvalues of A∗ are simple. There may be multiple
+eigenvalues of A∗, depending on the constants ¯ρ, ¯u, µ0, b, but that would be only finitely many. Thus,
+without loss of generality, we can assume that A∗ has simple eigenvalues. Indeed, for the case of finite
+number of multiple eigenvalues, one can adapt a similar approach as in Section 4.2 of [8] (by considering
+suitable generalized eigenfunctions) to prove the required observability inequality. One can also see [29,
+Remarks, page 178] for a version of Ingham inequality in the case of repeated eigenvalues.
+2.4. Observation Estimates. For any eigenvalue ν, let us denote the corresponding eigenfunction of
+A∗ by Φν and let E(A∗) be the set of all eigenfunctions of A∗. We now define the observation operators
+associated to the system (1.1) as follows:
+B∗
+ρΦν := ¯uξ(2π) + ¯ρη(2π),
+(2.17)
+B∗
+uΦν := bξ(2π) + ¯uη(2π) + µ0ηx(2π),
+(2.18)
+for all Φν = (ξ, η, ζ) ∈ E(A∗). The following result proves that these observation terms are non-zero for
+all Φ ∈ E(A∗) \ {Φ0}, and have positive lower bounds for all n ∈ Z∗.
+Lemma 2.8. For all Φν ∈ E(A∗) \ {Φ0}, the observation operators satisfy B∗
+ρΦν ̸= 0 and B∗
+uΦν ̸= 0.
+Moreover, we have the following estimates
+��B∗
+ρΦh
+n
+�� ≥ C,
+��B∗
+ρΦp
+n
+�� ≥ C,
+(2.19)
+��B∗
+uΦh
+n
+�� ≥ C
+|n|,
+|B∗
+uΦp
+n| ≥ C |n| ,
+(2.20)
+for some C > 0 and all n ∈ Z∗.
+Proof. Note that
+B∗
+ρΦh
+n = ¯uξh
+n(2π) + ¯ρηh
+n(2π) = ¯uαn
+1 + ¯ραn
+2 = νn
+2 αn
+1 ̸= 0,
+for all n ∈ Z∗, thanks to the equation (2.11). We similarly have from equation (2.14)
+B∗
+ρΦp
+n = ¯uξp
+n(2π) + ¯ρηp
+n(2π) = ¯uβn
+1 + ¯ρβn
+2 = νn
+1 βn
+1 ̸= 0,
+for all n ∈ Z∗. The estimates on B∗
+ρΦh
+n and B∗
+ρΦp
+n are now follows directly from the above expressions.
+We now compute
+B∗
+uΦh
+n = bξh
+n(2π) + ¯uηh
+n(2π) + µ0(ηh
+n)x(2π)
+= bαn
+1 + (¯u + µ0in)αn
+2 = νn
+2 αn
+2 ̸= 0,
+10
+
+for all n ∈ Z∗, thanks to the equation (2.12). Since |αn
+2 | ≥
+C
+|n| and νn
+2 is bounded for all n ∈ Z∗, the
+estimate on B∗
+uΦh
+n follows. For the last quantity, note that
+B∗
+uΦp
+n = bξp
+n(2π) + ¯uηp
+n(2π) + µ0(ηp
+n)x(2π)
+= bβn
+1 + (¯u + µ0in)βn
+2 = νn
+1 βn
+2 ̸= 0,
+for all n ∈ Z∗, thanks to the equation (2.15). The estimate on B∗
+uΦp
+n is now follows directly as |νn
+1 | ≥ C |n|
+for all n ∈ Z∗.
+□
+2.5. Observability Inequality. Since the eigenfunctions E(A∗)\{Φ0} form a Riesz basis in ( ˙L2(0, 2π))2,
+therefore any (σT , vT ) ∈ ( ˙L2(0, 2π))2 can be written as
+(σT , vT )† =
+�
+n∈Z∗
+�
+ah
+nΦh
+n + ap
+nΦp
+n
+�
+,
+for some (ah
+n)n∈Z∗, (ap
+n)n∈Z∗ ∈ ℓ2. Then the solution to the adjoint system (2.2) is
+(σ(t, x), v(t, x))† =
+�
+n∈Z∗
+ah
+neνh
+n(T −t)Φh
+n +
+�
+n∈Z∗
+ap
+neνp
+n(T −t)Φp
+n,
+for (t, x) ∈ (0, T ) × (0, 2π). Thus we get
+σ(t, x) = ¯ρ
+�
+n∈Z∗
+ah
+neνh
+n(T −t)einx +
+�
+n∈Z∗
+ap
+neνp
+n(T −t)
+¯ρ
+νn
+1 − ¯ueinx,
+and
+v(t, x) =
+�
+n∈Z∗
+ah
+neνh
+n(T −t)(νn
+2 − ¯u)einx +
+�
+n∈Z∗
+ap
+neνp
+n(T −t)einx,
+for all (t, x) ∈ (0, T ) × (0, 2π). To prove the observability inequality, we need an upper bound of the
+norm of (σ(0), v(0))† and a lower bound estimate for the respective observation terms. We first estimate
+the upper bounds of the norm of (σ(0), v(0))†. We have
+��(σ(0), v(0))†��2
+( ˙L2(0,2π))2
+(2.21)
+≤ C
+� �
+n∈Z∗
+��ah
+n
+��2 �
+1 + |νn
+2 − ¯u|2�
+e2ℜ(νh
+n)T ��einx��2
+˙L2(0,2π) +
+�
+n∈Z∗
+|ap
+n|2
+�
+1
+|νn
+1 − ¯u|2 + 1
+�
+e2ℜ(νp
+n)T ��einx��2
+˙L2(0,2π)
+�
+≤ C
+� �
+n∈Z∗
+��ah
+n
+��2 +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+,
+since the sequences 1 + |νn
+2 − ¯u|2 and 1 +
+1
+|νn
+1 −¯u|
+2 are bounded for all n ∈ Z∗. We also have
+��(σ(0), v(0))†��2
+˙H−s
+per(0,2π)× ˙L2(0,2π)
+(2.22)
+≤ C
+� �
+n∈Z∗
+��ah
+n
+��2 �
+1 + |νn
+2 − ¯u|2� ��einx��2
+˙H−s
+per(0,2π) +
+�
+n∈Z∗
+|ap
+n|2
+�
+1
+|νn
+1 − ¯u|2 + 1
+�
+e2ℜ(νp
+n)T ��einx��2
+˙L2(0,2π)
+�
+≤ C
+� �
+n∈Z∗
+��ah
+n
+��2
+1
+|n|2s +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+.
+We now find the lower bounds of the respective observation terms and prove our main results for the
+barotropic case. We use the Ingham-type inequality (Lemma 1.15) for the same. Note that, the eigen-
+values (νh
+n)n∈Z∗ satisfies hypotheses (H1)-(H2) with τ = ¯u, β = −ω0 and (νp
+n)n∈Z∗ satisfies hypotheses
+(P1)-(P4) with r = 2.
+11
+
+2.5.1. Proof of Theorem 1.1. Let T > 2π
+¯u . The observability inequality is given by
+(2.23)
+� T
+0
+|¯uσ(t, 2π) + ¯ρv(t, 2π)|2 dt ≥ C
+��(σ(0), v(0))†��2
+( ˙L2(0,2π))2 ,
+for all (σT , vT )† ∈ D(A∗). We have the observation term
+� T
+0
+|σ(t, 2π) + v(t, 2π)|2 dt =
+� T
+0
+� �
+n∈Z∗
+ah
+nB∗
+ρΦh
+neνh
+n(T −t) +
+�
+n∈Z∗
+ap
+nB∗
+ρΦp
+neνp
+n(T −t)
+�2
+dt.
+Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.15) and the observation
+estimates (2.19), we obtain
+� T
+0
+|σ(t, 2π) + v(t, 2π)|2 dt ≥ C
+� �
+n∈Z∗
+��ah
+n
+��2 ��B∗
+ρΦh
+n
+��2 e2ℜ(νh
+n)T +
+�
+n∈Z∗
+|ap
+n|2 ��B∗
+ρΦp
+n
+��2 e2ℜ(νp
+n)T
+�
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2 +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+This estimate together with the norm estimate (2.21), the observability inequality (2.23) is now follows
+directly. This completes the proof.
+2.5.2. Proof of Theorem 1.3. Let T > 2π
+¯u . The observability inequality is given by
+(2.24)
+� T
+0
+|bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt ≥ C
+��(σ(0), v(0))†��2
+˙H−s
+per(0,2π)× ˙L2(0,2π) ,
+for all (σT , vT )† ∈ D(A∗). We have
+� T
+0
+|bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt =
+� T
+0
+� �
+n∈Z∗
+ah
+kB∗
+uΦh
+neνh
+n(T −t) +
+�
+n∈Z∗
+ap
+nB∗
+uΦp
+neνp
+n(T −t)
+�2
+dt.
+Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.15), we obtain
+� T
+0
+|bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2 ��B∗
+uΦh
+n
+��2 e2ℜ(νh
+n)T +
+�
+n∈Z∗
+|ap
+n|2 |B∗
+uΦp
+n|2 e2ℜ(νp
+n)T
+�
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2
+1
+|n|2 +
+�
+n∈Z∗
+|ap
+n|2 |n|2 e2ℜ(νp
+n)T
+�
+,
+thanks to the estimate (2.20). For s ≥ 1, we have
+1
+|n|2 ≥
+1
+|n|2s for all n ∈ Z∗ and therefore
+� T
+0
+|bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt ≥ C
+� �
+n∈Z∗
+��ah
+n
+��2
+1
+|n|2s +
+�
+n∈Z∗
+|ap
+n|2 |n|2 e2ℜ(νp
+n)T
+�
+≥ C
+��(σ(0), v(0))†�� ˙H−s
+per(0,2π)× ˙L2(0,2π) .
+This proves the observability inequality (2.24).
+2.6. Lack of Null Controllability for Less Regular Initial States.
+2.6.1. Proof of Proposition 1.4. For (σT , vT )† = Φh
+n, the solution to the adjoint system (2.2) is
+(σ(t, x), v(t, x))† = eνh
+n(T −t)Φh
+n(x),
+for (t, x) ∈ (0, T ) × (0, 2π) and n ∈ Z∗. For all n ∈ Z∗, we have the following estimate
+��Φh
+n
+�� ˙H−s
+per(0,2π)× ˙L2(0,2π) ≥ C
+|n|s ,
+and therefore
+��(σ(0), v(0))†��2
+˙H−s
+per× ˙L2 ≥
+C
+|n|2s ,
+12
+
+for all n ∈ Z∗. On the other hand, we have the upper bound of the observation term
+� T
+0
+|bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt ≤
+C
+|n|2 ,
+for all n ∈ Z∗. Thus, if the observability inequality (2.24) holds, then we get
+C
+|n|2s ≤
+C
+|n|2 =⇒ |n|2−2s ≤ C,
+which is not possible since 0 ≤ s < 1. This completes the proof.
+2.7. Lack of Controllability at Small Time. We prove that the system (1.5) is not null control-
+lable when the time is small, that is, Proposition 1.8. We construct an approximate solution for the
+corresponding transport equation. The idea of constructing an approximate solution for the transport
+equation was addressed in [2], where the authors proved a lack of null controllability result at a small
+time in the case of an interior control (acts in the transport equation). Very recently, in [6, Section 6],
+this approach has been applied to a coupled transport-elliptic system in the case of a boundary control
+(acts in density). We will follow mainly the proof given in [6] to prove our lack of null controllability
+result when the time is small.
+2.7.1. Proof of Proposition 1.2. Let 0 < T < 2π
+¯u . Following the notations in the proof of Proposition
+1.2, Consider the transport equation
+(2.25)
+
+
+
+
+
+
+
+
+
+˜σt(t, x) + ¯u˜σx(t, x) − b¯ρ
+µ0
+˜σ(t, x) = 0,
+(t, x) ∈ (0, T ) × (0, 2π),
+˜σ(t, 0) = ˜σ(t, 2π),
+t ∈ (0, T ),
+˜σ(T, x) = ˜σT (x),
+x ∈ (0, 2π).
+Since ¯uT < 2π, there exists a nontrivial function ˜σT ∈ C∞(0, 2π) with supp(˜σT ) ⊂ (¯uT, 2π) such that
+supp(˜σ) ⊂ (0, T ) × (¯uT, 2π). Let N > 0 be a fixed integer. We define the polynomial
+P N(x) :=
+N
+�
+l=−N
+(x − l),
+x ∈ (0, 2π)
+and the function
+˜σN
+T := P N
+�
+−i d
+dx
+�
+˜σT .
+We write the terminal state as
+˜σT (x) :=
+�
+n∈Z
+aneinx,
+x ∈ (0, 2π).
+Then, the solution to the transport equation (2.25) is
+˜σN
+T (x) =
+�
+n∈Z
+an
+N
+�
+l=−N
+�
+−i d
+dx − l
+�
+einx =
+�
+n∈Z
+an
+N
+�
+l=−N
+(n − l) einx =
+�
+n∈Z
+anP N(n)einx,
+for (t, x) ∈ (0, T ) × (0, 2π). Note that P N(n) = 0 for all |n| ≤ N and therefore
+˜σN
+T (x) =
+�
+|n|≥N+1
+anP N(n)einx.
+With this ˜σN
+T , let us now consider the following system
+(2.26)
+
+
+
+
+
+
+
+
+
+˜σt(t, x) + ¯u˜σx(t, x) − b¯ρ
+µ0
+˜σ(t, x) = 0,
+(t, x) ∈ (0, T ) × (0, 2π),
+˜σ(t, 0) = ˜σ(t, 2π),
+t ∈ (0, T ),
+˜σ(T, x) = ˜σN
+T (x),
+x ∈ (0, 2π).
+13
+
+Since supp(˜σN
+T ) ⊂ supp(˜σT ) ⊂ (T, 2π
+¯u ), the solution satisfies ˜σN(t, 0) = ˜σN(t, 2π) = 0. We now consider
+the following adjoint system
+(2.27)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+σt(t, x) + ¯uσx(t, x) + ¯ρvx(t, x) = 0,
+(t, x) ∈ (0, T ) × (0, 2π),
+vt(t, x) − µ0vxx(t, x) + ¯uvx(t, x) + bσx(t, x) = 0,
+(t, x) ∈ (0, T ) × (0, 2π),
+σ(t, 0) = σ(t, 2π),
+t ∈ (0, T ),
+v(t, 0) = v(t, 2π),
+vx(t, 0) = vx(t, 2π),
+t ∈ (0, T ),
+σ(T, x) = ˜σN
+T (x),
+v(T, x) = vN
+T (x),
+x ∈ (0, 2π),
+where we choose vN
+T such that
+(˜σN
+T , vN
+T )† =
+�
+|n|≥N+1
+˜ah
+nΦh
+n
+with ˜ah
+n¯ρ := anP N(n) for all |n| ≥ N + 1. We write the solutions to the systems (2.26) and (2.27)
+respectively as
+˜σN(t, x) =
+�
+|n|≥N+1
+anP N(n)e(¯uin− b¯
+ρ
+µ0 )(T −t)einx,
+(2.28)
+σN(t, x) =
+�
+|n|≥N+1
+anP N(n)eνh
+n(T −t)einx,
+(2.29)
+vN(t, x) =
+�
+|n|≥N+1
+anP N(n)νn
+2 − ¯u
+¯ρ
+eνh
+n(T −t)einx,
+(2.30)
+for (t, x) ∈ [0, T ]×[0, 2π]. We prove that the solution component σN of (2.27) approximates the solution
+˜σN of (2.26). Indeed,
+��σN(·, x) − ˜σN(·, x)
+��2
+L2(0,T )
+≤
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2
+����eνh
+n(T −t) − e
+�
+¯uin− b¯
+ρ
+µ0
+�
+(T −t)
+����
+2
+L2(0,T )
+≤
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2
+������
+e
+− µ0n
+2
+�
+n−
+�
+n2− 4b¯
+ρ
+µ2
+0
+�
+(T −t)
+− e− b¯
+ρ
+µ0 (T −t)
+������
+2
+L2(0,T )
+≤
+�
+|n|≥N+1
+1
+|n|2 |an|2 ��P N(n)
+��2 ,
+for all x ∈ [0, 2π] and therefore
+��σN(·, x) − ˜σN(·, x)
+��2
+L2(0,T ) ≤
+C
+|N|2
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 ,
+for all x ∈ [0, 2π]. We also find L2- estimate of the solution component vN. We have for all x ∈ [0, 2π]
+��vN(·, x)
+��2
+L2(0,T ) ≤
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 |νn
+2 − ¯u|2
+¯ρ2
+���eνh
+n(T −t)���
+2
+L2(0,T )
+≤ C
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2
+1
+|n|2
+≤
+C
+|N|2
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 .
+Let us now suppose that the following observability inequality holds
+� T
+0
+��¯uσN(t, 2π) + ¯ρvN(t, 2π)
+��2 dt ≥ C
+��(σN(0), vN(0))
+��2
+(L2(0,2π))2 .
+14
+
+Then, we have
+��(σN(0), vN(0))
+��2
+(L2(0,2π))2 ≤ C
+� T
+0
+��¯uσN(t, 2π) + ¯ρvN(t, 2π)
+��2 dt
+≤ C
+� T
+0
+�
+¯u2 ��(σN(t, 2π) − ˜σN(t, 2π))
+��2 + ¯u2 ��˜σN(t, 2π)
+��2 + ¯ρ2 ��vN(t, 2π)
+��2�
+dt
+≤ C
+N 2
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 ,
+as we have ˜σN(t, 0) = 0 = ˜σN(t, 2π) for all t ∈ (0, T ). Thus we get
+��σN(0)
+��2
+L2(0,2π) ≤
+��(σN(0), vN(0))†��2
+(L2(0,2π))2 ≤ C
+N 2
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 ≤ C
+N 2
+��σN(0)
+��2
+L2(0,2π) ,
+since ℜ(νh
+n) is bounded. Therefore, 1 ≤
+C
+N 2 for all N and hence the above inequality cannot hold. This
+is a contradiction and the proof is complete.
+3. Controllability of Linearized Compressible Navier-Stokes System (Non-barotropic)
+We denote the positive constants
+λ0 := λ + 2µ
+¯ρ
+,
+κ0 :=
+κ
+¯ρcν
+,
+and from now on-wards, we denote cν by c0 to distinguish it from the eigenvalue ν.
+3.1. Functional Setting. We define the inner product in the space (L2(0, 2π))3 as follows
+�
+
+
+f1
+g1
+h1
+
+
+ ,
+
+
+
+f2
+g2
+h2
+
+
+
+�
+:= R¯θ
+� 2π
+0
+f1(x)f2(x)dx + ¯ρ2
+� 2π
+0
+g1(x)g2(x)dx + ¯ρ2c0
+¯θ
+� 2π
+0
+h1(x)h2(x)dx,
+for fi, gi, hi ∈ L2(0, 2π), i = 1, 2, 3. We write the system (1.5) in abstract differential equation
+(3.1)
+U ′(t) = AU(t),
+U(0) = U0,
+t ∈ (0, T ),
+where U := (ρ, u, θ)†, U0 := (ρ0, u0, θ0)† and the operator A is given by
+A :=
+
+
+
+
+−¯u∂x
+−¯ρ∂x
+0
+− R¯θ
+¯ρ ∂x
+λ0∂xx − ¯u∂x
+−R∂x
+0
+− R¯θ
+c0 ∂x
+κ0∂xx − ¯u∂x
+
+
+
+
+with the domain
+(3.2)
+D(A) := H1
+per(0, 2π) × (H2
+per(0, 2π))2.
+The adjoint of the operator A∗ is given by
+A∗ :=
+
+
+
+
+¯u∂x
+¯ρ∂x
+0
+R¯θ
+¯ρ ∂x
+λ0∂xx + ¯u∂x
+R∂x
+0
+R¯θ
+c0 ∂x
+κ0∂xx + ¯u∂x
+
+
+
+
+with the same domain D(A∗) = D(A). The adjoint system is given by
+(3.3)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−σt − ¯uσx − ¯ρvx = 0, in (0, T ) × (0, 2π),
+−vt − λ0vxx − R¯θ
+¯ρ σx − ¯uvx − Rϕx = 0, in (0, T ) × (0, 2π),
+−ϕt − κ0ϕxx − R¯θ
+c0
+vx − ¯uϕx = 0, in (0, T ) × (0, 2π),
+σ(t, 0) = σ(t, 2π),
+v(t, 0) = v(t, 2π),
+vx(t, 0) = vx(t, 2π),
+t ∈ (0, T ),
+ϕ(t, 0) = ϕ(t, 2π),
+ϕx(t, 0) = ϕx(t, 2π),
+t ∈ (0, T ),
+σ(T, x) = σT (x),
+v(T, x) = vT (x),
+ϕ(T, x) = ϕT (x),
+x ∈ (0, 2π),
+15
+
+with (σT , vT , ϕT ) is a terminal state. We also write the following system with source terms f, g, and h.
+(3.4)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−σt − ¯uσx − ¯ρvx = f, in (0, T ) × (0, 2π),
+−vt − λ0vxx − R¯θ
+¯ρ σx − ¯uvx − Rϕx = g, in (0, T ) × (0, 2π),
+−ϕt − κ0ϕxx − R¯θ
+c0
+vx − ¯uϕx = h, in (0, T ) × (0, 2π),
+σ(t, 0) = σ(t, 2π),
+v(t, 0) = v(t, 2π),
+vx(t, 0) = vx(t, 2π),
+t ∈ (0, T ),
+ϕ(t, 0) = ϕ(t, 2π),
+ϕx(t, 0) = ϕx(t, 2π),
+t ∈ (0, T ),
+σ(T, x) = σT (x),
+v(T, x) = vT (x),
+ϕ(T, x) = ϕT (x),
+x ∈ (0, 2π).
+3.2. Well-Posedness of the System.
+Lemma 3.1. The operator A generates a C0-semigroup of contractions on (L2(0, 2π))3. Moreover, for
+every U0 ∈ (L2(0, 2π))3 the system (2.1) admits a unique solution U in C0([0, T ]; (L2(0, 2π))3) and
+∥U(t)∥(L2(0,2π))3 ≤ C ∥U0∥(L2(0,2π))3
+for all t ≥ 0.
+Lemma 3.2 ( [31]). For any f, g, h ∈ L2(0, T ; (L2(0, 2π))3), the adjoint system (3.4) (with (σT , vT , ϕT ) =
+(0, 0, 0)) has a unique solution (σ, v, ϕ) in the space
+C0([0, T ]; L2(0, 2π)) × [C0([0, T ]; L2(0, 2π)) ∩ L2(0, T ; H1
+per(0, 2π))]2.
+Definition 3.3. We give the following definitions of solutions based on the act of the controls.
+• For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control p ∈ L2(0, T ), we
+say (ρ, u, θ) ∈ (L2(0, 2π))3 is a solution to the system (1.5)-(1.6)-(1.7) if for any (f, g, h) ∈
+L2(0, T ; (L2(0, 2π))3), the following identity holds.
+� T
+0
+� 1
+0
+ρ(t, x)f(t, x)dxdt +
+� T
+0
+� 1
+0
+u(t, x)g(t, x)dxdt +
+� T
+0
+� 1
+0
+θ(t, x)h(t, x)dxdt
+=
+� 1
+0
+ρ0(x)σ(0, x)dx +
+� 1
+0
+u0(x)v(0, x)dx +
+� 1
+0
+θ0(x)ϕ(0, x)dx + R¯θ
+� T
+0
+�
+¯uσ(t, 2π) + ¯ρv(t, 2π)
+�
+p(t)dt,
+where (σ, v, ϕ) is the solution to the adjoint system (3.4) with (σT , vT , ϕT ) = (0, 0, 0).
+• For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control q ∈ L2(0, T ), we
+say (ρ, u, θ) ∈ L2(0, T ; (H1(0, 2π))′) × L2(0, T ; (L2(0, 2π))2) is a solution to the system (1.5)-
+(1.6)-(1.8) if for any (f, g, h) ∈ L2(0, T ; H1(0, 2π))×L2(0, T ; (L2(0, 2π))2), the following identity
+holds.
+� T
+0
+⟨ρ(t, ·), f(t, ·)⟩(H1(0,2π))′,H1(0,2π) dxdt +
+� T
+0
+� 1
+0
+u(t, x)g(t, x)dxdt +
+� T
+0
+� 1
+0
+θ(t, x)h(t, x)dxdt
+=
+� 1
+0
+ρ0(x)σ(0, x)dx +
+� 1
+0
+u0(x)v(0, x)dx +
+� 1
+0
+θ0(x)ϕ(0, x)dx
++
+� T
+0
+�
+R¯θ¯ρσ(t, 2π) + λ0¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)
+�
+q(t)dt,
+where (σ, v, ϕ) is the solution to the adjoint system (3.4) with (σT , vT , ϕT ) = (0, 0, 0).
+• For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control r ∈ L2(0, T ), we
+say (ρ, u, θ) ∈ L2(0, T ; (H1(0, 2π))′) × L2(0, T ; (L2(0, 2π))2) is a solution to the system (1.5)-
+(1.6)-(1.8) if for any (f, g, h) ∈ L2(0, T ; H1(0, 2π))×L2(0, T ; (L2(0, 2π))2), the following identity
+holds.
+� T
+0
+⟨ρ(t, ·), f(t, ·)⟩(H1(0,2π))′,H1(0,2π) dxdt +
+� T
+0
+� 1
+0
+u(t, x)g(t, x)dxdt +
+� T
+0
+� 1
+0
+θ(t, x)h(t, x)dxdt
+=
+� 1
+0
+ρ0(x)σ(0, x)dx +
+� 1
+0
+u0(x)v(0, x)dx +
+� 1
+0
+θ0(x)ϕ(0, x)dx
++
+� T
+0
+�
+R¯ρ2v(t, 2π) + ¯ρ2c0¯u
+¯θ
+ϕ(t, 2π) + ¯ρ2c0κ0
+¯θ
+ϕx(t, 2π)
+�
+r(t)dt,
+where (σ, v, ϕ) is the solution to the adjoint system (3.4) with (σT , vT , ϕT ) = (0, 0, 0).
+16
+
+Proposition 3.4. For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control p ∈
+L2(0, T ), the system (1.5)-(1.6)-(1.7) admits a unique solution (ρ, u, θ) in the space
+C0([0, T ]; L2(0, 2π)) × [C0([0, T ]; L2(0, 2π)) ∩ L2(0, T ; H1
+per(0, 2π))]2.
+Proposition 3.5. For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control q ∈
+L2(0, T ), the system (1.5)-(1.6)-(1.8) admits a unique solution (ρ, u, θ) in the space
+L2(0, T ; (H1
+per(0, 2π))′) × L2(0, T ; (L2(0, 2π))2).
+Moreover, the operator q �→ (ρ, u, θ) is linear and continuous from L2(0, T ) into L2(0, T ; (H1
+per(0, 2π))′)×
+L2(0, T ; (L2(0, 2π))2).
+Proposition 3.6. For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control r ∈
+L2(0, T ), the system (1.5)-(1.6)-(1.9) admits a unique solution (ρ, u, θ) in the space
+L2(0, T ; (H1
+per(0, 2π))′) × L2(0, T ; (L2(0, 2π))2).
+Moreover, the operator r �→ (ρ, u, θ) is linear and continuous from L2(0, T ) into L2(0, T ; (H1
+per(0, 2π))′)×
+L2(0, T ; (L2(0, 2π))2).
+The proofs of Proposition 3.4, Proposition 3.5 and Proposition 3.6 can be done in a similar way
+( [3, Theorem 2.4] and [9, Theorem 3.2]) like the barotropic case and so we skip the proofs.
+3.3. Spectral Analysis of A∗. We first write the following lemma.
+Lemma 3.7. The following statements hold.
+(1) ker(A∗) = span
+
+
+
+
+
+
+
+
+−1
+1
+1
+
+
+ ,
+
+
+
+1
+−1
+1
+
+
+ ,
+
+
+
+1
+1
+−1
+
+
+
+
+
+
+
+
+.
+(2) sup {ℜ(ν) : ν ∈ σ(A∗), ν ̸= 0} < 0.
+(3) The spectrum of A∗ consists of the eigenvalue 0 and three branches of complex eigenvalues
+{νh
+n, νp
+n, νp1
+n }n∈Z∗ with the asymptotic expressions given as
+νh
+n = ¯uin − ¯ω + O(|n|−2),
+(3.5)
+νp1
+n = −λ0n2 + ¯uin + O(1),
+(3.6)
+νp2
+n = −κ0n2 + ¯uin + O(1),
+(3.7)
+for all |n| large, where ¯ω = R¯θ
+λ0 .
+(4) The eigenfunctions of A∗ corresponding to νh
+n and νp1
+n , νp2
+n are respectively
+(3.8)
+Φh
+n =
+
+
+
+ξh
+n
+ηh
+n
+ζh
+n
+
+
+ =
+
+
+
+αn
+1
+αn
+2
+αn
+3
+
+
+ einx,
+Φp1
+n =
+
+
+
+ξp1
+n
+ηp1
+n
+ζp1
+n
+
+
+ =
+
+
+
+βn
+1
+βn
+2
+βn
+3
+
+
+ einx,
+Φp2
+n =
+
+
+
+ξp2
+n
+ηp2
+n
+ζp2
+n
+
+
+ =
+
+
+
+γn
+1
+γn
+2
+γn
+3
+
+
+ einx,
+for all n ∈ Z∗, with the constants αn
+i , βn
+i and γn
+i (i = 1, 2, 3) given as
+(3.9)
+
+
+
+
+
+
+
+αn
+1 = R¯ρ,
+αn
+2 = −R(¯u − νn
+3 ),
+αn
+3 = (λ0in + ¯u − νn
+3 )(¯u − νn
+3 ) − R¯θ
+βn
+1 = −
+R¯ρ
+¯u−νn
+1 ,
+βn
+2 = R,
+βn
+3 =
+1
+¯u−νn
+1 [R¯θ − (λ0in + ¯u − νn
+1 )(¯u − νn
+1 )]
+γn
+1 = (λ0in + ¯u − νn
+2 )(κ0in + ¯u − νn
+2 ) − R2 ¯θ
+c0 ,
+γn
+2 = − R¯θ
+¯ρ (κ0in + ¯u − νn
+2 ),
+γn
+3 = R2 ¯θ2
+¯ρc0 ,
+for all n ∈ Z∗, where νn
+1 , νn
+2 and νn
+3 are roots of the cubic polynomial
+ν3 − [(λ0 + κ0)in + 3¯u]ν2 − [λ0κ0n2 − 2(λ0 + κ0)¯uin − 3¯u2 + R2¯θ
+c0
++ R¯θ]ν
+(3.10)
++λ0κ0¯un2 − (λ0 + κ0)¯u2in − ¯u3 + R2¯θ
+c0
+¯u + R¯θκ0in + R¯θ¯u = 0,
+for all n ∈ Z∗.
+(5) The eigenfunctions {Φh
+n, Φp1
+n , Φp2
+n
+: n ∈ Z∗} of A∗ forms a Riesz basis of ( ˙L2(0, 2π))3.
+17
+
+Remark 3.8. We have the asymptotic expressions of αi, βi, γi, i = 1, 2, 3 as follows.
+
+
+
+
+
+
+
+αn
+1 ∼+∞ 1,
+αn
+2 ∼+∞
+1
+|n|,
+αn
+3 ∼+∞
+1
+|n|,
+βn
+1 ∼+∞
+1
+|n|,
+βn
+2 ∼+∞ 1,
+βn
+3 ∼+∞
+1
+|n|,
+γn
+1 ∼+∞
+1
+|n|,
+γn
+2 ∼+∞
+1
+|n|,
+γn
+3 ∼+∞ 1.
+Proof. We will prove only the parts (2), (3) and (4).
+Let Φ = (ξ, η, ζ)† be the eigenfunction of A∗ corresponding to the eigenvalue ν ̸= 0. Then, we have
+�
+A∗
+
+
+
+ξ
+η
+ζ
+
+
+ ,
+
+
+
+ξ
+η
+ζ
+
+
+
+�
+=
+�
+ν
+
+
+
+ξ
+η
+ζ
+
+
+ ,
+
+
+
+ξ
+η
+ζ
+
+
+
+�
+,
+that is,
+R¯θ¯u
+� 2π
+0
+ξ(x)ξx(x)dx + R¯θ¯ρ
+� 2π
+0
+ξ(x)ηx(x)dx + λ0¯ρ2
+� 2π
+0
+η(x)ηxx(x)dx + ¯ρ2¯u
+� 2π
+0
+η(x)ηx(x)dx
++R¯θ¯ρ
+� 2π
+0
+ξx(x)η(x)dx + R¯ρ2
+� 2π
+0
+η(x)ζx(x)dx + ¯ρ2c0
+¯θ
+κ0
+� 2π
+0
+η(x)ζx(x)dx + ¯ρ2c0
+¯θ
+¯u
+� 2π
+0
+η(x)ζx(x)dx
++R¯ρ2
+� 2π
+0
+ηx(x)ζ(x)dx + b¯ρ
+� 2π
+0
+ξx(x)η(x)dx = ν
+� 2π
+0
+|ξ(x)|2 dx + ν
+� 2π
+0
+|η(x)|2 dx + ν
+� 2π
+0
+|ζ(x)|2 dx.
+An integration by parts yields
+ℜ(ν) = −
+∥ηx∥2
+L2(0,2π) + ∥ζx∥2
+L2(0,2π)
+∥ξ∥2
+L2(0,2π) + ∥η∥2
+L2(0,2π) + ∥ζ∥2
+L2(0,2π)
+< 0,
+which proves part (2).
+We denote
+ϕn(x) := einx,
+n ∈ Z.
+Then the set
+
+
+
+
+
+
+
+
+ϕn
+0
+0
+
+
+ ,
+
+
+
+0
+ϕn
+0
+
+
+ ,
+
+
+
+0
+0
+ϕn
+
+
+
+
+
+
+
+
+forms an orthogonal basis of (L2(0, 2π))3. Let us define
+En :=
+
+
+
+ϕn
+0
+0
+0
+ϕn
+0
+0
+0
+ϕn
+
+
+ ,
+and Φn := (ξn, ηn, ζn)†,
+for all n ∈ Z. Then, we have the following relation
+(3.11)
+A∗EnΦn = inEnRnΦn,
+n ∈ Z,
+where
+(3.12)
+Rn :=
+
+
+
+
+¯u
+¯ρ
+0
+R¯θ
+¯ρ
+λ0in + ¯u
+R
+0
+R¯θ
+c0
+κ0in + ¯u
+
+
+
+ ,
+n ∈ Z.
+Thus, if (αn, νn) is an eigenpair of Rn, then (Enαn, inνn) will be an eigenpair of A∗. Therefore, it’s
+remains to find the eigenvalues and eigenvectors of the matrix Rn for n ∈ Z. The characteristics equation
+of Rn is
+ν3 − [(λ0 + κ0)in + 3¯u]ν2 − [λ0κ0n2 − 2(λ0 + κ0)¯uin − 3¯u2 + R2¯θ
+c0
++ R¯θ]ν
+(3.13)
++λ0κ0¯un2 − (λ0 + κ0)¯u2in − ¯u3 + R2¯θ
+c0
+¯u + R¯θκ0in + R¯θ¯u = 0,
+for all n ∈ Z.
+Claim 1. 0 cannot be a root of the polynomial (3.13) for any n ∈ Z.
+18
+
+Proof of Claim 1. Let ν = 0 be a root of (3.13). Then, there exists some n ∈ Z such that
+λ0κ0¯un2 − (λ0 + κ0)¯u2in − ¯u3 + R2¯θ
+c0
+¯u + R¯θκ0in + R¯θ¯u = 0,
+which implies
+λ0κ0n2 − ¯u2 + R2¯θ
+c0
++ R¯θ = 0, and (λ0 + κ0)¯u2 = R¯θκ0.
+We then have
+λ0κ0n2 = ¯u2 − R2¯θ
+c0
+− R¯θ = ¯u2 − R2¯θ
+c0
+−
+�λ0
+κ0
++ 1
+�
+¯u2 = −R2¯θ
+c0
+− λ0
+κ0
+¯u2 < 0,
+a contradiction. This proves our first claim.
+Claim 2. ¯u cannot be a root of the polynomial (3.13) for any n ∈ Z∗.
+Proof of Claim 2. Observe that ¯u is a root of (3.13) if and only if R¯θκ0in = 0. Thus, for all n ∈ Z∗, ¯u
+cannot be a root of (3.13), which proves our second claim.
+For fixed n ∈ Z∗, let νn
+1 , νn
+2 and νn
+3 be the roots of this cubic polynomial. The relation between roots
+and coefficients are
+
+
+
+
+
+
+
+νn
+1 + νn
+2 + νn
+3 = (λ0 + κ0)in + 3¯u
+νn
+1 νn
+2 + νn
+2 νn
+3 + νn
+3 νn
+1 = −[λ0κ0n2 − 2(λ0 + κ0)¯uin − 3¯u2 + R2 ¯θ
+c0 + R¯θ]
+νn
+1 νn
+2 νn
+3 = −[λ0κ0¯un2 − (λ0 + κ0)¯u2in − ¯u3 + R2 ¯θ
+c0 ¯u + R¯θκ0in + R¯θ¯u].
+We will find the asymptotic expressions of roots of the cubic polynomial (3.13) for large values of |n|.
+The first relation between roots and coefficients tells us that ¯u is present in at least one of the roots of
+the cubic polynomial (3.13). Thus, using the transformation
+(3.14)
+ν = ¯u + ǫn,
+it is enough to find the roots of the transformed cubic equation in ǫn
+(3.15)
+ǫ3
+n − (λ0 + κ0)inǫ2
+n −
+�
+λ0κ0n2 + R2¯θ
+c0
++ R¯θ
+�
+ǫn + R¯θκ0in = 0
+for all n ∈ Z∗. We use the transformation ǫn = in˜ǫn, for n ∈ Z∗, to simplify the above equation and we
+get
+(3.16)
+˜ǫ3
+n − (λ0 + κ0)˜ǫ2
+n +
+�
+λ0k0 + 1
+n2
+�R2¯θ
+c0
++ R¯θ
+��
+˜ǫn − R¯θκ0
+n2
+= 0
+for all n ∈ Z∗. We now use Rouche’s Theorem to find the roots of this polynomial. Let us first state the
+Rouch´e’s Theorem, the proof of which can be found in [11].
+Theorem 3.9 (Rouch´e’s Theorem). Let Ω ⊂ C be an open connected set and f, g : Ω → C be holomorphic
+on Ω. Suppose there exists a ∈ Ω and R > 0 such that B(a, R) ⊂ Ω and
+|g(z) − f(z)| < |g(z)| for all z ∈ ∂B(a, R),
+then f and g have the same number of zeros inside B(a, R).
+We define the functions f, g : C → C by
+f(z) := z3 − (λ0 + κ0)z2 +
+�
+λ0k0 + 1
+n2
+�R2¯θ
+c0
++ R¯θ
+��
+z − R¯θκ0
+n2
+and
+g(z) := z3 − (λ0 + κ0)z2 + λ0k0z
+for all z ∈ C and n ∈ Z∗. The roots of g are 0, λ0 and κ0. We now choose R := 1
+2 min{λ0, κ0, |λ0 − κ0|}.
+Then, we have the following estimates
+|g(z) − f(z)| =
+����
+1
+n2
+�R2¯θ
+c0
++ R¯θ
+�
+z − R¯θκ0
+n2
+���� ≤ C
+n2 (|z| + 1)
+
+
+
+
+
+
+
+= C
+n2 (R + 1),
+for all z ∈ ∂B(0, R),
+≤ C
+n2 (λ0 + R + 1),
+for all z ∈ ∂B(λ0, R),
+≤ C
+n2 (κ0 + R + 1),
+for all z ∈ ∂B(κ0, R),
+19
+
+for all n ∈ Z∗. On the other hand, the choice of R tells us that the function g does not have any root
+on the sets ∂B(0, R), ∂B(λ0, R) and ∂B(κ0, R). Therefore, inf|z|=R |g(z)| > 0,inf|z−λ0|=R |g(z)| > 0 and
+inf|z−κ0|=R |g(z)| > 0. This implies, for |n| large enough, we have
+|g(z) − f(z)| < |g(z)| for all z ∈ ∂B(0, R) ∪ ∂B(λ0, R) ∪ ∂B(κ0, R).
+Thus, the function f has a unique root inside each of the sets B(0, R), B(λ0, R) and B(κ0, R). We denote
+these roots by zn
+1 , zn
+2 and zn
+3 respectively. We now find asymptotic expressions of these roots.
+Asymptotic expression of zn
+1 . Since zn
+1 ∈ B(0, R), we have
+zn
+1 =
+1
+(zn
+1 − λ0)(zn
+1 − κ0)
+�R¯θκ0
+n2
+− 1
+n2
+�R2¯θ
+c0
++ R¯θ
+�
+zn
+1
+�
+and therefore
+|zn
+1 | ≤
+1
+|zn
+1 − λ0| |zn
+1 − κ0|
+�����
+R¯θκ0
+n2
+���� +
+����
+1
+n2
+�R2¯θ
+c0
++ R¯θ
+�
+zn
+1
+����
+�
+≤
+C
+|n|2
+for |n| large enough. To find the asymptotic expression of zn
+1 , we write f(zn
+1 ) = 0 in the following way
+zn
+1 = R¯θκ0
+n2
+�
+λ0κ0 − (λ0 + κ0)zn
+1 + (zn
+1 )2 + 1
+n2
+�R¯θ
+c0
++ R¯θ
+��−1
+= R¯θκ0
+n2
+1
+λ0κ0
+�
+1 − (λ0 + κ0)
+λ0κ0
+zn
+1 +
+1
+λ0κ0n2
+�R¯θ
+c0
++ R¯θ
+�
++ O(|n|−4)
+�−1
+= ¯ω
+n2
+�
+1 + (λ0 + κ0)
+λ0κ0
+zn
+1 −
+1
+λ0κ0n2
+�R¯θ
+c0
++ R¯θ
+�
++ O(|n|−4)
+�
+= ¯ω
+n2 + O(|n|−4),
+since |zn
+1 | ≤ C
+n2 for all |n| large.
+Asymptotic expression of zn
+2 . Since zn
+2 ∈ B(λ0, R), we have
+zn
+2 − λ0 =
+1
+zn
+2 (zn
+2 − κ0)
+�R¯θκ0
+n2
+− 1
+n2
+�R2¯θ
+c0
++ R¯θ
+�
+zn
+1
+�
+and therefore
+|zn
+2 − λ0| ≤
+1
+|zn
+2 | |zn
+2 − κ0|
+�����
+R¯θκ0
+n2
+���� +
+����
+1
+n2
+�R2¯θ
+c0
++ R¯θ
+�
+zn
+1
+����
+�
+≤
+C
+|n|2
+for |n| large enough. Thus, we can write
+zn
+2 = λ0 + O(|n|−2)
+for all |n| large.
+Asymptotic expression of zn
+3 . Following the similar approach as mentioned above, we can get
+zn
+3 = κ0 + O(|n|−2)
+for all |n| large.
+Combining all of the above, we obtain the asymptotic expressions of the roots of (3.15) as
+ǫn
+1 := λ0in + O(|n|−1),
+ǫn
+2 := κ0in + O(|n|−1),
+ǫn
+3 := − ¯ω
+in + O(|n|−3)
+for all |n| large. Therefore, eigenvalues of the matrix Rn are νn
+1 , νn
+2 and νn
+3 with the asymptotic expressions
+νn
+1 = λ0in + ¯u + O(|n|−1),
+(3.17)
+νn
+2 = κ0in + ¯u + O(|n|−1),
+(3.18)
+νn
+3 = ¯u − ¯ω
+in + O(|n|−3),
+(3.19)
+for all |n| large.
+To find the eigenvectors of the matrix Rn, we now consider the equation
+Rnαn = νn
+3 αn,
+n ∈ Z∗,
+20
+
+where αn = (αn
+1 , αn
+2, αn
+3 )†, that is,
+(¯u − νn
+3 )αn
+1 + ¯ραn
+2 = 0,
+(3.20)
+R¯θ
+¯ρ αn
+1 + (λ0in + ¯u − νn
+3 )αn
+2 + Rαn
+3 = 0,
+(3.21)
+R¯θ
+c0
+αn
+2 + (κ0in + ¯u − νn
+3 )αn
+3 = 0,
+(3.22)
+for all n ∈ Z∗. One solution is given by
+αn
+1 = R¯ρ,
+αn
+2 = −R(¯u − νn
+3 ),
+αn
+3 = (λ0in + ¯u − νn
+3 )(¯u − νn
+3 ) − R¯θ,
+n ∈ Z∗.
+We next consider the equation
+Rnβn = νn
+1 βn,
+n ∈ Z∗,
+where βn = (βn
+1 , βn
+2 , βn
+3 )†, that is,
+(¯u − νn
+1 )βn
+1 + ¯ρβn
+2 = 0,
+(3.23)
+R¯θ
+¯ρ βn
+1 + (λ0in + ¯u − νn
+1 )βn
+2 + Rβn
+3 = 0,
+(3.24)
+R¯θ
+c0
+βn
+2 + (κ0in + ¯u − νn
+1 )βn
+3 = 0,
+(3.25)
+for all n ∈ Z∗. One solution is given by
+βn
+1 = −
+R¯ρ
+¯u − νn
+1
+,
+βn
+2 = R,
+βn
+3 =
+1
+¯u − νn
+1
+[R¯θ − (λ0in + ¯u − νn
+1 )(¯u − νn
+1 )],
+n ∈ Z∗.
+We finally consider the equation
+Rnγn = νn
+2 γn,
+n ∈ Z∗,
+where γn = (γn
+1 , γn
+2 , γn
+3 )†, that is,
+(¯u − νn
+2 )γn
+1 + ¯ργn
+2 = 0,
+(3.26)
+R¯θ
+¯ρ γn
+1 + (λ0in + ¯u − νn
+2 )γn
+2 + Rγn
+3 = 0,
+(3.27)
+R¯θ
+c0
+γn
+2 + (κ0in + ¯u − νn
+2 )γn
+3 = 0,
+(3.28)
+for all n ∈ Z∗. One solution is given by
+γn
+1 = (λ0in + ¯u − νn
+2 )(κ0in + ¯u − νn
+2 ) − R2¯θ
+c0
+,
+γn
+2 = −R¯θ
+¯ρ (κ0in + ¯u − νn
+2 ),
+γn
+3 = R2¯θ2
+¯ρc0
+,
+n ∈ Z∗.
+Therefore, the eigenvectors of Rn corresponding to the eigenvalues νn
+3 , νn
+1 and νn
+2 are respectively αn, βn
+and γn, where
+αn =
+
+
+
+αn
+1
+αn
+2
+αn
+3
+
+
+ ,
+βn =
+
+
+
+βn
+1
+βn
+2
+βn
+3
+
+
+ ,
+γn =
+
+
+
+γn
+1
+γn
+2
+γn
+3
+
+
+ ,
+for all n ∈ Z∗. Hence, the eigenvalues of the operator A∗ are νh
+n := inν3
+n, νp1
+n := inν2
+n and νp2
+n := inν1
+n
+for all n ∈ Z∗ with the asymptotic expressions
+νh
+n = ¯uin − ¯ω + O(|n|−1),
+νp1
+n = −λ0n2 + ¯uin + O(1),
+νp2
+n = −κ0n2 + ¯uin + O(1),
+for |n| large enough and the corresponding eigenfunctions are
+Φh
+n(x) := En(x)αn = αneinx,
+Φp1
+n (x) := En(x)βn = βneinx,
+Φp2
+n (x) := En(x)γn = γneinx,
+for all n ∈ Z∗ and x ∈ (0, 2π). This completes the proof.
+□
+Remark 3.10. Note that, all the eigenvalues of A∗ are simple at least for |n| large enough. Depending
+on the constants ¯ρ, ¯u, ¯θ, λ0, κ0, R and c0, there may be multiple eigenvalues, but that would be only finitely
+many of them. Therefore, in this case also, we assume that all the eigenvalues of A∗ are simple.
+21
+
+3.4. Observation Estimates. For any eigenvalue ν, let us denote the corresponding eigenfunction of
+A∗ by Φν and let E(A∗) be the set of all eigenfunctions of A∗. We define the observation operator
+corresponding to the system (1.5) as follows:
+B∗
+ρΦ := R¯θ¯uξ(2π) + R¯θ¯ρη(2π),
+(3.29)
+B∗
+uΦ := R¯ρ¯θξ(2π) + ¯ρ2¯uη(2π) + λ0¯ρ2ηx(2π) + R¯ρ2ζ(2π),
+(3.30)
+B∗
+θΦ := R¯ρ2η(2π) + ¯ρ2c0¯u
+¯θ
+ζ(2π) + ¯ρ2c0κ0
+¯θ
+ζx(2π),
+(3.31)
+for all Φ = (ξ, η, ζ)† ∈ E(A∗). Then, we have the following estimates:
+Lemma 3.11. For all Φ ∈ E(A∗) \ {Φ0}, the observation operators satisfies B∗
+ρΦ ̸= 0 and B∗
+uΦ ̸= 0.
+Moreover, we have the following estimates
+��B∗
+ρΦh
+n
+�� ≥ C,
+��B∗
+ρΦp1
+n
+�� ≥ C,
+��B∗
+ρΦp2
+n
+�� ≥ C,
+(3.32)
+��B∗
+uΦh
+n
+�� ≥ C
+|n|,
+|B∗
+uΦp1
+n | ≥ C |n| , |B∗
+uΦp2
+n | ≥ C,
+(3.33)
+��B∗
+θΦh
+n
+�� ≥ C
+|n|,
+|B∗
+uΦp2
+n | ≥ C
+��B∗
+ρΦp2
+n
+�� ≥ C |n| ,
+(3.34)
+for some C > 0 and all n ∈ Z∗.
+Proof. Recall from the proof of Lemma 3.7 that ν1, ν2, ν3 ̸= 0 (Claim 1). We consider the following cases:
+Case 1. (Control acts in density) We have
+B∗
+ρΦh
+n = R¯θ¯uξh
+n(2π) + R¯ρ¯θηh
+n(2π) = R¯θ(¯uαn
+1 + ¯ραn
+2) = R¯θνn
+3 αn
+1 ̸= 0,
+for all n ∈ Z∗, thanks to the eigenvector equation (3.20). We similarly have
+B∗
+ρΦp1
+n = R¯θ¯uξp1
+n (2π) + R¯ρ¯θηp1
+n (2π) = R¯θ(¯uβn
+1 + ¯ρβn
+2 ) = R¯θνn
+1 βn
+1 ̸= 0,
+and
+B∗
+ρΦp2
+n = R¯θ¯uξp2
+n (2π) + R¯ρ¯θηp2
+n (2π) = R¯θ(¯uγn
+1 + ¯ργn
+2 ) = R¯θνn
+2 γn
+1 ̸= 0,
+for all n ∈ Z∗, thanks to the equations (3.23) and (3.26).
+Case 2. (Control acts in Velocity) We have
+B∗
+uΦh
+n = R¯ρ¯θξh
+n(2π) + λ0¯ρ2(ηh
+n)x(2π) + ¯ρ2¯uηh
+n(2π) + R¯ρ2ζh
+n(2π)
+= R¯ρ¯θαn
+1 + λ0¯ρ2inαn
+2 + ¯ρ2¯uαn
+2 + R¯ρ2αn
+3
+= ¯ρ2νn
+3 αn
+2 ̸= 0,
+for all n ∈ Z∗, thanks to the equation (3.21). We similarly have
+B∗
+uΦp1
+n = R¯ρ¯θξp1
+n (2π) + ν0¯ρ2(ηp1
+n )x(2π) + ¯ρ2¯uηp1
+n (2π) + R¯ρ2ζp1
+n (2π)
+= R¯ρ¯θβn
+1 + ν0¯ρ2inβn
+2 + ¯ρ2¯uβn
+2 + R¯ρ2βn
+3
+= ¯ρ2νn
+1 βn
+2 ̸= 0.
+and
+B∗
+uΦp2
+n = R¯ρ¯θξp2
+n (2π) + λ0¯ρ2(ηp2
+n )x(2π) + ¯ρ2¯uηp2
+n (2π) + R¯ρ2ζp2
+n (2π)
+= R¯ρ¯θγn
+1 + λ0 ¯ρ2inγn
+2 + ¯ρ2¯uγn
+2 + R¯ρ2γn
+3
+= ¯ρ2νn
+2 γn
+2 ̸= 0,
+for all n ∈ Z∗, thanks to (3.24) and (3.27).
+Case 3. (Control acts in Temperature) We have
+B∗
+θΦh
+n = R¯ρ2ηh
+n(2π) + ¯ρ2c0¯u
+¯θ
+ζh
+n(2π) + ¯ρ2c0κ0
+¯θ
+(ζh
+n)x(2π)
+= R¯ρ2αn
+2 + ¯ρ2c0
+¯θ
+(¯u + κ0in)αn
+3 = ¯ρ2c0
+¯θ
+νn
+3 αn
+3 ̸= 0,
+22
+
+for all n ∈ Z∗, thanks to (3.22). Similarly, we have
+B∗
+θΦp1
+n = R¯ρ2ηp1
+n (2π) + ¯ρ2c0¯u
+¯θ
+ζp1
+n (2π) + ¯ρ2c0κ0
+¯θ
+(ζp1
+n )x(2π)
+= R¯ρ2βn
+2 + ¯ρ2c0
+¯θ
+(¯u + κ0in)βn
+3 = ¯ρ2c0
+¯θ
+νn
+1 βn
+3 ̸= 0,
+and
+B∗
+θΦp2
+n = R¯ρ2ηp2
+n (2π) + ¯ρ2c0¯u
+¯θ
+ζp2
+n (2π) + ¯ρ2c0κ0
+¯θ
+(ζp2
+n )x(2π)
+= R¯ρ2γn
+2 + ¯ρ2c0
+¯θ
+(¯u + κ0in)γn
+3 = ¯ρ2c0
+¯θ
+νn
+2 γn
+3 ̸= 0,
+for all n ∈ Z∗, thanks to (3.25) and (3.28). The estimates on the observation terms are then follows
+directly from the asymptotic expressions (3.17)-(3.18)-(3.19) and Remark 3.8.
+□
+3.5. Observability Inequality. Let us rewrite the eigenvalues as {νh
+n, νp
+n}n∈Z∗, where
+νp
+n =
+�
+νp1
+k ,
+if n = 2k − 1, k ∈ Z
+νp2
+k ,
+if n = 2k, k ∈ Z∗,
+for all n ∈ Z∗ and νh
+n is as defined earlier. We also denote the observation term
+B∗Φp
+n =
+�
+B∗Φp1
+k ,
+if n = 2k − 1, k ∈ Z,
+B∗Φp2
+n ,
+if n = 2k, k ∈ Z∗,
+for all n ∈ Z∗. Recall that, we have defined the set
+S :=
+�
+(λ0, κ0) :
+�
+λ0
+κ0
+/∈ Q
+�
+.
+Then, the eigenvalues (νh
+n)n∈Z∗ satisfies hypotheses (H1)-(H2) with τ = ¯u, β = −¯ω and for all (λ0, κ0) ∈
+S, the sequence (νp
+n)n∈Z∗ satisfies hypotheses (P1)-(P2)-(P3)-(P4) of Lemma 1.15.
+Since the eigenfunctions E(A∗)\{Φ0} of A∗ forms a Riesz basis in ( ˙L2(0, 2π))3, therefore any (σT , vT , ϕT )† ∈
+( ˙L2(0, 2π))3 can be written as
+(σT , vT , ϕT )† =
+�
+n∈Z∗
+ah
+nΦh
+n +
+�
+n∈Z∗
+ap1
+n Φp1
+n +
+�
+n∈Z∗
+ap2
+n Φp2
+n ,
+for some (ah
+n)n∈Z∗, (ap1
+n )n∈Z∗, (ap2
+n )n∈Z∗ ∈ ℓ2. Then, the solution to the adjoint system (3.3) is
+(σ(t, x), v(t, x), ϕ(t, x))† =
+�
+n∈Z∗
+ah
+neνh
+n(T −t)Φh
+n(x) +
+�
+n∈Z∗
+ap1
+n eνp1
+n (T −t)Φp1
+n (x) +
+�
+n∈Z∗
+ap2
+n eνp2
+n (T −t)Φp2
+n (x),
+for (t, x) ∈ (0, T ) × (0, 2π), that is,
+σ(t, x) =
+�
+n∈Z∗
+ah
+neνh
+n(T −t)αn
+1einx +
+�
+n∈Z∗
+ap1
+n eνp1
+n (T −t)βn
+1 einx +
+�
+n∈Z∗
+ap2
+n eνp2
+n (T −t)γn
+1 einx,
+v(t, x) =
+�
+n∈Z∗
+ah
+neνh
+n(T −t)αn
+2einx +
+�
+n∈Z∗
+ap1
+n eνp1
+n (T −t)βn
+2 einx +
+�
+n∈Z∗
+ap2
+n eνp2
+n (T −t)γn
+2 einx,
+ϕ(t, x) =
+�
+n∈Z∗
+ah
+neνh
+n(T −t)αn
+3einx +
+�
+n∈Z∗
+ap1
+n eνp1
+n (T −t)βn
+3 einx +
+�
+n∈Z∗
+ap2
+n eνp2
+n (T −t)γn
+3 einx,
+for (t, x) ∈ (0, T ) × (0, 2π). We then have
+��(σ(0), v(0), ϕ(0))†��2
+( ˙L2(0,2π))3 ≤ C
+� �
+n∈Z∗
+��ah
+n
+��2 ��Φh
+n
+��2
+L2(0,2π) +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T ∥Φp
+n∥2
+L2(0,2π)
+�
+(3.35)
+≤ C
+� �
+n∈Z∗
+��ah
+n
+��2 +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+23
+
+and
+��(σ(0), v(0), ϕ(0))†��2
+˙H−s
+per(0,2π)×( ˙L2(0,2π))2 ≤ C
+� �
+n∈Z∗
+��ah
+n
+��2 ��Φh
+n
+��2
+H−s
+per(0,2π) +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T ∥Φp
+n∥2
+L2(0,2π)
+�
+(3.36)
+≤ C
+� �
+n∈Z∗
+��ah
+n
+��2
+1
+|n|2s +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+.
+3.5.1. Proof of Theorem 1.7. Let T > 2π
+¯u . The observability inequality is given by
+(3.37)
+� T
+0
+��R¯θ¯uσ(t, 2π) + R¯θ¯ρv(t, 2π)
+��2 dt ≥ C
+��(σ(0), v(0), ϕ(0))†��2
+( ˙L2(0,2π))3 ,
+for all (σT , vT , ϕT )† ∈ ( ˙L2(0, 2π))3. We have the observation term
+� T
+0
+��R¯θ¯uσ(t, 2π) + R¯θ¯ρv(t, 2π)
+��2 dt
+=
+� T
+0
+�����
+�
+n∈Z∗
+ah
+nB∗
+ρΦh
+neνh
+n(T −t) +
+�
+n∈Z∗
+ap1
+n B∗
+ρΦp1
+n eνp1
+n (T −t) +
+�
+n∈Z∗
+ap2
+n B∗
+ρΦp2
+n eνp2
+n (T −t)
+�����
+2
+dt
+=
+� T
+0
+�����
+�
+n∈Z∗
+ah
+nB∗
+ρΦh
+neνh
+n(T −t) +
+�
+n∈Z∗
+ap
+nB∗
+ρΦp
+neνp
+n(T −t)
+�����
+2
+dt.
+Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.15), we have
+� T
+0
+��R¯θ¯uσ(t, 2π) + R¯θ¯ρv(t, 2π)
+��2 dt ≥ C
+� �
+n∈Z∗
+��ah
+n
+��2 ��B∗
+ρΦh
+n
+��2 e2ℜ(νh
+n)(T −t) +
+�
+n∈Z∗
+|ap
+n|2 ��B∗
+ρΦp
+n
+��2 e2ℜ(νp
+n)T
+�
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2 +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+,
+thanks to the estimate 3.32. This estimate together with the norm estimate (3.35), the observability
+inequality (3.37) is now follows. This completes the proof.
+3.5.2. Proof of Theorem 1.9. Let T > 2π
+¯u . The observability inequality is given by
+(3.38)
+� T
+0
+��R¯ρ¯θσ(t, 2π) + λ0¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)
+��2 dt ≥ C
+��(σ(0), v(0), ϕ(0))†��2
+˙H−s
+per(0,2π)×( ˙L2(0,2π))2 ,
+for all (σT , vT , ϕT )† ∈ ˙H−s
+per(0, 2π) × ( ˙L2(0, 2π))2. We have
+� T
+0
+��R¯ρ¯θσ(t, 2π) + λ0¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)
+��2 dt
+=
+� T
+0
+�����
+�
+n∈Z∗
+ah
+nB∗
+uΦh
+neνh
+n(T −t) +
+�
+n∈Z∗
+ap1
+n B∗
+uΦp1
+n eνp1
+n (T −t) +
+�
+n∈Z∗
+ap2
+n B∗
+uΦp2
+n eνp2
+n (T −t)
+�����
+2
+dt
+=
+� T
+0
+�����
+�
+n∈Z∗
+ah
+nB∗
+uΦh
+neνh
+n(T −t) +
+�
+n∈Z∗
+ap
+nB∗
+uΦp
+neνp
+n(T −t)
+�����
+2
+dt.
+Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.15) and the observation
+estimates ((3.33)), we obtain
+� T
+0
+��R¯ρ¯θσ(t, 2π) + λ0 ¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)
+��2 dt
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2 ��B∗
+uΦh
+n
+��2 e2ℜ(νh
+n)(T −t) +
+�
+n∈Z∗
+|ap
+n|2 |B∗
+uΦp
+n|2 e2ℜ(νp
+n)T
+�
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2
+1
+|n|2 +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+.
+24
+
+Since s ≥ 1, we have
+1
+|n|2 ≥
+1
+|n|2s for all n ∈ Z∗ and therefore
+� T
+0
+��R¯ρ¯θσ(t, 2π) + λ0 ¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)
+��2 dt
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2
+1
+|n|2s +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+≥ C
+��(σ(0), v(0), ϕ(0))†�� ˙H−s
+per(0,2π)×( ˙L2(0,2π))2 ,
+thanks to the estimate (3.35). This proves the observability inequality (3.38).
+3.5.3. Proof of Theorem 1.11. Let T > 2π
+¯u . The observability inequality is given by
+(3.39)
+� T
+0
+����R¯ρ2v(t, 2π) + ¯ρ2c0¯u
+¯θ
+ϕ(t, 2π) + ¯ρ2c0κ0
+¯θ
+ϕx(t, 2π)
+����
+2
+dt ≥ C
+��(σ(0), v(0), ϕ(0))†��2
+˙H−s
+per(0,2π)×( ˙L2(0,2π))2 ,
+for all (σT , vT , ϕT )† ∈ ˙H−s
+per(0, 2π) × ( ˙L2(0, 2π))2. We have
+� T
+0
+����R¯ρ2v(t, 2π) + ¯ρ2c0¯u
+¯θ
+ϕ(t, 2π) + ¯ρ2c0κ0
+¯θ
+ϕx(t, 2π)
+����
+2
+dt
+=
+� T
+0
+�����
+�
+n∈Z∗
+ah
+nB∗
+θΦh
+neνh
+n(T −t) +
+�
+n∈Z∗
+ap1
+n B∗
+θΦp1
+n eνp1
+n (T −t) +
+�
+n∈Z∗
+ap2
+n B∗
+θΦp2
+n eνp2
+n (T −t)
+�����
+2
+dt
+=
+� T
+0
+�����
+�
+n∈Z∗
+ah
+nB∗
+θΦh
+neνh
+n(T −t) +
+�
+n∈Z∗
+ap
+nB∗
+θΦp
+neνp
+n(T −t)
+�����
+2
+dt.
+Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.15) and the observation
+estimates ((3.34)), we have
+� T
+0
+����R¯ρ2v(t, 2π) + ¯ρ2c0¯u
+¯θ
+ϕ(t, 2π) + ¯ρ2c0κ0
+¯θ
+ϕx(t, 2π)
+����
+2
+dt
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2 ��B∗
+θΦh
+n
+��2 e2ℜ(νh
+n)(T −t) +
+�
+n∈Z∗
+|ap
+n|2 |B∗
+θΦp
+n|2 e2ℜ(νp
+n)T
+�
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2
+1
+|n|2 +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+Since s ≥ 1, therefore
+1
+|n|2 ≥
+1
+|n|2s for all n ∈ Z∗. Thus we get
+� T
+0
+����R¯ρ2v(t, 2π) + ¯ρ2c0¯u
+¯θ
+ϕ(t, 2π) + ¯ρ2c0κ0
+¯θ
+ϕx(t, 2π)
+����
+2
+dt
+≥ C
+� �
+n∈Z∗
+��ah
+n
+��2
+1
+|n|2s +
+�
+n∈Z∗
+|ap
+n|2 e2ℜ(νp
+n)T
+�
+≥ C
+��(σ(0), v(0), ϕ(0))†�� ˙H−s
+per(0,2π)×( ˙L2(0,2π))2 ,
+and the observability inequality (3.39) follows.
+3.6. Lack of Null Controllability for Less Regular Initial States.
+3.6.1. Proof of Proposition 1.10. For (σT , vT , ϕT )† = Φh
+n, the solution to the adjoint system is
+(σ(t, x), v(t, x), ϕ(t, x))† = eνh
+n(T −t)Φh
+n(x),
+for (t, x) ∈ (0, T ) × (0, 2π). For large n, we have the following estimate
+��Φh
+n
+��
+H−s
+per(0,2π)×(L2(0,2π))2 ≥
+C
+|n|s ,
+and therefore
+��(σ(0), v(0), ϕ(0))†��2
+H−s
+per(0,2π)×(L2(0,2π))2 ≥
+C
+|n|2s .
+25
+
+We also have
+� T
+0
+��B∗
+uΦh
+n
+��2 dt ≤
+C
+|n|2 .
+Thus if the observability inequality (3.38) holds, then we get
+C
+|n|2s ≤
+C
+|n|2 =⇒ |n|2−2s ≤ C,
+which is not possible since 0 ≤ s < 1. This completes the proof.
+Remark 3.12. Since the observation terms B∗
+θΦh
+n and B∗
+uΦh
+n have same upper bounds, proof of Propo-
+sition 1.12 will be similar to above, so we omit the details.
+3.7. Lack of Controllability at Small Time. The proof will be similar to the barotropic case, that
+is, the proof of Proposition 1.2.
+3.7.1. Proof of Proposition 1.8. Let 0 < T < 2π
+¯u . Following the notations in the proof of Proposition 1.2,
+we consider the system
+(3.40)
+
+
+
+
+
+˜σt(t, x) + ¯u˜σx(t, x) − ¯ω˜σ(t, x) = 0,
+(t, x) ∈ (0, T ) × (0, 2π),
+˜σ(t, 0) = ˜σ(t, 2π),
+t ∈ (0, T ),
+˜σ(T, x) = ˜σN
+T (x),
+x ∈ (0, 2π).
+Since supp(˜σN
+T ) ⊂ supp(˜σT ) ⊂ (T, 2π), the solution satisfies ˜σN(t, 0) = ˜σN(t, 2π) = 0. We now consider
+the adjoint to our main system
+(3.41)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−σt(t, x) − ¯uσx(t, x) − ¯ρvx(t, x) = 0, in (0, T ) × (0, 2π),
+−vt(t, x) − λ0vxx(t, x) − R¯θ
+¯ρ σx(t, x) − ¯uvx(t, x) − Rϕx(t, x) = 0, in (0, T ) × (0, 2π),
+−ϕt(t, x) − κ0ϕxx(t, x) − R¯θ
+c0
+vx(t, x) − ¯uϕx(t, x) = 0, in (0, T ) × (0, 2π),
+σ(t, 0) = σ(t, 2π),
+v(t, 0) = v(t, 2π),
+vx(t, 0) = vx(t, 2π),
+t ∈ (0, T ),
+ϕ(t, 0) = ϕ(t, 2π),
+ϕx(t, 0) = ϕx(t, 2π),
+t ∈ (0, T ),
+σ(T, x) = σN
+T (x),
+v(T, x) = vN
+T (x),
+ϕ(T, x) = ϕN
+T (x),
+x ∈ (0, 2π),
+where we choose vN
+T and ϕN
+T such that
+(˜σN
+T , vN
+T , ϕN
+T )† =
+�
+|n|≥N+1
+˜ah
+nΦh
+n
+with ˜ah
+nαn
+1 := anP N(n) for all |n| ≥ N + 1. We write the solutions to the systems (3.40) and (3.41)
+respectively as
+˜σN(t, x) =
+�
+|n|≥N+1
+anP N(n)e(¯uin−¯ω)(T −t)einx,
+(3.42)
+σN(t, x) =
+�
+|n|≥N+1
+anP N(n)eνh
+n(T −t)einx,
+(3.43)
+vN(t, x) =
+�
+|n|≥N+1
+anP N(n)βn
+1
+αn
+1
+eνh
+n(T −t)einx,
+(3.44)
+ϕN(t, x) =
+�
+|n|≥N+1
+anP N(n) γn
+1
+αn
+1
+eνh
+n(T −t)einx,
+(3.45)
+26
+
+for all (t, x) ∈ [0, T ] × [0, 2π]. Similar to the barotropic case, we prove that the solution component σN
+approximates the solution ˜σN. Indeed, we have
+��σN(·, x) − ˜σN(·, x)
+��2
+L2(0,T ) ≤
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 ���eνh
+n(T −t) − e(¯uin−¯ω)(T −t)���
+2
+L2(0,T )
+≤
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 ���e(¯uin−¯ω)(T −t)eO(|n|−1)(T −t) − 1
+���
+2
+L2(0,T )
+≤
+C
+|n|2
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 ,
+for all x ∈ [0, 2π]. We also have for all x ∈ [0, 2π]
+��vN(·, x)
+��2
+L2(0,T ) ≤
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 |βn
+1 |2
+|αn
+1 |2
+���eνh
+n(T −t)���
+2
+L2(0,T )
+≤ C
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2
+1
+|n|2
+≤
+C
+|N|2
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2
+We suppose that the following observability inequality holds
+� T
+0
+��R¯θ¯uσN(t, 2π) + R¯θ¯ρvN(t, 2π)
+��2 dt ≥ C
+��(σN(0), vN(0), ϕN(0))†��2
+(L2(0,2π))3 .
+Then, we have
+��(σN(0), vN(0), ϕN(0))†��2
+(L2(0,2π))3
+≤ C
+� T
+0
+��R¯θ¯uσN(t, 2π) + R¯θ¯ρvN(t, 2π)
+��2 dt
+≤ C
+� T
+0
+�
+¯u2 ��(σN(t, 2π) − ˜σN(t, 2π))
+��2 + ¯u2 ��˜σN(t, 2π)
+��2 + ¯ρ2 ��vN(t, 2π)
+��2�
+dt
+≤ C
+N 2
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 ,
+since ˜σN(t, 0) = 0 = ˜σN(t, 2π) for all t ∈ (0, T ). Thus, we get
+��σN(0)
+��2
+L2(0,2π) ≤
+��(σN(0), vN(0), ϕN(0))†��2
+(L2(0,2π))3
+≤ C
+N 2
+�
+|n|≥N+1
+|an|2 ��P N(n)
+��2 ≤ C
+N 2
+��σN(0)
+��2
+L2(0,2π) ,
+since ℜ(νh
+n) is bounded. Therefore, 1 ≤
+C
+N 2 for all N and hence the above inequality cannot hold. This
+is a contradiction and the proof is complete.
+4. Further Comments and Conclusions
+4.1. Controllability Results Using Neumann Boundary Conditions. We consider the system
+(1.1) with the initial state (1.2) and the boundary conditions
+ρ(t, 0) = ρ(t, 2π),
+u(t, 0) = u(t, 2π),
+ux(t, 0) = ux(t, 2π) + q1(t),
+t ∈ (0, T ),
+(4.1)
+where q1 is a boundary control that acts on the velocity through Neumann conditions. Since the obser-
+vation terms satisfies similar estimates as in (2.20), following the proof of Theorem 1.3 and Proposition
+1.4, we can obtain the null controllability of the system (1.1)-(1.2)-(4.1) at time T > 2π
+¯u in the space
+˙Hs
+per(0, 2π) × ˙L2(0, 2π) for s ≥ 1, and the null controllability fails in the space Hs
+per(0, 2π) × L2(0, 2π)
+for 0 ≤ s < 1. In this case also, null controllability of the system (1.1)-(1.2)-(4.1) is inconclusive when
+the time is small (0 < T ≤ 2π
+¯u ).
+27
+
+Similar to the barotropic case, we consider the system (1.5) with the initial state (1.6) and the
+boundary conditions
+ρ(t, 0) = ρ(t, 2π),
+u(t, 0) = u(t, 2π),
+ux(t, 0) = ux(t, 2π) + q2(t),
+(4.2)
+θ(t, 0) = θ(t, 2π),
+θx(t, 0) = θx(t, 2π),
+t ∈ (0, T ).
+In this case also, following the proof of Theorem 1.9 and Proposition 1.10, we get null controllability of
+the system (1.5)-(1.6)-(4.2) at time T > 2π
+¯u in the space ˙Hs
+per(0, 2π) × ( ˙L2(0, 2π))2 for s ≥ 1, and null
+controllability fails in the space Hs
+per(0, 2π) × (L2(0, 2π))2 for 0 ≤ s < 1.
+We next consider the system (1.5) with the initial state (1.6) and the boundary conditions
+ρ(t, 0) = ρ(t, 2π),
+u(t, 0) = u(t, 2π),
+ux(t, 0) = ux(t, 2π),
+(4.3)
+θ(t, 0) = θ(t, 2π),
+θx(t, 0) = θx(t, 2π) + q3(t),
+t ∈ (0, T ).
+Similar to the previous case, following the proof of Theorem 1.11 and Proposition 1.12, we get null
+controllability of the system (1.5)-(1.6)-(4.3) at time T > 2π
+¯u in the space ˙Hs
+per(0, 2π) × ( ˙L2(0, 2π))2 for
+s ≥ 1, and null controllability fails in the space Hs
+per(0, 2π) × (L2(0, 2π))2 for 0 ≤ s < 1.
+For both systems (1.5)-(1.6)-(4.2) and (1.5)-(1.6)-(4.3), null controllability is inconclusive for a small
+time 0 < T ≤ 2π
+¯u .
+4.2. More number of controls. Adding controls in both velocity and temperature components through
+periodic boundary conditions does not improve the null controllability result of the system (1.5) with
+respect to the regularity of the initial states. Estimates of the observation terms remain the same as in
+the control acts in velocity or temperature.
+4.3. Controllability under Dirichlet Boundary Conditions. Let us consider the system (1.1) in
+the interval (0, 1) with the initial state (1.2) and the following boundary conditions
+(4.4)
+ρ(t, 0) = p(t),
+u(t, 0) = 0,
+u(t, 1) = q(t),
+t ∈ (0, T ),
+where p and q are boundary controls. It is known in [3] that the system (1.1)-(1.2)-(4.4) (with q = 0) is
+null controllability at a large time T using only one boundary control p ∈ L2(0, T ) provided the initial
+states belong to the space Hs
+per(0, 1) × L2(0, 1) with s > 1
+2. Null controllability of the system (1.1) using
+a boundary control acts only in velocity through Dirichlet conditions (that is, p = 0 in (4.4)) is still an
+open problem.
+In the case of non-barotropic fluids, null controllability of the system (1.5) at large time T using only
+one boundary control acts either in density, velocity or temperature through Dirichlet conditions is also
+an open problem.
+Appendix A. Proof of the Well-Posedness Results
+A.1. Existence of semigroup: proof of Lemma 3.1. The proof is divided into several parts.
+Part 1. The operator A is dissipative. Indeed, for all (ξ, η, ζ)† ∈ D(A)
+ℜ ⟨AU, U⟩(L2(0,2π))3 = ℜ
+�
+
+
+
+
+−¯uξx − ¯ρηx
+− R¯θ
+¯ρ ξx + λ0ηxx − ¯uηx − Rζx
+− R¯θ
+c0 ηx + κ0ζxx − ¯uζx
+
+
+
+ ,
+
+
+
+ξ
+η
+ζ
+
+
+
+�
+(L2(0,2π))3
+= ℜ
+�
+−R¯θ¯u
+� 2π
+0
+¯ξξxdx − R¯θ¯ρ
+� 2π
+0
+¯ξηxdx − R¯θ¯ρ
+� 2π
+0
+ξx¯ηdx + λ0¯ρ2
+� 2π
+0
+¯ηηxxdx − ¯ρ2¯u
+� 2π
+0
+¯ηηxdx
+−R¯ρ2
+� 2π
+0
+¯ηζxdx − R¯ρ2
+� 2π
+0
+ηx¯ζdx + κ0
+¯ρ2c0
+¯θ
+� 2π
+0
+¯ζζxxdx − ¯u ¯ρ2c0
+¯θ
+� 2π
+0
+¯ζζxdx
+�
+= −R¯θ¯u
+2
+� 2π
+0
+d
+dx(|ξ|2)dx − λ0¯ρ2
+� 2π
+0
+¯ηxηxdx − ¯ρ2¯u
+2
+� 2π
+0
+d
+dx(|η|2)dx − κ0
+¯ρ2c0
+¯θ
+� 2π
+0
+¯ζxζxdx
+− ¯u
+2
+¯ρ2c0
+¯θ
+� 2π
+0
+d
+dx(|ζ|2)dx − λ0¯ρ2
+� 2π
+0
+|ux|2 dx − κ0
+¯ρ2c0
+¯θ
+� 2π
+0
+|ζx|2 dx ≤ 0.
+28
+
+Part 2.
+The operator A is maximal.
+This is equivalent to the following.
+For any ν > 0 and any
+
+
+
+f
+g
+h
+
+
+ ∈ (L2(0, 2π))3, we can find a
+
+
+
+ξ
+η
+ζ
+
+
+ ∈ D(A) such that
+(νI − A)
+
+
+
+ξ
+η
+ζ
+
+
+ =
+
+
+
+f
+g
+h
+
+
+ ,
+that is,
+νξ + ¯uξx + ¯ρηx = f,
+νη + R¯θ
+¯ρ ξx − λ0ηxx + ¯uηx + Rζx = g,
+νζ + R¯θ
+c0
+ηx − κ0ζxx + ¯uζx = h.
+Let ǫ > 0. Instead of solving the above problem, we will solve the following regularized problem
+(A.1)
+
+
+
+
+
+
+
+νξ + ¯uξx − ǫξxx + ¯ρηx = f,
+νη + R¯θ
+¯ρ ξx − λ0ηxx + ¯uηx + Rζx = g,
+νζ + R¯θ
+c0 ηx − κ0ζxx + ¯uζx = h.
+with the following boundary conditions
+ξ(0) = ξ(2π),
+ξx(0) = ξx(2π),
+η(0) = η(2π),
+ηx(0) = ηx(2π),
+ζ(0) = ζ(2π),
+ζx(0) = ζx(2π).
+We now proceed through the following steps.
+Step 1.
+Using Lax-Milgram theorem, we first prove that the system (A.1) has a unique solution in
+(H1
+per(0, 2π))3. Define the operator B : (H1
+per(0, 2π))3 × (H1
+per(0, 2π))3 → C by
+B
+
+
+
+
+
+
+ξ
+η
+ζ
+
+
+ ,
+
+
+
+ξ1
+η1
+ζ1
+
+
+
+
+
+
+= ǫ
+� 2π
+0
+ξx( ¯ξ1)xdx + ¯ρ
+� 2π
+0
+ηx ¯ξ1dx + ¯u
+� 2π
+0
+ξx ¯ξ1dx + ν
+� 2π
+0
+ξ ¯ξ1dx
++ λ0
+� 2π
+0
+ηx( ¯η1)xdx + ¯u
+� 2π
+0
+ηx ¯η1dx + R¯θ
+¯ρ
+� 1
+0
+ξx ¯η1dx + R
+� 2π
+0
+ζx ¯η1dx + ν
+� 2π
+0
+η ¯η1dx
++ κ0
+� 2π
+0
+ζx( ¯ζ1)xdx + ¯u
+� 2π
+0
+ζx ¯ζ1dx + R¯θ
+c0
+� 2π
+0
+ηx ¯ζ1dx + ν
+� 2π
+0
+ζ ¯ζ1dx,
+for all
+
+
+
+ξ
+η
+ζ
+
+
+ ,
+
+
+
+ξ1
+η1
+ζ1
+
+
+ ∈ (H1
+per(0, 2π))3. Then, one can show that B is continuous and coercive. Thus, by
+Lax-Milgram theorem, for every ǫ > 0, there exists a unique solution (ξǫ, ηǫ, ζǫ)† ∈ (H1
+per(0, 2π))3 such
+that
+B
+
+
+
+
+
+
+ξǫ
+ηǫ
+ζǫ
+
+
+ ,
+
+
+
+ξ
+η
+ζ
+
+
+
+
+
+ = F
+
+
+
+
+
+
+ξ
+η
+ζ
+
+
+
+
+
+ ,
+∀
+
+
+
+ξ
+η
+ζ
+
+
+ ∈ (H1
+per(0, 2π))3,
+where F : (H1
+per(0, 2π))3 → C is the linear functional given by
+F
+
+
+
+
+
+
+ξ
+η
+ζ
+
+
+
+
+
+ :=
+� 2π
+0
+f ¯ξdx +
+� 2π
+0
+g¯ηdx +
+� 2π
+0
+h¯ζdx.
+29
+
+Step 2. Observe that
+ℜ
+
+
+B
+
+
+
+
+
+
+ξǫ
+ηǫ
+ζǫ
+
+
+ ,
+
+
+
+ξǫ
+ηǫ
+ζǫ
+
+
+
+
+
+
+
+
+ ≤
+� 2π
+0
+��fξǫ�� dx +
+� 2π
+0
+|gηǫ| dx +
+� 2π
+0
+��hζǫ�� dx
+≤ 1
+2
+� 2π
+0
+�
+|f|2 + |g|2 + |h|2�
+dx + 1
+2
+� 2π
+0
+���ξǫ��2 + |ηǫ|2 +
+��ζǫ��2�
+dx,
+which yields
+ǫ
+� 2π
+0
+|ξǫ
+x|2+ν
+2
+� 2π
+0
+|ξǫ|2+λ0
+� 2π
+0
+|ηǫ
+x|2+ν
+2
+� 2π
+0
+|ηǫ|2+κ0
+� 2π
+0
+|ζǫ
+x|2+ν
+2
+� 2π
+0
+|ζǫ|2 ≤ 1
+2
+� 2π
+0
+(|f|2+|g|2+|h|2)
+This shows that the sequences (ηǫ) and (ζǫ) are bounded in H1(0, 2π) and the sequences (ξǫ) and (√ǫξǫ
+x)
+are bounded in L2(0, 2π). Since the spaces H1(0, 2π) and L2(0, 2π) are reflexive, there exist subsequences,
+still denoted by (ηǫ), (ζǫ), (ξǫ), and functions ξ ∈ L2(0, 2π) and η ∈ H1(0, 2π) such that
+ηǫ ⇀ η in H1(0, 2π), and ξǫ ⇀ ξ in L2(0, 2π).
+Furthermore, we have
+� 2π
+0
+|ǫξǫ
+x|2 = ǫ
+� 1
+0
+��√ǫξǫ��2 → 0, as ǫ → 0.
+Now, since B
+
+
+
+
+
+
+ξǫ
+ηǫ
+ζǫ
+
+
+ ,
+
+
+
+ξ
+η
+ζ
+
+
+
+
+
+ = F
+
+
+
+
+
+
+ξ
+η
+ζ
+
+
+
+
+
+, for all
+
+
+
+ξ
+η
+ζ
+
+
+ ∈ (H1
+per(0, 2π))3, we may take
+
+
+
+ξ1
+0
+0
+
+
+ ∈
+(H1
+per(0, 2π))3, so that we obtain
+(A.2)
+ǫ
+� 2π
+0
+ξǫ
+x( ¯ξ1)xdx + ¯ρ
+� 2π
+0
+ηǫ
+x ¯ξ1dx + ¯u
+� 2π
+0
+ξǫ
+x ¯ξ1dx + ν
+� 2π
+0
+ξǫ ¯ξ1dx =
+� 2π
+0
+f ¯ξ1dx.
+Similarly, by taking
+
+
+
+0
+η1
+0
+
+
+ ,
+
+
+
+0
+0
+ζ1
+
+
+ ∈ (H1
+per(0, 2π))3, we get
+(A.3) λ0
+� 2π
+0
+ηǫ
+x( ¯η1)xdx+ ¯u
+� 2π
+0
+ηǫ
+x ¯η1dx+ R¯θ
+¯ρ
+� 1
+0
+ξǫ
+x ¯η1dx+R
+� 2π
+0
+ζǫ
+x ¯η1dx+ν
+� 2π
+0
+ηǫ ¯η1dx =
+� 2π
+0
+g ¯η1dx,
+and
+(A.4)
+κ0
+� 2π
+0
+ζǫ
+x( ¯ζ1)xdx + ¯u
+� 2π
+0
+ζǫ
+x ¯ζ1dx + R¯θ
+c0
+� 2π
+0
+ηǫ
+x ¯ζ1dx + ν
+� 2π
+0
+ζǫ ¯ζ1dx =
+� 2π
+0
+h ¯ζ1dx
+Integrating by parts, we get from equation (A.2) that,
+ǫ
+� 2π
+0
+ξǫ
+x( ¯ξ1)xdx + ¯ρ
+� 2π
+0
+ηǫ
+x ¯ξ1dx − ¯u
+� 2π
+0
+ξǫ( ¯ξ1)xdx + ν
+� 2π
+0
+ξǫ ¯ξ1dx =
+� 2π
+0
+f ¯ξ1dx.
+Then, passing to the limit ǫ → 0, we obtain
+¯ρ
+� 2π
+0
+ηx ¯ξ1dx + ¯u
+� 2π
+0
+ξx ¯ξ1dx + ν
+� 2π
+0
+ξ ¯ξ1dx =
+� 2π
+0
+f ¯ξ1dx,
+and the above relation is true ∀ξ1 ∈ C∞
+c (0, 2π). As a consequence,
+¯ρηx + ¯uξx + νξ = f,
+in the sense of distribution and therefore ¯uξx = f − ¯ρηx − νξ ∈ L2(0, 2π); in other words, ξ ∈ H1(0, 2π).
+We similarly have from identities (A.3) and (A.4)
+νη + R¯θ
+¯ρ ξx − λ0ηxx + ¯uηx + Rζx = g,
+νζ + R¯θ
+c0
+ηx − κ0ζxx + ¯uζx = h,
+in the sense of distribution and therefore η, ζ ∈ H2(0, 2π).
+30
+
+Step 3. We now show η(0) = η(2π) and ηx(0) = ηx(2π). Since the inclusion map i : H1(0, 2π) → C0( ¯
+0, 2π)
+is compact and ηǫ ⇀ η in H1(0, 2π), we obtain
+ηǫ → η
+in C0[0, 2π].
+Thus, (ηǫ(0), ηǫ(2π)) → (η(0), η(2π)). Since ηǫ(0) = ηǫ(2π) for all ǫ > 0, we have
+η(0) = η(2π).
+From (A.3), we have after passing the limit as ǫ → 0
+λ0
+� 2π
+0
+ηx( ¯η1)xdx + ¯u
+� 2π
+0
+ηx ¯η1dx + R¯θ
+¯ρ
+� 1
+0
+ξx ¯η1dx + R
+� 2π
+0
+ζx ¯η1dx + ν
+� 2π
+0
+η ¯η1dx =
+� 2π
+0
+g ¯η1dx.
+Integrating by parts, we get
+−λ0
+� 2π
+0
+ηxx ¯η1dx + λ0(ηx(2π) ¯η1(2π) − ηx(0) ¯η1(0)) + ¯u
+� 2π
+0
+ηx ¯η1dx + R¯θ
+¯ρ
+� 1
+0
+ξx ¯η1dx
++R
+� 2π
+0
+ζx ¯η1dx + ν
+� 2π
+0
+η ¯η1dx =
+� 2π
+0
+g ¯η1dx,
+and therefore
+ηx(2π) ¯η1(2π) − ηx(0) ¯η1(0) = 0
+that is ηx(0) = ηx(2π). In a similar way, we can obtain ζ(0) = ζ(2π) and ζx(0) = ζx(2π).
+We now show ξ(0) = ξ(2π). Recall that we have after taking limit as ǫ → 0
+¯ρ
+� 2π
+0
+ηx ¯ξ1dx − ¯u
+� 2π
+0
+ξ( ¯ξ1)xdx + ν
+� 2π
+0
+ξ ¯ξ1dx =
+� 2π
+0
+f ¯ξ1dx.
+Integrating by parts, we get
+(A.5)
+¯ρ
+� 2π
+0
+ηx ¯ξ1dx + ¯u
+� 2π
+0
+ξx ¯ξ1dx − ¯u(ξ(2π)ξ1(2π) − ξ(0)ξ1(0)) + ν
+� 2π
+0
+ξ ¯ξ1dx =
+� 2π
+0
+f ¯ξ1dx,
+and therefore
+ξ(0) = ξ(2π).
+So, we get
+
+
+
+ξ
+η
+ζ
+
+
+ ∈ D(A). Hence, the operator A is maximal.
+References
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+Navier-Stokes equations, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 33 (2016), pp. 529–574.
+[2] K. Beauchard, A. Koenig, and K. Le Balc’h, Null-controllability of linear parabolic transport systems, J. ´Ec.
+polytech. Math., 7 (2020), pp. 743–802.
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+
diff --git a/W9E2T4oBgHgl3EQfuggX/content/tmp_files/load_file.txt b/W9E2T4oBgHgl3EQfuggX/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1485 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf,len=1484
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='04080v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='AP]  10 Jan 2023  NULL CONTROLLABILITY OF ONE-DIMENSIONAL LINEARIZED  COMPRESSIBLE NAVIER-STOKES SYSTEM IN PERIODIC SETUP USING ONE  BOUNDARY CONTROL  JITEN KUMBHAKAR†  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In this article, we study boundary null controllability properties of the linearized compress-  ible Navier-Stokes equations in one dimension for both barotropic and non-barotropic fluids using only  one boundary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The control acts through periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We prove null control-  lability of the linearized compressible Navier-Stokes system (for both barotropic and non-barotropic  fluids) at large time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We also prove that our result is sharp with respect to the regularity of the  initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Finally, for both barotropic and non-barotropic fluids, we prove a lack of controllability  at small time T when a control acts in density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Introduction and Main Results  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Controllability of Linearized Compressible Navier-Stokes System (Barotropic)  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Controllability of Linearized Compressible Navier-Stokes System (Non-barotropic)  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Further Comments and Conclusions  27  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proof of the Well-Posedness Results  28  References  31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Introduction and Main Results  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Linearized Compressible Navier-Stokes System in 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let I = (0, +∞) be the time interval  and Ω ⊂ R be a spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For a compressible, isentropic (barotropic) fluid, that is, when the  pressure depends only on the density and the temperature is constant, the Navier-Stokes system in I ×Ω  consists of the equation of continuity and the momentum equation  ρt(t, x) + (ρu)x(t, x) = 0,  ρ(t, x)[ut(t, x) + u(t, x)ux(t, x)] + px(t, x) − (λ + 2µ)uxx(t, x) = 0,  where ρ denotes the density of the fluid, u is the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The constants λ, µ are called the viscosity  coefficients that satisfy µ > 0, λ + µ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The pressure p satisfies the following constitutive equation in  I × Ω  p(t, x) = aργ(t, x),  (t, x) ∈ I × Ω,  for some constants a > 0, γ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In the case of non-barotropic fluids, that is, when the pressure is a  function of both density and temperature of the fluid, the Navier-Stokes system consists of the equation  of continuity, the momentum equation, and an additional thermal energy equation  cνρ(t, x)[θt(t, x) + u(t, x)θx(t, x)] + θ(t, x)pθ(t, x)ux(t, x) − κθxx(t, x) − (λ + 2µ)u2  x(t, x) = 0,  where θ is the temperature of the fluid, cν is the specific heat constant, and κ is the heat conductivity  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For an ideal gas, Boyles law gives the pressure p(t, x) = Rρ(t, x)θ(t, x) in I × Ω with R as the  universal gas constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' See [19, Chapter 1] for more about compressible flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 35M30, 35Q30, 76N25, 93B05, 93B07, 93C20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Linearized compressible Navier-Stokes system, null-controllability, observability, boundary  control, Ingham-type inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  †Indian Institute of Science Education and Research Kolkata, Campus road, Mohanpur, West Bengal 741246, India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  jk17ip021@iiserkol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The Barotropic Case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let T > 0 be a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We first consider the Navier-Stokes system  for compressible, isentropic (barotropic) fluids linearized around some constant steady state (¯ρ, ¯u) with  ¯ρ > 0 and ¯u > 0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)  \uf8f1  \uf8f2  \uf8f3  ρt(t, x) + ¯uρx(t, x) + ¯ρux(t, x) = 0,  in (0, T ) × (0, 2π),  ut(t, x) − µ  ¯ρ uxx(t, x) + ¯uux(t, x) + aγ¯ργ−2ρx(t, x) = 0,  in (0, T ) × (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The initial conditions are  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)  ρ(0, x) = ρ0(x),  u(0, x) = u0(x),  x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We will consider two different problems, based on the act of control, by imposing any one of the following  boundary conditions on the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Control in Density:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3)  ρ(t, 0) = ρ(t, 2π) + p(t),  u(t, 0) = u(t, 2π),  ux(t, 0) = ux(t, 2π),  t ∈ (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Control in Velocity:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4)  ρ(t, 0) = ρ(t, 2π),  u(t, 0) = u(t, 2π) + q(t),  ux(t, 0) = ux(t, 2π),  t ∈ (0, T ),  where p and q are boundary controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Our main goal is to study null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)  at a given time T > 0 with the initial condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2) and one of the boundary conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  More precisely, given any initial state (ρ0, u0) in some suitable Hilbert space, we want to find a boundary  control p (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' q) such that the solution (ρ, u) to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4))  satisfies  (ρ(T, x), u(T, x)) = (0, 0) in (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Before stating our main results, we first introduce the Sobolev space for any s > 0  Hs  per(0, 2π) =  �  ϕ : ϕ =  �  n∈Z  cneinx,  �  n∈Z  |n|2s |cn|2 < ∞  �  ,  with the norm  ∥ϕ∥Hs  per(0,2π) :=  ��  n∈Z  (1 + |n|2)s |cn|2  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  For s > 0, we denote H−s  per(0, 2π) to be the dual of the Sobolev space Hs  per(0, 2π) with respect to the  pivot space L2(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We also define the space  ˙L2(0, 2π) :=  �  ϕ ∈ L2(0, 2π) :  � 2π  0  ϕ(x)dx = 0  �  and  ˙Hs  per(0, 2π) :=  �  ϕ ∈ Hs  per(0, 2π) :  � 2π  0  ϕ(x)dx = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  If the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) is null controllable in time T by using a boundary control p, then integrating both  equations in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1), we get a compatibility condition on the initial states  aγ¯ργ−2  � 2π  0  ρ0(x)dx = ¯u  � 2π  0  u0(x)dx = −aγ¯ργ−2¯u  � T  0  p(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  If the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) is null controllable in time T by using a boundary control q, then also we will get  a similar compatibility condition on the initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since every initial state (ρ0, u0) in (L2(0, 2π))2  will not satisfy this compatibility condition, we will work on the Hilbert space ( ˙L2(0, 2π))2 to avoid this  difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  When a boundary control q acts in the velocity component, it is known in [7] that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) is null controllable at time T > 2π  ¯u provided that the initial state is regular enough, in particular,  lies in the space ˙Hs+1  per (0, 2π) × ˙Hs  per(0, 2π) for s > 9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In the first part of our article, we generalize this  result (with respect to the regularity of initial states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We also prove null controllability of the system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) when there is a boundary control p acts in the density component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We write all the statements  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given time T >  2π  ¯u  and initial state (ρ0, u0) ∈ ( ˙L2(0, 2π))2, there exists a  boundary control p ∈ L2(0, T ) such that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) is null controllable at time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2  For small time, we have a lack of null controllability result for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2, the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) is not  null controllable at small time 0 < T < 2π  ¯u by means of any boundary control p ∈ L2(0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We have a similar null controllability result for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) when a boundary control acts in the  velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given time T > 2π  ¯u and initial state (ρ0, u0) ∈ ˙Hs  per(0, 2π)× ˙L2(0, 2π),  there exists a boundary control q ∈ L2(0, T ) such that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) is null controllable at  time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The following result shows that the null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) at time T > 2π  ¯u is  optimal in the space ˙H1  per(0, 2π) × ˙L2(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0) ∈ Hs  per(0, 2π) × L2(0, 2π) with 0 ≤ s < 1, the  system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) is not null controllable at any time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Following the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2, the lack of null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) cannot be obtained when the time is small, in particular, when 0 < T < 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Lack of con-  trollability at small time may be possible to obtain by constructing a Gaussian beam, as mentioned  in [31, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5] for the interior control case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) at time T =  2π  ¯u  is inconclusive in both cases,  whether there is a control act in density or velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The Non-Barotropic Case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We next consider the Navier-Stokes system for compressible non-  barotropic fluids linearized around some constant steady state (¯ρ, ¯u, ¯θ) with ¯ρ, ¯u, ¯θ > 0  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ρt(t, x) + ¯uρx(t, x) + ¯ρux(t, x) = 0, in (0, T ) × (0, 2π),  ut(t, x) − λ + 2µ  ¯ρ  uxx(t, x) + R¯θ  ¯ρ ρx(t, x) + ¯uux(t, x) + Rθx(t, x) = 0, in (0, T ) × (0, 2π),  θt(t, x) − κ  ¯ρcν  θxx(t, x) + R¯θ  cν  ux(t, x) + ¯uθx(t, x) = 0, in (0, T ) × (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)  The initial conditions are  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)  ρ(0, x) = ρ0(x),  u(0, x) = u0(x),  θ(0, x) = θ0(x),  x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  In this case, we will consider three different problems, based on the act of control, by imposing any one  of the following boundary conditions on the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Control in Density:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7)  ρ(t, 0) = ρ(t, 2π) + p(t), u(t, 0) = u(t, 2π), ux(t, 0) = ux(t, 2π), θ(t, 0) = θ(t, 2π), θx(t, 0) = θx(t, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Control in Velocity:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8)  ρ(t, 0) = ρ(t, 2π), u(t, 0) = u(t, 2π) + q(t), ux(t, 0) = ux(t, 2π), θ(t, 0) = θ(t, 2π), θx(t, 0) = θx(t, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Control in Temperature:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9)  ρ(t, 0) = ρ(t, 2π), u(t, 0) = u(t, 2π), ux(t, 0) = ux(t, 2π), θ(t, 0) = θ(t, 2π) + r(t), θx(t, 0) = θx(t, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  for t ∈ (0, T ), where p, q and r are boundary controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  In this case also, we want to prove null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) at any given time T > 0  depending on the act of the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Similar to the barotropic case, we will work on the Hilbert space  ( ˙L2(0, 2π))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We denote the positive constants  λ0 := λ + 2µ  ¯ρ  ,  κ0 :=  κ  ¯ρcν  ,  and define the set  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='10)  S :=  �  (λ0, κ0) :  �  λ0  κ0  /∈ Q  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We prove the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given time T > 2π  ¯u , (λ0, κ0) ∈ S, and initial state (ρ0, u0, θ0) ∈ ( ˙L2(0, 2π))3,  there exists a boundary control p ∈ L2(0, T ) such that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7) is null controllable at  time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The following proposition gives a lack of null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7) when the time  is small enough, that is, 0 < T < 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3, the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7) is  not null controllable at a small time 0 < T < 2π  ¯u using any boundary control p ∈ L2(0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Similar to the barotropic case, we have the null controllability result of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) at time T (large  enough) when there is a boundary control acts in the velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given time T > 2π  ¯u , (λ0, κ0) ∈ S, and initial state (ρ0, u0, θ0) ∈  ˙Hs  per(0, 2π) × ( ˙L2(0, 2π))2, there exists a boundary control q ∈ L2(0, T ) such that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8) is null controllable at time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The following result proves that the space ˙H1  per(0, 2π) × ( ˙L2(0, 2π))2 is optimal for null controllability of  the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0, θ0) ∈ Hs  per(0, 2π) × (L2(0, 2π))2 with 0 ≤ s < 1,  the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8) is not null controllable at any time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We have similar result when there is a control act in temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given time T > 2π  ¯u , (λ0, κ0) ∈ S and initial state (ρ0, u0, θ0) ∈  ˙Hs  per(0, 2π) × ( ˙L2(0, 2π))2, there exists a boundary control r ∈ L2(0, T ) such that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9) is null controllable at time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Similar to the previous one (control acts in velocity), we have optimality of the space  ˙H1  per(0, 2π) ×  ( ˙L2(0, 2π))2 in the sense below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0, θ0) ∈ Hs  per(0, 2π) × (L2(0, 2π))2 with 0 ≤ s < 1,  the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9) is not null controllable at any time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Following the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8, lack of null controllability of the systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9) cannot be obtained when the time is small, in particular, when 0 <  T < 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' However, this result may be possible to obtain by constructing a Gaussian beam, as mentioned  in [31, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5] for the interior control case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) at time T = 2π  ¯u is inconclusive in both cases,  whether there is a control act in density, velocity, or temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' An Ingham-Type Inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' To prove the null controllability results for both barotropic and  non-barotropic systems, we need the following Ingham-type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15 ( [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let {νh  n}n∈Z and {νp  n}n∈Z be two sequences in C with the following  properties: there is N ∈ N, such that  (H1) for all n, l ∈ Z, νh  n ̸= νh  l unless n = l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (H2) νh  n = β + τni + en for all |n| ≥ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  where τ > 0, β ∈ C and {en}|n|≥N ∈ ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Also, there exists constants A0 ≥ 0, B0 ≥ δ with δ > 0 and some ǫ > 0, r > 1 for which {νp  n}n∈Z  satisfies  (P1) for all n, l ∈ Z, νp  n ̸= νp  l unless n = l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (P2)  −ℜ(νp  n)  |ℑ(νp  n)| ≥ �c for some �c > 0 and for all |n| ≥ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (P3) |νp  n − νp  l | ≥ δ |nr − lr| for all n ̸= l with |n| , |l| ≥ N and  (P4) ǫ(A0 + B0nr) ≤ |νp  n| ≤ A0 + B0nr for all |n| ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We also assume that the families are disjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=',  �  νh  n, n ∈ Z  �  ∩ {νp  n, n ∈ Z} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Then, for any time T > 2π  τ , there exists a positive constant C depending only on T such that  � T  0  �����  �  n∈Z  aneνp  nt +  �  n∈Z  bneνh  nt  �����  2  dt ≥ C  ��  n∈Z  |an|2e2ℜ(νp  n)T +  �  n∈Z  |bn|2  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11)  4  for all sequences {an}n∈Z and {bn}n∈Z in ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any vector v, we denote its transpose by v† (instead of vT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Throughout the article,  C > 0 denotes a generic constant that may depend on the time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proving null controllability of the systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) using a boundary control is equivalent to  proving an observability inequality for the corresponding adjoint systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Spectrum of the associated  linearized operators (for the adjoint systems) and the above Ingham-type inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11) plays a crucial  role to prove such observability inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) (barotropic fluids), spectrum of the  associated adjoint operator consists of two branches of complex eigenvalues, namely, the hyperbolic and  parabolic branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The hyperbolic branch has eigenvalues with the real part converging to − b¯ρ  µ0 , whereas  real part of the parabolic branch diverges to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have obtained explicit expressions of eigenvalues  and eigenfunctions in terms of a Riesz basis (See Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For the non-barotropic fluids  (that is, system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)), we get three branches of complex eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' one is of the hyperbolic type,  and two are parabolic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Similar to the barotropic case, the real part of the hyperbolic branch  converges to − R¯θ  λ0 and the real parts of both parabolic branches diverge to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In this case, we have  obtained explicit expressions of eigenfunctions and asymptotic behavior of the eigenvalues (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We also proved that the eigenfunctions form a Riesz basis in ( ˙L2(0, 2π))2 for the barotropic system  and ( ˙L2(0, 2π))3 for the non-barotropic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then, by writing the solutions to the corresponding  adjoint systems in terms of the eigenfunctions, the null controllability results have been proved using the  combined parabolic-hyperbolic Ingham type inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  A vast amount of literature is available on the controllability of Navier-Stokes equations for incom-  pressible fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For instance, one can see the works of Coron [12], Coron and Fursikov [13], Fursikov  and Imanuvilov [22, 23], Imanuvilov [26, 27], Fern´andez-Cara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' [20, 21], Guerrero [25], Coron and  Guerrero [14], Chapouly [4], Coron and Lissy [15], Badra, Ervedoza and Guerrero [1], Coron, Marbach  and Sueur [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In comparison, for compressible fluids, less works are available on the Navier-Stokes  system’s controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In this context, we first mention the work of Ervedoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' [17], where the  authors established local exact controllability of one dimensional compressible Navier-Stokes system at  a large time T in the space H3(0, L) × H3(0, L) using two boundary controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This result has been  improved in [18] where the null controllability is achieved in the space H1(0, L) × H1(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  It is known in [10] that, for barotropic fluids, the one-dimensional compressible Navier-Stokes system  linearized around (¯ρ, 0) (with ¯ρ > 0) cannot be null controllable at any time T > 0 by using a boundary  control or a localized distributed control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For the linearized system around (¯ρ, ¯u) (with ¯ρ, ¯u > 0), the  authors in [8] proved null controllability of the Navier-Stokes equations (with homogeneous periodic  boundary conditions) for viscous, compressible isothermal barotropic fluids at time T (large) in the  space ˙H1  per(0, 2π) × L2(0, 2π), when there is an interior control act only in the velocity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' They  also proved that the space ˙H1  per(0, 2π) × L2(0, 2π) is optimal in the sense that if we choose the initial  state from  ˙Hs  per(0, 2π) × L2(0, 2π) with 0 ≤ s < 1, the linearized system cannot be null controllable  at any time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In the case of linearization around (¯ρ, ¯u) with ¯ρ, ¯u > 0, the compressible Navier-  Stokes system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) is equivalent (in some sense) to the transformed system in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Using a moving  distributed control, the authors in [32] proved the null controllability of a one-dimensional structurally  damped wave equation in the space Hs+2 × Hs for s >  15  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' There is a generalization to this result  in higher dimensions by Chaves-Silva, Rosier, and Zuazua [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Inspired by the work of Martin, Rosier  and Rouchon [32], Chowdhury and Mitra in [7] proved the null controllability of the same compressible  Navier-Stokes system linearized around (¯ρ, ¯u) at time T (large) by using a boundary control that acts  on the velocity component through periodic conditions, provided the initial states are regular enough,  more precisely, in the space  ˙H1+s  per (0, 2π) × ˙Hs  per(0, 2π) with s > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  However, the question of null  controllability at a large time T in the space ˙H1+s  per (0, 2π)× ˙Hs  per(0, 2π) with 0 < s ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5 was unaddressed  in [7], and up to the author’s knowledge, there has been no improvement in this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In this article, we  have answered this question (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have proved null controllability of  the linearized compressible Navier-Stokes system for barotropic fluids (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) at large time T in the space  ˙Hs  per(0, 2π) × ˙L2(0, 2π) with s ≥ 1 by using one boundary control acting in the velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We  have also proved that our result is optimal in the sense that the system cannot be null controllable by a  boundary control (acts in velocity) when the initial states belong to the space ˙Hs  per(0, 2π)× ˙L2(0, 2π) with  0 ≤ s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' When a control acts only in the density component through periodic boundary conditions,  we have established null controllability of the linearized system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) at large time T in the  space ˙L2(0, 2π) × ˙L2(0, 2π) and that null controllability fails at small time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  5  For the non-barotropic fluids, it is known in [31] that the compressible Navier-Stokes system linearized  around (¯ρ, 0, ¯θ) (with ¯ρ, ¯θ > 0) is not null controllable at any time T > 0 by using a boundary control  or a localized distributed control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For the linearization around (¯ρ, ¯u, ¯θ) with ¯ρ, ¯u, ¯θ > 0, it is only known  that the system is not null controllable at small time by a localized interior control or a boundary control  acting on the velocity component (see [31, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5] for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' To the author’s knowledge, no  controllability result is known for the linearized system around (¯ρ, ¯u, ¯θ), that is, the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5), when  the time is large, which is studied for the first time in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The main difficulty in the linearized compressible Navier-Stokes system is the presence of transport  and parabolic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The thermoelasticity system is also an example involving both transport and  parabolic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Lebeau and Zuazua [30] have studied distributed Controllability for thermoelasticity  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Following [30], Beauchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' in [2] proved null controllability for some coupled transport-  parabolic systems when an interior control acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' They proved null controllability at large time T in  the space L2(0, 2π) × ˙L2(0, 2π) by one interior control acts in the density equation and in the space  ˙H2(0, 2π) × H2(0, 2π) when only one interior control acts in the velocity equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' see also [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The main contribution of this article is that we prove the null controllability of the one-dimensional  linearized compressible Navier-Stokes system for both barotropic and non-barotropic fluids by using only  one boundary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We consider all the possible cases of the act of control for both systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)  and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We get better regularity of the initial states for the controllability of barotropic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)  compared to [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In the case of non-barotropic fluids, since the transport equation does not affect the  temperature equation, it is pretty natural to obtain similar spaces of null controllability of the system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The combined parabolic-hyperbolic Ingham type inequality (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15) helps us obtain each  case’s best possible results (with respect to the state space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Our results cannot be obtained as a  consequence of interior control results by the extension method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' When the boundary control acts in the  density component, we prove that both systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) are not null controllable at small time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The proof is inspired from [2] and is independent of that in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The result stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1 is similar to the results in [2], showing that we can achieve the space  ( ˙L2(0, 2π))2 in the case of only one boundary control (acts in density) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Likewise the case of interior  control [8], we also obtain similar results for our boundary control case (acts in velocity) (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3  and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The rest of the article is organized as follows:  – In Section 2, we prove the null controllability of the linearized compressible Navier-Stokes system  for barotropic fluids (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) at a large time T using a boundary control that acts either in density  or velocity, that is, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The proofs of lack of null  controllability at small time T (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2) and at any time T with less regular initial states  (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) are also included in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  – In Section 3, we give all the null controllability results of linearized compressible Navier-Stokes  system for non-barotropic fluids (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) based on the act of the control, namely the proofs of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have also included the proofs of lack of null  controllability results at small time T (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8) and at any time T when the initial states  are less regular (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='10 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  – In section 4, we give few comments and discuss some open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  – For the sake of completeness, we give the proof of well-posedness result (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) for the  non-barotropic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The author would like to thank his PhD supervisor Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Shirshendu Chowdhury  for suggesting this problem and fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The author would also like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Rajib  Dutta for careful reading and improvement of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  This work is supported by the Prime Minister’s Research Fellowship (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 41-1/2018-TS-1/PMRF),  Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Controllability of Linearized Compressible Navier-Stokes System (Barotropic)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Functional Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We denote the positive constants  µ0 := µ  ¯ρ ,  b := aγ¯ργ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  6  We define the inner product in the space (L2(0, 2π))2 as follows  ��  f1  g1  �  ,  �  f2  g2  ��  := b  � 2π  0  f1(x)f2(x)dx + ¯ρ  � 2π  0  g1(x)g2(x)dx,  for fi, gi ∈ L2(0, 2π), i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We write the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) in abstract differential equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)  U ′(t) = AU(t),  U(0) = U0,  t ∈ (0, T ),  where U := (ρ, u)†, U0 := (ρ0, u0)† and the operator A is given by  A :=  �  −¯u∂x  −¯ρ∂x  −b∂x  µ0∂xx − ¯u∂x  �  with the domain  D(A) := H1  per(0, 2π) × H2  per(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The adjoint of the operator A∗ is given by  A∗ :=  �  ¯u∂x  ¯ρ∂x  b∂x  µ0∂xx + ¯u∂x  �  with the same domain D(A∗) = D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The adjoint system is then given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −σt(t, x) − ¯uσx(t, x) − ¯ρvx(t, x) = 0, in (0, T ) × (0, 2π),  −vt(t, x) − µ0vxx(t, x) − ¯uvx(t, x) − bσx(t, x) = 0, in (0, T ) × (0, 2π),  σ(t, 0) = σ(t, 2π),  v(t, 0) = v(t, 2π),  vx(t, 0) = vx(t, 2π), t ∈ (0, T ),  σ(T, x) = σT (x),  v(T, x) = vT (x),  x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We now write the adjoint system with source terms f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −σt(t, x) − ¯uσx(t, x) − ¯ρvx(t, x) = f, in (0, T ) × (0, 2π),  −vt(t, x) − µ0vxx(t, x) − ¯uvx(t, x) − bσx(t, x) = g, in (0, T ) × (0, 2π),  σ(t, 0) = σ(t, 2π),  v(t, 0) = v(t, 2π),  vx(t, 0) = vx(t, 2π), t ∈ (0, T ),  σ(T, x) = σT (x),  v(T, x) = vT (x),  x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Well-Posedness of the System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This section devotes to the well-posedness of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)  under the boundary conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) and the initial conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2), and the adjoint system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  When there is no control act on the system, we have the existence of solutions to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) using  semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1 ( [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The operator A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' A∗) generates a C0-semigroup of contractions  on (L2(0, 2π))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Moreover, for every U0 ∈ (L2(0, 2π))2 the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) admits a unique solution U in  C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2) and  ∥U(t)∥(L2(0,2π))2 ≤ C ∥U0∥(L2(0,2π))2  for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The following lemma shows the existence of a unique solution to the adjoint system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2 ( [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given source term (f, g) ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2), the adjoint system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3)  (with (σT , vT ) = (0, 0)) has a unique solution (σ, v) in the space  C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)) × [C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' H1  per(0, 2π))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Once we have the existence results of the homogeneous system (without any boundary control) associated  to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1), we can now guarantee the existence of a unique solution to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) (in the  sense of transposition) when there is a boundary control p (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' q) act in density (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' velocity) in the  space L2(0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Before writing the statements, let us first define the notion of a solution in the sense of  transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  7  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (1) For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2 and boundary control p ∈  L2(0, T ), a function (ρ, u) ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2) is a solution to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) if  for any given (f, g) ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2), the following identity holds true:  � T  0  � 2π  0  ρ(t, x)f(t, x)dxdt +  � T  0  � 2π  0  u(t, x)g(t, x)dxdt  = ⟨(ρ0(·), u0(·)), (σ(0, ·), v(0, ·))⟩L2×L2 +  � T  0  �  ¯uσ(t, 2π) + ¯ρv(t, 2π)  �  p(t)dt,  where (σ, v) is the unique weak solution to the adjoint system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) with (σt, vT ) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (2) For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2 and boundary control q ∈ L2(0, T ), a function  (ρ, u) ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (H1(0, 2π))′ × L2(0, 2π)) is a solution to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) if for any  given (f, g) ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' H1(0, 2π) × L2(0, 2π)), the following identity holds true:  � T  0  ⟨ρ(t, ·), f(t, ·)⟩(H1)′,H1 dt +  � T  0  � 2π  0  u(t, x)g(t, x)dxdt  = ⟨(ρ0(·), u0(·)), (σ(0, ·), v(0, ·))⟩L2×L2 +  � T  0  �  bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)  �  q(t)dt,  (σ, v) is the unique weak solution to the adjoint system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) with (σT , vT ) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4 ( [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2 and boundary  control p ∈ L2(0, T ), the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) admits a unique solution (ρ, u) in the space  C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)) × [C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' H1  per(0, 2π))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5 ( [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0) ∈ (L2(0, 2π))2 and boundary  control q ∈ L2(0, T ), the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) admits a unique solution (ρ, u) in the space  L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (H1  per(0, 2π))′) × L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Moreover, the operator q �→ (ρ, u) is linear and continuous from L2(0, T ) into L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (H1  per(0, 2π))′) ×  L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Spectral Analysis of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We denote the spectrum of A∗ by σ(A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The following lemma gives  behavior of the spectrum of the operator A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The following statements holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (1) ker(A∗) = span  ��  1  1  �  ,  �  1  −1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (2) sup {ℜ(ν) : ν ∈ σ(A∗), ν ̸= 0} < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (3) The spectrum of A∗ consists of the eigenvalue 0 and pairs of complex eigenvalues {νh  n, νp  n}n∈Z∗  given as  νh  n = −1  2  �  µ0n2 − n  �  µ2  0n2 − 4b¯ρ − 2¯uin  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4)  νp  n = −1  2  �  µ0n2 + n  �  µ2  0n2 − 4b¯ρ − 2¯uin  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (4) The eigenvalues satisfy the following properties  \uf8f1  \uf8f2  \uf8f3  lim|n|→∞ ℜ(νh  n) = −ω0,  lim|n|→∞  ℜ(νp  n)  n2  = −µ0  lim|n|→∞  ℑ(νh  n)  n  = ¯u,  lim|n|→∞  ℑ(νp  n)  n  = ¯u  with ω0 = b¯ρ  µ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (5) The eigenfunctions of A∗ corresponding to νh  n and νp  n are respectively  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)  Φh  n :=  �  ξh  n  ηh  n  �  =  �  ¯ρ  νn  2 − ¯u  �  einx,  Φp  n :=  �  ξp  n  ηp  n  �  =  �  ¯ρ  νn  1 −¯u  1  �  einx,  for n ∈ Z, where  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7)  νn  1 := 1  2  �  µ0in + 2¯u + i  �  µ2  0n2 − 4b¯ρ  �  ,  νn  2 := 1  2  �  µ0in + 2¯u − i  �  µ2  0n2 − 4b¯ρ  �  ,  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  8  (6) The eigenfunctions {Φh  n, Φp  n : n ∈ Z∗} of A∗ forms a Riesz basis of ( ˙L2(0, 2π))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We will prove only the parts (2), (3) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Part (4) can be proved using part (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Let Φ = (ξ, η)† be the eigenfunction of A∗ corresponding to the eigenvalue ν ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then, we have  �  A∗  �  ξ  η  �  ,  �  ξ  η  ��  =  �  ν  �  ξ  η  �  ,  �  ξ  η  ��  ,  that is,  b¯u  � 2π  0  ξ(x)ξx(x)dx + b¯ρ  � 2π  0  ξ(x)ηx(x)dx + µ0¯ρ  � 2π  0  η(x)ηxx(x)dx + ¯ρ¯u  � 2π  0  η(x)ηx(x)dx  +b¯ρ  � 2π  0  ξx(x)η(x)dx = ν  � 2π  0  |ξ(x)|2 dx + ν  � 2π  0  |η(x)|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  An integration by parts yields  ℜ(ν) = −  ∥ηx∥2  L2(0,2π)  |ξ|2  L2(0,2π) + ∥η∥2  L2(0,2π)  < 0,  which proves part (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We denote  ϕn(x) := einx,  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Then the set  ��  ϕn  0  �  ,  �  0  ϕn  ��  forms an orthogonal basis of (L2(0, 2π))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let us define  En :=  �  ϕn  0  0  ϕn  �  ,  and Φn := (ξn, ηn)†,  for all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then, we have the following relation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8)  A∗EnΦn = inEnRnΦn,  n ∈ Z,  where the matrix Rn for n ∈ Z is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9)  Rn :=  �  ¯u  ¯ρ  b  µ0in + ¯u  �  ,  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Thus, if (αn, νn) is an eigenpair of Rn, then (Enαn, inνn) will be an eigenpair of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Therefore, it’s  remains to find the eigenvalues and eigenvectors of the matrix Rn for n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The characteristics equation  of Rn is  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='10)  ν2 − (µ0in + 2¯u)ν + µ0¯uin + ¯u2 − b¯ρ = 0,  for all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Therefore, the eigenvalues of the matrix Rn are  νn  1 := 1  2  �  µ0in + 2¯u + i  �  µ2  0n2 − 4b¯ρ  �  ,  νn  2 := 1  2  �  µ0in + 2¯u − i  �  µ2  0n2 − 4b¯ρ  �  ,  for all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Note that, 0 and ¯u cannot be an eigenvalue of the matrix Rn for all n ∈ Z∗ and ¯u cannot  be an eigenvalue of Rn for all n ∈ Z, because b, ¯ρ, µ0, ¯u > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' To find the eigenvectors of the matrix Rn,  we first consider the equation  Rnαh  n = νn  2 αh  n,  n ∈ Z,  where αh  n := (αn  1 , αn  2 )†, that is,  (¯u − νn  2 )αn  1 + ¯ραn  2 = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11)  bαn  1 + (µ0in + ¯u − νn  2 )αn  2 = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='12)  for all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' One solution is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13)  αh  n =  �  αn  1  αn  2  �  :=  �  ¯ρ  νn  2 − ¯u  �  ,  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We next consider the equation  Rnαp  n = νn  1 αp  n,  n ∈ Z,  9  where αp  n := (βn  1 , βn  2 )†, that is,  (¯u − νn  1 )βn  1 + ¯ρβn  2 = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='14)  bβn  1 + (µ0in + ¯u − νn  1 )βn  2 = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15)  for all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' One solution is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='16)  αp  n =  �  βn  1  βn  2  �  :=  �  ¯ρ  νn  1 −¯u  1  �  ,  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Thus, the eigenvectors of Rn corresponding to the eigenvalues νn  2 and νn  1 are respectively  αh  n =  �  αn  1  αn  2  �  =  �  ¯ρ  νn  2 − ¯u  �  ,  αp  n =  �  βn  1  βn  2  �  =  �  ¯ρ  νn  1 −¯u  1  �  ,  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Hence, the eigenvalues of the operator A∗ are  νh  n := inνn  2 = 1  2  �  −µ0n2 + n  �  µ2  0n2 − 4b¯ρ + 2¯uin  �  , νp  n := inνn  1 = 1  2  �  −µ0n2 − n  �  µ2  0n2 − 4b¯ρ + 2¯uin  �  for n ∈ Z and the corresponding eigenfunctions are respectively  Φh  n :=  �  ξh  n  ηh  n  �  = Enαh  n = αh  neinx,  Φp  n :=  �  ξp  n  ηp  n  �  = Enαp  n = αp  neinx,  for all n ∈ Z and x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Note that, for all |n| large, all the eigenvalues of A∗ are simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' There may be multiple  eigenvalues of A∗, depending on the constants ¯ρ, ¯u, µ0, b, but that would be only finitely many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus,  without loss of generality, we can assume that A∗ has simple eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Indeed, for the case of finite  number of multiple eigenvalues, one can adapt a similar approach as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2 of [8] (by considering  suitable generalized eigenfunctions) to prove the required observability inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' One can also see [29,  Remarks, page 178] for a version of Ingham inequality in the case of repeated eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Observation Estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any eigenvalue ν, let us denote the corresponding eigenfunction of  A∗ by Φν and let E(A∗) be the set of all eigenfunctions of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We now define the observation operators  associated to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) as follows:  B∗  ρΦν := ¯uξ(2π) + ¯ρη(2π),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='17)  B∗  uΦν := bξ(2π) + ¯uη(2π) + µ0ηx(2π),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='18)  for all Φν = (ξ, η, ζ) ∈ E(A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The following result proves that these observation terms are non-zero for  all Φ ∈ E(A∗) \\ {Φ0}, and have positive lower bounds for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For all Φν ∈ E(A∗) \\ {Φ0}, the observation operators satisfy B∗  ρΦν ̸= 0 and B∗  uΦν ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Moreover, we have the following estimates  ��B∗  ρΦh  n  �� ≥ C,  ��B∗  ρΦp  n  �� ≥ C,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='19)  ��B∗  uΦh  n  �� ≥ C  |n|,  |B∗  uΦp  n| ≥ C |n| ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='20)  for some C > 0 and all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Note that  B∗  ρΦh  n = ¯uξh  n(2π) + ¯ρηh  n(2π) = ¯uαn  1 + ¯ραn  2 = νn  2 αn  1 ̸= 0,  for all n ∈ Z∗, thanks to the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We similarly have from equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='14)  B∗  ρΦp  n = ¯uξp  n(2π) + ¯ρηp  n(2π) = ¯uβn  1 + ¯ρβn  2 = νn  1 βn  1 ̸= 0,  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The estimates on B∗  ρΦh  n and B∗  ρΦp  n are now follows directly from the above expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We now compute  B∗  uΦh  n = bξh  n(2π) + ¯uηh  n(2π) + µ0(ηh  n)x(2π)  = bαn  1 + (¯u + µ0in)αn  2 = νn  2 αn  2 ̸= 0,  10  for all n ∈ Z∗, thanks to the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since |αn  2 | ≥  C  |n| and νn  2 is bounded for all n ∈ Z∗, the  estimate on B∗  uΦh  n follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For the last quantity, note that  B∗  uΦp  n = bξp  n(2π) + ¯uηp  n(2π) + µ0(ηp  n)x(2π)  = bβn  1 + (¯u + µ0in)βn  2 = νn  1 βn  2 ̸= 0,  for all n ∈ Z∗, thanks to the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The estimate on B∗  uΦp  n is now follows directly as |νn  1 | ≥ C |n|  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Observability Inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since the eigenfunctions E(A∗)\\{Φ0} form a Riesz basis in ( ˙L2(0, 2π))2,  therefore any (σT , vT ) ∈ ( ˙L2(0, 2π))2 can be written as  (σT , vT )† =  �  n∈Z∗  �  ah  nΦh  n + ap  nΦp  n  �  ,  for some (ah  n)n∈Z∗, (ap  n)n∈Z∗ ∈ ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then the solution to the adjoint system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2) is  (σ(t, x), v(t, x))† =  �  n∈Z∗  ah  neνh  n(T −t)Φh  n +  �  n∈Z∗  ap  neνp  n(T −t)Φp  n,  for (t, x) ∈ (0, T ) × (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus we get  σ(t, x) = ¯ρ  �  n∈Z∗  ah  neνh  n(T −t)einx +  �  n∈Z∗  ap  neνp  n(T −t)  ¯ρ  νn  1 − ¯ueinx,  and  v(t, x) =  �  n∈Z∗  ah  neνh  n(T −t)(νn  2 − ¯u)einx +  �  n∈Z∗  ap  neνp  n(T −t)einx,  for all (t, x) ∈ (0, T ) × (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' To prove the observability inequality, we need an upper bound of the  norm of (σ(0), v(0))† and a lower bound estimate for the respective observation terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We first estimate  the upper bounds of the norm of (σ(0), v(0))†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have  ��(σ(0), v(0))†��2  ( ˙L2(0,2π))2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='21)  ≤ C  � �  n∈Z∗  ��ah  n  ��2 �  1 + |νn  2 − ¯u|2�  e2ℜ(νh  n)T ��einx��2  ˙L2(0,2π) +  �  n∈Z∗  |ap  n|2  �  1  |νn  1 − ¯u|2 + 1  �  e2ℜ(νp  n)T ��einx��2  ˙L2(0,2π)  �  ≤ C  � �  n∈Z∗  ��ah  n  ��2 +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  ,  since the sequences 1 + |νn  2 − ¯u|2 and 1 +  1  |νn  1 −¯u|  2 are bounded for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We also have  ��(σ(0), v(0))†��2  ˙H−s  per(0,2π)× ˙L2(0,2π)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='22)  ≤ C  � �  n∈Z∗  ��ah  n  ��2 �  1 + |νn  2 − ¯u|2� ��einx��2  ˙H−s  per(0,2π) +  �  n∈Z∗  |ap  n|2  �  1  |νn  1 − ¯u|2 + 1  �  e2ℜ(νp  n)T ��einx��2  ˙L2(0,2π)  �  ≤ C  � �  n∈Z∗  ��ah  n  ��2  1  |n|2s +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We now find the lower bounds of the respective observation terms and prove our main results for the  barotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We use the Ingham-type inequality (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15) for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Note that, the eigen-  values (νh  n)n∈Z∗ satisfies hypotheses (H1)-(H2) with τ = ¯u, β = −ω0 and (νp  n)n∈Z∗ satisfies hypotheses  (P1)-(P4) with r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let T > 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The observability inequality is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='23)  � T  0  |¯uσ(t, 2π) + ¯ρv(t, 2π)|2 dt ≥ C  ��(σ(0), v(0))†��2  ( ˙L2(0,2π))2 ,  for all (σT , vT )† ∈ D(A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have the observation term  � T  0  |σ(t, 2π) + v(t, 2π)|2 dt =  � T  0  � �  n∈Z∗  ah  nB∗  ρΦh  neνh  n(T −t) +  �  n∈Z∗  ap  nB∗  ρΦp  neνp  n(T −t)  �2  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15) and the observation  estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='19), we obtain  � T  0  |σ(t, 2π) + v(t, 2π)|2 dt ≥ C  � �  n∈Z∗  ��ah  n  ��2 ��B∗  ρΦh  n  ��2 e2ℜ(νh  n)T +  �  n∈Z∗  |ap  n|2 ��B∗  ρΦp  n  ��2 e2ℜ(νp  n)T  �  ≥ C  � �  n∈Z∗  ��ah  n  ��2 +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  This estimate together with the norm estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='21), the observability inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='23) is now follows  directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let T > 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The observability inequality is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='24)  � T  0  |bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt ≥ C  ��(σ(0), v(0))†��2  ˙H−s  per(0,2π)× ˙L2(0,2π) ,  for all (σT , vT )† ∈ D(A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have  � T  0  |bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt =  � T  0  � �  n∈Z∗  ah  kB∗  uΦh  neνh  n(T −t) +  �  n∈Z∗  ap  nB∗  uΦp  neνp  n(T −t)  �2  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15), we obtain  � T  0  |bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt  ≥ C  � �  n∈Z∗  ��ah  n  ��2 ��B∗  uΦh  n  ��2 e2ℜ(νh  n)T +  �  n∈Z∗  |ap  n|2 |B∗  uΦp  n|2 e2ℜ(νp  n)T  �  ≥ C  � �  n∈Z∗  ��ah  n  ��2  1  |n|2 +  �  n∈Z∗  |ap  n|2 |n|2 e2ℜ(νp  n)T  �  ,  thanks to the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For s ≥ 1, we have  1  |n|2 ≥  1  |n|2s for all n ∈ Z∗ and therefore  � T  0  |bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt ≥ C  � �  n∈Z∗  ��ah  n  ��2  1  |n|2s +  �  n∈Z∗  |ap  n|2 |n|2 e2ℜ(νp  n)T  �  ≥ C  ��(σ(0), v(0))†�� ˙H−s  per(0,2π)× ˙L2(0,2π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  This proves the observability inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Lack of Null Controllability for Less Regular Initial States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For (σT , vT )† = Φh  n, the solution to the adjoint system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2) is  (σ(t, x), v(t, x))† = eνh  n(T −t)Φh  n(x),  for (t, x) ∈ (0, T ) × (0, 2π) and n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For all n ∈ Z∗, we have the following estimate  ��Φh  n  �� ˙H−s  per(0,2π)× ˙L2(0,2π) ≥ C  |n|s ,  and therefore  ��(σ(0), v(0))†��2  ˙H−s  per× ˙L2 ≥  C  |n|2s ,  12  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' On the other hand, we have the upper bound of the observation term  � T  0  |bσ(t, 2π) + ¯uv(t, 2π) + µ0vx(t, 2π)|2 dt ≤  C  |n|2 ,  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus, if the observability inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='24) holds, then we get  C  |n|2s ≤  C  |n|2 =⇒ |n|2−2s ≤ C,  which is not possible since 0 ≤ s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Lack of Controllability at Small Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We prove that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) is not null control-  lable when the time is small, that is, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We construct an approximate solution for the  corresponding transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The idea of constructing an approximate solution for the transport  equation was addressed in [2], where the authors proved a lack of null controllability result at a small  time in the case of an interior control (acts in the transport equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Very recently, in [6, Section 6],  this approach has been applied to a coupled transport-elliptic system in the case of a boundary control  (acts in density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We will follow mainly the proof given in [6] to prove our lack of null controllability  result when the time is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let 0 < T < 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Following the notations in the proof of Proposition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2, Consider the transport equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='25)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ˜σt(t, x) + ¯u˜σx(t, x) − b¯ρ  µ0  ˜σ(t, x) = 0,  (t, x) ∈ (0, T ) × (0, 2π),  ˜σ(t, 0) = ˜σ(t, 2π),  t ∈ (0, T ),  ˜σ(T, x) = ˜σT (x),  x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Since ¯uT < 2π, there exists a nontrivial function ˜σT ∈ C∞(0, 2π) with supp(˜σT ) ⊂ (¯uT, 2π) such that  supp(˜σ) ⊂ (0, T ) × (¯uT, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let N > 0 be a fixed integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We define the polynomial  P N(x) :=  N  �  l=−N  (x − l),  x ∈ (0, 2π)  and the function  ˜σN  T := P N  �  −i d  dx  �  ˜σT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We write the terminal state as  ˜σT (x) :=  �  n∈Z  aneinx,  x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Then, the solution to the transport equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='25) is  ˜σN  T (x) =  �  n∈Z  an  N  �  l=−N  �  −i d  dx − l  �  einx =  �  n∈Z  an  N  �  l=−N  (n − l) einx =  �  n∈Z  anP N(n)einx,  for (t, x) ∈ (0, T ) × (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Note that P N(n) = 0 for all |n| ≤ N and therefore  ˜σN  T (x) =  �  |n|≥N+1  anP N(n)einx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  With this ˜σN  T , let us now consider the following system  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='26)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ˜σt(t, x) + ¯u˜σx(t, x) − b¯ρ  µ0  ˜σ(t, x) = 0,  (t, x) ∈ (0, T ) × (0, 2π),  ˜σ(t, 0) = ˜σ(t, 2π),  t ∈ (0, T ),  ˜σ(T, x) = ˜σN  T (x),  x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  13  Since supp(˜σN  T ) ⊂ supp(˜σT ) ⊂ (T, 2π  ¯u ), the solution satisfies ˜σN(t, 0) = ˜σN(t, 2π) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We now consider  the following adjoint system  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='27)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  σt(t, x) + ¯uσx(t, x) + ¯ρvx(t, x) = 0,  (t, x) ∈ (0, T ) × (0, 2π),  vt(t, x) − µ0vxx(t, x) + ¯uvx(t, x) + bσx(t, x) = 0,  (t, x) ∈ (0, T ) × (0, 2π),  σ(t, 0) = σ(t, 2π),  t ∈ (0, T ),  v(t, 0) = v(t, 2π),  vx(t, 0) = vx(t, 2π),  t ∈ (0, T ),  σ(T, x) = ˜σN  T (x),  v(T, x) = vN  T (x),  x ∈ (0, 2π),  where we choose vN  T such that  (˜σN  T , vN  T )† =  �  |n|≥N+1  ˜ah  nΦh  n  with ˜ah  n¯ρ := anP N(n) for all |n| ≥ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We write the solutions to the systems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='26) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='27)  respectively as  ˜σN(t, x) =  �  |n|≥N+1  anP N(n)e(¯uin− b¯  ρ  µ0 )(T −t)einx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='28)  σN(t, x) =  �  |n|≥N+1  anP N(n)eνh  n(T −t)einx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='29)  vN(t, x) =  �  |n|≥N+1  anP N(n)νn  2 − ¯u  ¯ρ  eνh  n(T −t)einx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='30)  for (t, x) ∈ [0, T ]×[0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We prove that the solution component σN of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='27) approximates the solution  ˜σN of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Indeed,  ��σN(·, x) − ˜σN(·, x)  ��2  L2(0,T )  ≤  �  |n|≥N+1  |an|2 ��P N(n)  ��2  ����eνh  n(T −t) − e  �  ¯uin− b¯  ρ  µ0  �  (T −t)  ����  2  L2(0,T )  ≤  �  |n|≥N+1  |an|2 ��P N(n)  ��2  ������  e  − µ0n  2  �  n−  �  n2− 4b¯  ρ  µ2  0  �  (T −t)  − e− b¯  ρ  µ0 (T −t)  ������  2  L2(0,T )  ≤  �  |n|≥N+1  1  |n|2 |an|2 ��P N(n)  ��2 ,  for all x ∈ [0, 2π] and therefore  ��σN(·, x) − ˜σN(·, x)  ��2  L2(0,T ) ≤  C  |N|2  �  |n|≥N+1  |an|2 ��P N(n)  ��2 ,  for all x ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We also find L2- estimate of the solution component vN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have for all x ∈ [0, 2π]  ��vN(·, x)  ��2  L2(0,T ) ≤  �  |n|≥N+1  |an|2 ��P N(n)  ��2 |νn  2 − ¯u|2  ¯ρ2  ���eνh  n(T −t)���  2  L2(0,T )  ≤ C  �  |n|≥N+1  |an|2 ��P N(n)  ��2  1  |n|2  ≤  C  |N|2  �  |n|≥N+1  |an|2 ��P N(n)  ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Let us now suppose that the following observability inequality holds  � T  0  ��¯uσN(t, 2π) + ¯ρvN(t, 2π)  ��2 dt ≥ C  ��(σN(0), vN(0))  ��2  (L2(0,2π))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  14  Then, we have  ��(σN(0), vN(0))  ��2  (L2(0,2π))2 ≤ C  � T  0  ��¯uσN(t, 2π) + ¯ρvN(t, 2π)  ��2 dt  ≤ C  � T  0  �  ¯u2 ��(σN(t, 2π) − ˜σN(t, 2π))  ��2 + ¯u2 ��˜σN(t, 2π)  ��2 + ¯ρ2 ��vN(t, 2π)  ��2�  dt  ≤ C  N 2  �  |n|≥N+1  |an|2 ��P N(n)  ��2 ,  as we have ˜σN(t, 0) = 0 = ˜σN(t, 2π) for all t ∈ (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus we get  ��σN(0)  ��2  L2(0,2π) ≤  ��(σN(0), vN(0))†��2  (L2(0,2π))2 ≤ C  N 2  �  |n|≥N+1  |an|2 ��P N(n)  ��2 ≤ C  N 2  ��σN(0)  ��2  L2(0,2π) ,  since ℜ(νh  n) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Therefore, 1 ≤  C  N 2 for all N and hence the above inequality cannot hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This  is a contradiction and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Controllability of Linearized Compressible Navier-Stokes System (Non-barotropic)  We denote the positive constants  λ0 := λ + 2µ  ¯ρ  ,  κ0 :=  κ  ¯ρcν  ,  and from now on-wards, we denote cν by c0 to distinguish it from the eigenvalue ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Functional Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We define the inner product in the space (L2(0, 2π))3 as follows  �\uf8eb  \uf8ec  \uf8ed  f1  g1  h1  \uf8f6  \uf8f7  \uf8f8 ,  \uf8eb  \uf8ec  \uf8ed  f2  g2  h2  \uf8f6  \uf8f7  \uf8f8  �  := R¯θ  � 2π  0  f1(x)f2(x)dx + ¯ρ2  � 2π  0  g1(x)g2(x)dx + ¯ρ2c0  ¯θ  � 2π  0  h1(x)h2(x)dx,  for fi, gi, hi ∈ L2(0, 2π), i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We write the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) in abstract differential equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)  U ′(t) = AU(t),  U(0) = U0,  t ∈ (0, T ),  where U := (ρ, u, θ)†, U0 := (ρ0, u0, θ0)† and the operator A is given by  A :=  \uf8eb  \uf8ec  \uf8ec  \uf8ed  −¯u∂x  −¯ρ∂x  0  − R¯θ  ¯ρ ∂x  λ0∂xx − ¯u∂x  −R∂x  0  − R¯θ  c0 ∂x  κ0∂xx − ¯u∂x  \uf8f6  \uf8f7  \uf8f7  \uf8f8  with the domain  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)  D(A) := H1  per(0, 2π) × (H2  per(0, 2π))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The adjoint of the operator A∗ is given by  A∗ :=  \uf8eb  \uf8ec  \uf8ec  \uf8ed  ¯u∂x  ¯ρ∂x  0  R¯θ  ¯ρ ∂x  λ0∂xx + ¯u∂x  R∂x  0  R¯θ  c0 ∂x  κ0∂xx + ¯u∂x  \uf8f6  \uf8f7  \uf8f7  \uf8f8  with the same domain D(A∗) = D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The adjoint system is given by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −σt − ¯uσx − ¯ρvx = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ) × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  −vt − λ0vxx − R¯θ  ¯ρ σx − ¯uvx − Rϕx = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ) × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  −ϕt − κ0ϕxx − R¯θ  c0  vx − ¯uϕx = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ) × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  σ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = σ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  vx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = vx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  t ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  ϕx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = ϕx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  t ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  σ(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) = σT (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  v(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) = vT (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  ϕ(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) = ϕT (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  x ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  15  with (σT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' vT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' ϕT ) is a terminal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We also write the following system with source terms f, g, and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −σt − ¯uσx − ¯ρvx = f, in (0, T ) × (0, 2π),  −vt − λ0vxx − R¯θ  ¯ρ σx − ¯uvx − Rϕx = g, in (0, T ) × (0, 2π),  −ϕt − κ0ϕxx − R¯θ  c0  vx − ¯uϕx = h, in (0, T ) × (0, 2π),  σ(t, 0) = σ(t, 2π),  v(t, 0) = v(t, 2π),  vx(t, 0) = vx(t, 2π),  t ∈ (0, T ),  ϕ(t, 0) = ϕ(t, 2π),  ϕx(t, 0) = ϕx(t, 2π),  t ∈ (0, T ),  σ(T, x) = σT (x),  v(T, x) = vT (x),  ϕ(T, x) = ϕT (x),  x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Well-Posedness of the System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The operator A generates a C0-semigroup of contractions on (L2(0, 2π))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Moreover, for  every U0 ∈ (L2(0, 2π))3 the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) admits a unique solution U in C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))3) and  ∥U(t)∥(L2(0,2π))3 ≤ C ∥U0∥(L2(0,2π))3  for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2 ( [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any f, g, h ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))3), the adjoint system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) (with (σT , vT , ϕT ) =  (0, 0, 0)) has a unique solution (σ, v, ϕ) in the space  C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)) × [C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' H1  per(0, 2π))]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We give the following definitions of solutions based on the act of the controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control p ∈ L2(0, T ), we  say (ρ, u, θ) ∈ (L2(0, 2π))3 is a solution to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7) if for any (f, g, h) ∈  L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))3), the following identity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  � T  0  � 1  0  ρ(t, x)f(t, x)dxdt +  � T  0  � 1  0  u(t, x)g(t, x)dxdt +  � T  0  � 1  0  θ(t, x)h(t, x)dxdt  =  � 1  0  ρ0(x)σ(0, x)dx +  � 1  0  u0(x)v(0, x)dx +  � 1  0  θ0(x)ϕ(0, x)dx + R¯θ  � T  0  �  ¯uσ(t, 2π) + ¯ρv(t, 2π)  �  p(t)dt,  where (σ, v, ϕ) is the solution to the adjoint system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) with (σT , vT , ϕT ) = (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control q ∈ L2(0, T ), we  say (ρ, u, θ) ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (H1(0, 2π))′) × L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2) is a solution to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8) if for any (f, g, h) ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' H1(0, 2π))×L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2), the following identity  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  � T  0  ⟨ρ(t, ·), f(t, ·)⟩(H1(0,2π))′,H1(0,2π) dxdt +  � T  0  � 1  0  u(t, x)g(t, x)dxdt +  � T  0  � 1  0  θ(t, x)h(t, x)dxdt  =  � 1  0  ρ0(x)σ(0, x)dx +  � 1  0  u0(x)v(0, x)dx +  � 1  0  θ0(x)ϕ(0, x)dx  +  � T  0  �  R¯θ¯ρσ(t, 2π) + λ0¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)  �  q(t)dt,  where (σ, v, ϕ) is the solution to the adjoint system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) with (σT , vT , ϕT ) = (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control r ∈ L2(0, T ), we  say (ρ, u, θ) ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (H1(0, 2π))′) × L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2) is a solution to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8) if for any (f, g, h) ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' H1(0, 2π))×L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2), the following identity  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  � T  0  ⟨ρ(t, ·), f(t, ·)⟩(H1(0,2π))′,H1(0,2π) dxdt +  � T  0  � 1  0  u(t, x)g(t, x)dxdt +  � T  0  � 1  0  θ(t, x)h(t, x)dxdt  =  � 1  0  ρ0(x)σ(0, x)dx +  � 1  0  u0(x)v(0, x)dx +  � 1  0  θ0(x)ϕ(0, x)dx  +  � T  0  �  R¯ρ2v(t, 2π) + ¯ρ2c0¯u  ¯θ  ϕ(t, 2π) + ¯ρ2c0κ0  ¯θ  ϕx(t, 2π)  �  r(t)dt,  where (σ, v, ϕ) is the solution to the adjoint system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) with (σT , vT , ϕT ) = (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  16  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control p ∈  L2(0, T ), the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7) admits a unique solution (ρ, u, θ) in the space  C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)) × [C0([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' L2(0, 2π)) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' H1  per(0, 2π))]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control q ∈  L2(0, T ), the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8) admits a unique solution (ρ, u, θ) in the space  L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (H1  per(0, 2π))′) × L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Moreover, the operator q �→ (ρ, u, θ) is linear and continuous from L2(0, T ) into L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (H1  per(0, 2π))′)×  L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any given initial state (ρ0, u0, θ0) ∈ (L2(0, 2π))3 and boundary control r ∈  L2(0, T ), the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9) admits a unique solution (ρ, u, θ) in the space  L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (H1  per(0, 2π))′) × L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Moreover, the operator r �→ (ρ, u, θ) is linear and continuous from L2(0, T ) into L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (H1  per(0, 2π))′)×  L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (L2(0, 2π))2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The proofs of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6 can be done in a similar way  ( [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4] and [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2]) like the barotropic case and so we skip the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Spectral Analysis of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We first write the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (1) ker(A∗) = span  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  \uf8eb  \uf8ec  \uf8ed  −1  1  1  \uf8f6  \uf8f7  \uf8f8 ,  \uf8eb  \uf8ec  \uf8ed  1  −1  1  \uf8f6  \uf8f7  \uf8f8 ,  \uf8eb  \uf8ec  \uf8ed  1  1  −1  \uf8f6  \uf8f7  \uf8f8  \uf8fc  \uf8f4  \uf8fd  \uf8f4  \uf8fe  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (2) sup {ℜ(ν) : ν ∈ σ(A∗), ν ̸= 0} < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (3) The spectrum of A∗ consists of the eigenvalue 0 and three branches of complex eigenvalues  {νh  n, νp  n, νp1  n }n∈Z∗ with the asymptotic expressions given as  νh  n = ¯uin − ¯ω + O(|n|−2),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)  νp1  n = −λ0n2 + ¯uin + O(1),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)  νp2  n = −κ0n2 + ¯uin + O(1),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7)  for all |n| large, where ¯ω = R¯θ  λ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (4) The eigenfunctions of A∗ corresponding to νh  n and νp1  n , νp2  n are respectively  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8)  Φh  n =  \uf8eb  \uf8ec  \uf8ed  ξh  n  ηh  n  ζh  n  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  αn  1  αn  2  αn  3  \uf8f6  \uf8f7  \uf8f8 einx,  Φp1  n =  \uf8eb  \uf8ec  \uf8ed  ξp1  n  ηp1  n  ζp1  n  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  βn  1  βn  2  βn  3  \uf8f6  \uf8f7  \uf8f8 einx,  Φp2  n =  \uf8eb  \uf8ec  \uf8ed  ξp2  n  ηp2  n  ζp2  n  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  γn  1  γn  2  γn  3  \uf8f6  \uf8f7  \uf8f8 einx,  for all n ∈ Z∗, with the constants αn  i , βn  i and γn  i (i = 1, 2, 3) given as  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9)  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  αn  1 = R¯ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  αn  2 = −R(¯u − νn  3 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  αn  3 = (λ0in + ¯u − νn  3 )(¯u − νn  3 ) − R¯θ  βn  1 = −  R¯ρ  ¯u−νn  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  βn  2 = R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  βn  3 =  1  ¯u−νn  1 [R¯θ − (λ0in + ¯u − νn  1 )(¯u − νn  1 )]  γn  1 = (λ0in + ¯u − νn  2 )(κ0in + ¯u − νn  2 ) − R2 ¯θ  c0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  γn  2 = − R¯θ  ¯ρ (κ0in + ¯u − νn  2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  γn  3 = R2 ¯θ2  ¯ρc0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  for all n ∈ Z∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' where νn  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' νn  2 and νn  3 are roots of the cubic polynomial  ν3 − [(λ0 + κ0)in + 3¯u]ν2 − [λ0κ0n2 − 2(λ0 + κ0)¯uin − 3¯u2 + R2¯θ  c0  + R¯θ]ν  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='10)  +λ0κ0¯un2 − (λ0 + κ0)¯u2in − ¯u3 + R2¯θ  c0  ¯u + R¯θκ0in + R¯θ¯u = 0,  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  (5) The eigenfunctions {Φh  n, Φp1  n , Φp2  n  : n ∈ Z∗} of A∗ forms a Riesz basis of ( ˙L2(0, 2π))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  17  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have the asymptotic expressions of αi, βi, γi, i = 1, 2, 3 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  αn  1 ∼+∞ 1,  αn  2 ∼+∞  1  |n|,  αn  3 ∼+∞  1  |n|,  βn  1 ∼+∞  1  |n|,  βn  2 ∼+∞ 1,  βn  3 ∼+∞  1  |n|,  γn  1 ∼+∞  1  |n|,  γn  2 ∼+∞  1  |n|,  γn  3 ∼+∞ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We will prove only the parts (2), (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Let Φ = (ξ, η, ζ)† be the eigenfunction of A∗ corresponding to the eigenvalue ν ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' we have  �  A∗  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8  �  =  �  ν  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  R¯θ¯u  � 2π  0  ξ(x)ξx(x)dx + R¯θ¯ρ  � 2π  0  ξ(x)ηx(x)dx + λ0¯ρ2  � 2π  0  η(x)ηxx(x)dx + ¯ρ2¯u  � 2π  0  η(x)ηx(x)dx  +R¯θ¯ρ  � 2π  0  ξx(x)η(x)dx + R¯ρ2  � 2π  0  η(x)ζx(x)dx + ¯ρ2c0  ¯θ  κ0  � 2π  0  η(x)ζx(x)dx + ¯ρ2c0  ¯θ  ¯u  � 2π  0  η(x)ζx(x)dx  +R¯ρ2  � 2π  0  ηx(x)ζ(x)dx + b¯ρ  � 2π  0  ξx(x)η(x)dx = ν  � 2π  0  |ξ(x)|2 dx + ν  � 2π  0  |η(x)|2 dx + ν  � 2π  0  |ζ(x)|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  An integration by parts yields  ℜ(ν) = −  ∥ηx∥2  L2(0,2π) + ∥ζx∥2  L2(0,2π)  ∥ξ∥2  L2(0,2π) + ∥η∥2  L2(0,2π) + ∥ζ∥2  L2(0,2π)  < 0,  which proves part (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We denote  ϕn(x) := einx,  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Then the set  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  \uf8eb  \uf8ec  \uf8ed  ϕn  0  0  \uf8f6  \uf8f7  \uf8f8 ,  \uf8eb  \uf8ec  \uf8ed  0  ϕn  0  \uf8f6  \uf8f7  \uf8f8 ,  \uf8eb  \uf8ec  \uf8ed  0  0  ϕn  \uf8f6  \uf8f7  \uf8f8  \uf8fc  \uf8f4  \uf8fd  \uf8f4  \uf8fe  forms an orthogonal basis of (L2(0, 2π))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let us define  En :=  \uf8eb  \uf8ec  \uf8ed  ϕn  0  0  0  ϕn  0  0  0  ϕn  \uf8f6  \uf8f7  \uf8f8 ,  and Φn := (ξn, ηn, ζn)†,  for all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then, we have the following relation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11)  A∗EnΦn = inEnRnΦn,  n ∈ Z,  where  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='12)  Rn :=  \uf8eb  \uf8ec  \uf8ec  \uf8ed  ¯u  ¯ρ  0  R¯θ  ¯ρ  λ0in + ¯u  R  0  R¯θ  c0  κ0in + ¯u  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Thus, if (αn, νn) is an eigenpair of Rn, then (Enαn, inνn) will be an eigenpair of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Therefore, it’s  remains to find the eigenvalues and eigenvectors of the matrix Rn for n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The characteristics equation  of Rn is  ν3 − [(λ0 + κ0)in + 3¯u]ν2 − [λ0κ0n2 − 2(λ0 + κ0)¯uin − 3¯u2 + R2¯θ  c0  + R¯θ]ν  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13)  +λ0κ0¯un2 − (λ0 + κ0)¯u2in − ¯u3 + R2¯θ  c0  ¯u + R¯θκ0in + R¯θ¯u = 0,  for all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0 cannot be a root of the polynomial (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13) for any n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  18  Proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let ν = 0 be a root of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then, there exists some n ∈ Z such that  λ0κ0¯un2 − (λ0 + κ0)¯u2in − ¯u3 + R2¯θ  c0  ¯u + R¯θκ0in + R¯θ¯u = 0,  which implies  λ0κ0n2 − ¯u2 + R2¯θ  c0  + R¯θ = 0, and (λ0 + κ0)¯u2 = R¯θκ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We then have  λ0κ0n2 = ¯u2 − R2¯θ  c0  − R¯θ = ¯u2 − R2¯θ  c0  −  �λ0  κ0  + 1  �  ¯u2 = −R2¯θ  c0  − λ0  κ0  ¯u2 < 0,  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This proves our first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' ¯u cannot be a root of the polynomial (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13) for any n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Observe that ¯u is a root of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13) if and only if R¯θκ0in = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus, for all n ∈ Z∗, ¯u  cannot be a root of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13), which proves our second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  For fixed n ∈ Z∗, let νn  1 , νn  2 and νn  3 be the roots of this cubic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The relation between roots  and coefficients are  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  νn  1 + νn  2 + νn  3 = (λ0 + κ0)in + 3¯u  νn  1 νn  2 + νn  2 νn  3 + νn  3 νn  1 = −[λ0κ0n2 − 2(λ0 + κ0)¯uin − 3¯u2 + R2 ¯θ  c0 + R¯θ]  νn  1 νn  2 νn  3 = −[λ0κ0¯un2 − (λ0 + κ0)¯u2in − ¯u3 + R2 ¯θ  c0 ¯u + R¯θκ0in + R¯θ¯u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We will find the asymptotic expressions of roots of the cubic polynomial (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13) for large values of |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The first relation between roots and coefficients tells us that ¯u is present in at least one of the roots of  the cubic polynomial (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus, using the transformation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='14)  ν = ¯u + ǫn,  it is enough to find the roots of the transformed cubic equation in ǫn  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15)  ǫ3  n − (λ0 + κ0)inǫ2  n −  �  λ0κ0n2 + R2¯θ  c0  + R¯θ  �  ǫn + R¯θκ0in = 0  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We use the transformation ǫn = in˜ǫn, for n ∈ Z∗, to simplify the above equation and we  get  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='16)  ˜ǫ3  n − (λ0 + κ0)˜ǫ2  n +  �  λ0k0 + 1  n2  �R2¯θ  c0  + R¯θ  ��  ˜ǫn − R¯θκ0  n2  = 0  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We now use Rouche’s Theorem to find the roots of this polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let us first state the  Rouch´e’s Theorem, the proof of which can be found in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9 (Rouch´e’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let Ω ⊂ C be an open connected set and f, g : Ω → C be holomorphic  on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Suppose there exists a ∈ Ω and R > 0 such that B(a, R) ⊂ Ω and  |g(z) − f(z)| < |g(z)| for all z ∈ ∂B(a, R),  then f and g have the same number of zeros inside B(a, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We define the functions f, g : C → C by  f(z) := z3 − (λ0 + κ0)z2 +  �  λ0k0 + 1  n2  �R2¯θ  c0  + R¯θ  ��  z − R¯θκ0  n2  and  g(z) := z3 − (λ0 + κ0)z2 + λ0k0z  for all z ∈ C and n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The roots of g are 0, λ0 and κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We now choose R := 1  2 min{λ0, κ0, |λ0 − κ0|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Then, we have the following estimates  |g(z) − f(z)| =  ����  1  n2  �R2¯θ  c0  + R¯θ  �  z − R¯θκ0  n2  ���� ≤ C  n2 (|z| + 1)  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  = C  n2 (R + 1),  for all z ∈ ∂B(0, R),  ≤ C  n2 (λ0 + R + 1),  for all z ∈ ∂B(λ0, R),  ≤ C  n2 (κ0 + R + 1),  for all z ∈ ∂B(κ0, R),  19  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' On the other hand, the choice of R tells us that the function g does not have any root  on the sets ∂B(0, R), ∂B(λ0, R) and ∂B(κ0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Therefore, inf|z|=R |g(z)| > 0,inf|z−λ0|=R |g(z)| > 0 and  inf|z−κ0|=R |g(z)| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This implies, for |n| large enough, we have  |g(z) − f(z)| < |g(z)| for all z ∈ ∂B(0, R) ∪ ∂B(λ0, R) ∪ ∂B(κ0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Thus, the function f has a unique root inside each of the sets B(0, R), B(λ0, R) and B(κ0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We denote  these roots by zn  1 , zn  2 and zn  3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We now find asymptotic expressions of these roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Asymptotic expression of zn  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since zn  1 ∈ B(0, R), we have  zn  1 =  1  (zn  1 − λ0)(zn  1 − κ0)  �R¯θκ0  n2  − 1  n2  �R2¯θ  c0  + R¯θ  �  zn  1  �  and therefore  |zn  1 | ≤  1  |zn  1 − λ0| |zn  1 − κ0|  �����  R¯θκ0  n2  ���� +  ����  1  n2  �R2¯θ  c0  + R¯θ  �  zn  1  ����  �  ≤  C  |n|2  for |n| large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' To find the asymptotic expression of zn  1 , we write f(zn  1 ) = 0 in the following way  zn  1 = R¯θκ0  n2  �  λ0κ0 − (λ0 + κ0)zn  1 + (zn  1 )2 + 1  n2  �R¯θ  c0  + R¯θ  ��−1  = R¯θκ0  n2  1  λ0κ0  �  1 − (λ0 + κ0)  λ0κ0  zn  1 +  1  λ0κ0n2  �R¯θ  c0  + R¯θ  �  + O(|n|−4)  �−1  = ¯ω  n2  �  1 + (λ0 + κ0)  λ0κ0  zn  1 −  1  λ0κ0n2  �R¯θ  c0  + R¯θ  �  + O(|n|−4)  �  = ¯ω  n2 + O(|n|−4),  since |zn  1 | ≤ C  n2 for all |n| large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Asymptotic expression of zn  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since zn  2 ∈ B(λ0, R), we have  zn  2 − λ0 =  1  zn  2 (zn  2 − κ0)  �R¯θκ0  n2  − 1  n2  �R2¯θ  c0  + R¯θ  �  zn  1  �  and therefore  |zn  2 − λ0| ≤  1  |zn  2 | |zn  2 − κ0|  �����  R¯θκ0  n2  ���� +  ����  1  n2  �R2¯θ  c0  + R¯θ  �  zn  1  ����  �  ≤  C  |n|2  for |n| large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus, we can write  zn  2 = λ0 + O(|n|−2)  for all |n| large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Asymptotic expression of zn  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Following the similar approach as mentioned above, we can get  zn  3 = κ0 + O(|n|−2)  for all |n| large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Combining all of the above, we obtain the asymptotic expressions of the roots of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15) as  ǫn  1 := λ0in + O(|n|−1),  ǫn  2 := κ0in + O(|n|−1),  ǫn  3 := − ¯ω  in + O(|n|−3)  for all |n| large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Therefore, eigenvalues of the matrix Rn are νn  1 , νn  2 and νn  3 with the asymptotic expressions  νn  1 = λ0in + ¯u + O(|n|−1),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='17)  νn  2 = κ0in + ¯u + O(|n|−1),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='18)  νn  3 = ¯u − ¯ω  in + O(|n|−3),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='19)  for all |n| large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  To find the eigenvectors of the matrix Rn, we now consider the equation  Rnαn = νn  3 αn,  n ∈ Z∗,  20  where αn = (αn  1 , αn  2, αn  3 )†, that is,  (¯u − νn  3 )αn  1 + ¯ραn  2 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='20)  R¯θ  ¯ρ αn  1 + (λ0in + ¯u − νn  3 )αn  2 + Rαn  3 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='21)  R¯θ  c0  αn  2 + (κ0in + ¯u − νn  3 )αn  3 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='22)  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' One solution is given by  αn  1 = R¯ρ,  αn  2 = −R(¯u − νn  3 ),  αn  3 = (λ0in + ¯u − νn  3 )(¯u − νn  3 ) − R¯θ,  n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We next consider the equation  Rnβn = νn  1 βn,  n ∈ Z∗,  where βn = (βn  1 , βn  2 , βn  3 )†, that is,  (¯u − νn  1 )βn  1 + ¯ρβn  2 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='23)  R¯θ  ¯ρ βn  1 + (λ0in + ¯u − νn  1 )βn  2 + Rβn  3 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='24)  R¯θ  c0  βn  2 + (κ0in + ¯u − νn  1 )βn  3 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='25)  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' One solution is given by  βn  1 = −  R¯ρ  ¯u − νn  1  ,  βn  2 = R,  βn  3 =  1  ¯u − νn  1  [R¯θ − (λ0in + ¯u − νn  1 )(¯u − νn  1 )],  n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We finally consider the equation  Rnγn = νn  2 γn,  n ∈ Z∗,  where γn = (γn  1 , γn  2 , γn  3 )†, that is,  (¯u − νn  2 )γn  1 + ¯ργn  2 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='26)  R¯θ  ¯ρ γn  1 + (λ0in + ¯u − νn  2 )γn  2 + Rγn  3 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='27)  R¯θ  c0  γn  2 + (κ0in + ¯u − νn  2 )γn  3 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='28)  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' One solution is given by  γn  1 = (λ0in + ¯u − νn  2 )(κ0in + ¯u − νn  2 ) − R2¯θ  c0  ,  γn  2 = −R¯θ  ¯ρ (κ0in + ¯u − νn  2 ),  γn  3 = R2¯θ2  ¯ρc0  ,  n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Therefore, the eigenvectors of Rn corresponding to the eigenvalues νn  3 , νn  1 and νn  2 are respectively αn, βn  and γn, where  αn =  \uf8eb  \uf8ec  \uf8ed  αn  1  αn  2  αn  3  \uf8f6  \uf8f7  \uf8f8 ,  βn =  \uf8eb  \uf8ec  \uf8ed  βn  1  βn  2  βn  3  \uf8f6  \uf8f7  \uf8f8 ,  γn =  \uf8eb  \uf8ec  \uf8ed  γn  1  γn  2  γn  3  \uf8f6  \uf8f7  \uf8f8 ,  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Hence, the eigenvalues of the operator A∗ are νh  n := inν3  n, νp1  n := inν2  n and νp2  n := inν1  n  for all n ∈ Z∗ with the asymptotic expressions  νh  n = ¯uin − ¯ω + O(|n|−1),  νp1  n = −λ0n2 + ¯uin + O(1),  νp2  n = −κ0n2 + ¯uin + O(1),  for |n| large enough and the corresponding eigenfunctions are  Φh  n(x) := En(x)αn = αneinx,  Φp1  n (x) := En(x)βn = βneinx,  Φp2  n (x) := En(x)γn = γneinx,  for all n ∈ Z∗ and x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Note that, all the eigenvalues of A∗ are simple at least for |n| large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Depending  on the constants ¯ρ, ¯u, ¯θ, λ0, κ0, R and c0, there may be multiple eigenvalues, but that would be only finitely  many of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Therefore, in this case also, we assume that all the eigenvalues of A∗ are simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Observation Estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For any eigenvalue ν, let us denote the corresponding eigenfunction of  A∗ by Φν and let E(A∗) be the set of all eigenfunctions of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We define the observation operator  corresponding to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) as follows:  B∗  ρΦ := R¯θ¯uξ(2π) + R¯θ¯ρη(2π),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='29)  B∗  uΦ := R¯ρ¯θξ(2π) + ¯ρ2¯uη(2π) + λ0¯ρ2ηx(2π) + R¯ρ2ζ(2π),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='30)  B∗  θΦ := R¯ρ2η(2π) + ¯ρ2c0¯u  ¯θ  ζ(2π) + ¯ρ2c0κ0  ¯θ  ζx(2π),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='31)  for all Φ = (ξ, η, ζ)† ∈ E(A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then, we have the following estimates:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For all Φ ∈ E(A∗) \\ {Φ0}, the observation operators satisfies B∗  ρΦ ̸= 0 and B∗  uΦ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Moreover, we have the following estimates  ��B∗  ρΦh  n  �� ≥ C,  ��B∗  ρΦp1  n  �� ≥ C,  ��B∗  ρΦp2  n  �� ≥ C,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='32)  ��B∗  uΦh  n  �� ≥ C  |n|,  |B∗  uΦp1  n | ≥ C |n| , |B∗  uΦp2  n | ≥ C,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='33)  ��B∗  θΦh  n  �� ≥ C  |n|,  |B∗  uΦp2  n | ≥ C  ��B∗  ρΦp2  n  �� ≥ C |n| ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='34)  for some C > 0 and all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Recall from the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7 that ν1, ν2, ν3 ̸= 0 (Claim 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We consider the following cases:  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (Control acts in density) We have  B∗  ρΦh  n = R¯θ¯uξh  n(2π) + R¯ρ¯θηh  n(2π) = R¯θ(¯uαn  1 + ¯ραn  2) = R¯θνn  3 αn  1 ̸= 0,  for all n ∈ Z∗, thanks to the eigenvector equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We similarly have  B∗  ρΦp1  n = R¯θ¯uξp1  n (2π) + R¯ρ¯θηp1  n (2π) = R¯θ(¯uβn  1 + ¯ρβn  2 ) = R¯θνn  1 βn  1 ̸= 0,  and  B∗  ρΦp2  n = R¯θ¯uξp2  n (2π) + R¯ρ¯θηp2  n (2π) = R¯θ(¯uγn  1 + ¯ργn  2 ) = R¯θνn  2 γn  1 ̸= 0,  for all n ∈ Z∗, thanks to the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='23) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (Control acts in Velocity) We have  B∗  uΦh  n = R¯ρ¯θξh  n(2π) + λ0¯ρ2(ηh  n)x(2π) + ¯ρ2¯uηh  n(2π) + R¯ρ2ζh  n(2π)  = R¯ρ¯θαn  1 + λ0¯ρ2inαn  2 + ¯ρ2¯uαn  2 + R¯ρ2αn  3  = ¯ρ2νn  3 αn  2 ̸= 0,  for all n ∈ Z∗, thanks to the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We similarly have  B∗  uΦp1  n = R¯ρ¯θξp1  n (2π) + ν0¯ρ2(ηp1  n )x(2π) + ¯ρ2¯uηp1  n (2π) + R¯ρ2ζp1  n (2π)  = R¯ρ¯θβn  1 + ν0¯ρ2inβn  2 + ¯ρ2¯uβn  2 + R¯ρ2βn  3  = ¯ρ2νn  1 βn  2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  and  B∗  uΦp2  n = R¯ρ¯θξp2  n (2π) + λ0¯ρ2(ηp2  n )x(2π) + ¯ρ2¯uηp2  n (2π) + R¯ρ2ζp2  n (2π)  = R¯ρ¯θγn  1 + λ0 ¯ρ2inγn  2 + ¯ρ2¯uγn  2 + R¯ρ2γn  3  = ¯ρ2νn  2 γn  2 ̸= 0,  for all n ∈ Z∗, thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='24) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' (Control acts in Temperature) We have  B∗  θΦh  n = R¯ρ2ηh  n(2π) + ¯ρ2c0¯u  ¯θ  ζh  n(2π) + ¯ρ2c0κ0  ¯θ  (ζh  n)x(2π)  = R¯ρ2αn  2 + ¯ρ2c0  ¯θ  (¯u + κ0in)αn  3 = ¯ρ2c0  ¯θ  νn  3 αn  3 ̸= 0,  22  for all n ∈ Z∗, thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Similarly, we have  B∗  θΦp1  n = R¯ρ2ηp1  n (2π) + ¯ρ2c0¯u  ¯θ  ζp1  n (2π) + ¯ρ2c0κ0  ¯θ  (ζp1  n )x(2π)  = R¯ρ2βn  2 + ¯ρ2c0  ¯θ  (¯u + κ0in)βn  3 = ¯ρ2c0  ¯θ  νn  1 βn  3 ̸= 0,  and  B∗  θΦp2  n = R¯ρ2ηp2  n (2π) + ¯ρ2c0¯u  ¯θ  ζp2  n (2π) + ¯ρ2c0κ0  ¯θ  (ζp2  n )x(2π)  = R¯ρ2γn  2 + ¯ρ2c0  ¯θ  (¯u + κ0in)γn  3 = ¯ρ2c0  ¯θ  νn  2 γn  3 ̸= 0,  for all n ∈ Z∗, thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The estimates on the observation terms are then follows  directly from the asymptotic expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='17)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='18)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='19) and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Observability Inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let us rewrite the eigenvalues as {νh  n, νp  n}n∈Z∗, where  νp  n =  �  νp1  k ,  if n = 2k − 1, k ∈ Z  νp2  k ,  if n = 2k, k ∈ Z∗,  for all n ∈ Z∗ and νh  n is as defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We also denote the observation term  B∗Φp  n =  �  B∗Φp1  k ,  if n = 2k − 1, k ∈ Z,  B∗Φp2  n ,  if n = 2k, k ∈ Z∗,  for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Recall that, we have defined the set  S :=  �  (λ0, κ0) :  �  λ0  κ0  /∈ Q  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Then, the eigenvalues (νh  n)n∈Z∗ satisfies hypotheses (H1)-(H2) with τ = ¯u, β = −¯ω and for all (λ0, κ0) ∈  S, the sequence (νp  n)n∈Z∗ satisfies hypotheses (P1)-(P2)-(P3)-(P4) of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Since the eigenfunctions E(A∗)\\{Φ0} of A∗ forms a Riesz basis in ( ˙L2(0, 2π))3, therefore any (σT , vT , ϕT )† ∈  ( ˙L2(0, 2π))3 can be written as  (σT , vT , ϕT )† =  �  n∈Z∗  ah  nΦh  n +  �  n∈Z∗  ap1  n Φp1  n +  �  n∈Z∗  ap2  n Φp2  n ,  for some (ah  n)n∈Z∗, (ap1  n )n∈Z∗, (ap2  n )n∈Z∗ ∈ ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then, the solution to the adjoint system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) is  (σ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x))† =  �  n∈Z∗  ah  neνh  n(T −t)Φh  n(x) +  �  n∈Z∗  ap1  n eνp1  n (T −t)Φp1  n (x) +  �  n∈Z∗  ap2  n eνp2  n (T −t)Φp2  n (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  for (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ) × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  σ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) =  �  n∈Z∗  ah  neνh  n(T −t)αn  1einx +  �  n∈Z∗  ap1  n eνp1  n (T −t)βn  1 einx +  �  n∈Z∗  ap2  n eνp2  n (T −t)γn  1 einx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) =  �  n∈Z∗  ah  neνh  n(T −t)αn  2einx +  �  n∈Z∗  ap1  n eνp1  n (T −t)βn  2 einx +  �  n∈Z∗  ap2  n eνp2  n (T −t)γn  2 einx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) =  �  n∈Z∗  ah  neνh  n(T −t)αn  3einx +  �  n∈Z∗  ap1  n eνp1  n (T −t)βn  3 einx +  �  n∈Z∗  ap2  n eνp2  n (T −t)γn  3 einx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  for (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ) × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We then have  ��(σ(0), v(0), ϕ(0))†��2  ( ˙L2(0,2π))3 ≤ C  � �  n∈Z∗  ��ah  n  ��2 ��Φh  n  ��2  L2(0,2π) +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T ∥Φp  n∥2  L2(0,2π)  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='35)  ≤ C  � �  n∈Z∗  ��ah  n  ��2 +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  23  and  ��(σ(0), v(0), ϕ(0))†��2  ˙H−s  per(0,2π)×( ˙L2(0,2π))2 ≤ C  � �  n∈Z∗  ��ah  n  ��2 ��Φh  n  ��2  H−s  per(0,2π) +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T ∥Φp  n∥2  L2(0,2π)  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='36)  ≤ C  � �  n∈Z∗  ��ah  n  ��2  1  |n|2s +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let T > 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The observability inequality is given by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='37)  � T  0  ��R¯θ¯uσ(t, 2π) + R¯θ¯ρv(t, 2π)  ��2 dt ≥ C  ��(σ(0), v(0), ϕ(0))†��2  ( ˙L2(0,2π))3 ,  for all (σT , vT , ϕT )† ∈ ( ˙L2(0, 2π))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have the observation term  � T  0  ��R¯θ¯uσ(t, 2π) + R¯θ¯ρv(t, 2π)  ��2 dt  =  � T  0  �����  �  n∈Z∗  ah  nB∗  ρΦh  neνh  n(T −t) +  �  n∈Z∗  ap1  n B∗  ρΦp1  n eνp1  n (T −t) +  �  n∈Z∗  ap2  n B∗  ρΦp2  n eνp2  n (T −t)  �����  2  dt  =  � T  0  �����  �  n∈Z∗  ah  nB∗  ρΦh  neνh  n(T −t) +  �  n∈Z∗  ap  nB∗  ρΦp  neνp  n(T −t)  �����  2  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15), we have  � T  0  ��R¯θ¯uσ(t, 2π) + R¯θ¯ρv(t, 2π)  ��2 dt ≥ C  � �  n∈Z∗  ��ah  n  ��2 ��B∗  ρΦh  n  ��2 e2ℜ(νh  n)(T −t) +  �  n∈Z∗  |ap  n|2 ��B∗  ρΦp  n  ��2 e2ℜ(νp  n)T  �  ≥ C  � �  n∈Z∗  ��ah  n  ��2 +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  ,  thanks to the estimate 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This estimate together with the norm estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='35), the observability  inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='37) is now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let T > 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The observability inequality is given by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='38)  � T  0  ��R¯ρ¯θσ(t, 2π) + λ0¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)  ��2 dt ≥ C  ��(σ(0), v(0), ϕ(0))†��2  ˙H−s  per(0,2π)×( ˙L2(0,2π))2 ,  for all (σT , vT , ϕT )† ∈ ˙H−s  per(0, 2π) × ( ˙L2(0, 2π))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have  � T  0  ��R¯ρ¯θσ(t, 2π) + λ0¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)  ��2 dt  =  � T  0  �����  �  n∈Z∗  ah  nB∗  uΦh  neνh  n(T −t) +  �  n∈Z∗  ap1  n B∗  uΦp1  n eνp1  n (T −t) +  �  n∈Z∗  ap2  n B∗  uΦp2  n eνp2  n (T −t)  �����  2  dt  =  � T  0  �����  �  n∈Z∗  ah  nB∗  uΦh  neνh  n(T −t) +  �  n∈Z∗  ap  nB∗  uΦp  neνp  n(T −t)  �����  2  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15) and the observation  estimates ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='33)), we obtain  � T  0  ��R¯ρ¯θσ(t, 2π) + λ0 ¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)  ��2 dt  ≥ C  � �  n∈Z∗  ��ah  n  ��2 ��B∗  uΦh  n  ��2 e2ℜ(νh  n)(T −t) +  �  n∈Z∗  |ap  n|2 |B∗  uΦp  n|2 e2ℜ(νp  n)T  �  ≥ C  � �  n∈Z∗  ��ah  n  ��2  1  |n|2 +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  24  Since s ≥ 1, we have  1  |n|2 ≥  1  |n|2s for all n ∈ Z∗ and therefore  � T  0  ��R¯ρ¯θσ(t, 2π) + λ0 ¯ρ2vx(t, 2π) + ¯ρ2¯uv(t, 2π) + R¯ρ2ϕ(t, 2π)  ��2 dt  ≥ C  � �  n∈Z∗  ��ah  n  ��2  1  |n|2s +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  ≥ C  ��(σ(0), v(0), ϕ(0))†�� ˙H−s  per(0,2π)×( ˙L2(0,2π))2 ,  thanks to the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This proves the observability inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let T > 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The observability inequality is given by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='39)  � T  0  ����R¯ρ2v(t, 2π) + ¯ρ2c0¯u  ¯θ  ϕ(t, 2π) + ¯ρ2c0κ0  ¯θ  ϕx(t, 2π)  ����  2  dt ≥ C  ��(σ(0), v(0), ϕ(0))†��2  ˙H−s  per(0,2π)×( ˙L2(0,2π))2 ,  for all (σT , vT , ϕT )† ∈ ˙H−s  per(0, 2π) × ( ˙L2(0, 2π))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We have  � T  0  ����R¯ρ2v(t, 2π) + ¯ρ2c0¯u  ¯θ  ϕ(t, 2π) + ¯ρ2c0κ0  ¯θ  ϕx(t, 2π)  ����  2  dt  =  � T  0  �����  �  n∈Z∗  ah  nB∗  θΦh  neνh  n(T −t) +  �  n∈Z∗  ap1  n B∗  θΦp1  n eνp1  n (T −t) +  �  n∈Z∗  ap2  n B∗  θΦp2  n eνp2  n (T −t)  �����  2  dt  =  � T  0  �����  �  n∈Z∗  ah  nB∗  θΦh  neνh  n(T −t) +  �  n∈Z∗  ap  nB∗  θΦp  neνp  n(T −t)  �����  2  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Using the combined parabolic-hyperbolic Ingham type inequality (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='15) and the observation  estimates ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='34)), we have  � T  0  ����R¯ρ2v(t, 2π) + ¯ρ2c0¯u  ¯θ  ϕ(t, 2π) + ¯ρ2c0κ0  ¯θ  ϕx(t, 2π)  ����  2  dt  ≥ C  � �  n∈Z∗  ��ah  n  ��2 ��B∗  θΦh  n  ��2 e2ℜ(νh  n)(T −t) +  �  n∈Z∗  |ap  n|2 |B∗  θΦp  n|2 e2ℜ(νp  n)T  �  ≥ C  � �  n∈Z∗  ��ah  n  ��2  1  |n|2 +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  Since s ≥ 1, therefore  1  |n|2 ≥  1  |n|2s for all n ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus we get  � T  0  ����R¯ρ2v(t, 2π) + ¯ρ2c0¯u  ¯θ  ϕ(t, 2π) + ¯ρ2c0κ0  ¯θ  ϕx(t, 2π)  ����  2  dt  ≥ C  � �  n∈Z∗  ��ah  n  ��2  1  |n|2s +  �  n∈Z∗  |ap  n|2 e2ℜ(νp  n)T  �  ≥ C  ��(σ(0), v(0), ϕ(0))†�� ˙H−s  per(0,2π)×( ˙L2(0,2π))2 ,  and the observability inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='39) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Lack of Null Controllability for Less Regular Initial States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For (σT , vT , ϕT )† = Φh  n, the solution to the adjoint system is  (σ(t, x), v(t, x), ϕ(t, x))† = eνh  n(T −t)Φh  n(x),  for (t, x) ∈ (0, T ) × (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' For large n, we have the following estimate  ��Φh  n  ��  H−s  per(0,2π)×(L2(0,2π))2 ≥  C  |n|s ,  and therefore  ��(σ(0), v(0), ϕ(0))†��2  H−s  per(0,2π)×(L2(0,2π))2 ≥  C  |n|2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  25  We also have  � T  0  ��B∗  uΦh  n  ��2 dt ≤  C  |n|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Thus if the observability inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='38) holds, then we get  C  |n|2s ≤  C  |n|2 =⇒ |n|2−2s ≤ C,  which is not possible since 0 ≤ s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since the observation terms B∗  θΦh  n and B∗  uΦh  n have same upper bounds, proof of Propo-  sition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='12 will be similar to above, so we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Lack of Controllability at Small Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The proof will be similar to the barotropic case, that  is, the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let 0 < T < 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Following the notations in the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2,  we consider the system  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='40)  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ˜σt(t, x) + ¯u˜σx(t, x) − ¯ω˜σ(t, x) = 0,  (t, x) ∈ (0, T ) × (0, 2π),  ˜σ(t, 0) = ˜σ(t, 2π),  t ∈ (0, T ),  ˜σ(T, x) = ˜σN  T (x),  x ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Since supp(˜σN  T ) ⊂ supp(˜σT ) ⊂ (T, 2π), the solution satisfies ˜σN(t, 0) = ˜σN(t, 2π) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We now consider  the adjoint to our main system  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='41)  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  −σt(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) − ¯uσx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) − ¯ρvx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ) × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  −vt(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) − λ0vxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) − R¯θ  ¯ρ σx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) − ¯uvx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) − Rϕx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ) × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  −ϕt(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) − κ0ϕxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) − R¯θ  c0  vx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) − ¯uϕx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ) × (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  σ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = σ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  vx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = vx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  t ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  ϕx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 0) = ϕx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  t ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  σ(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) = σN  T (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  v(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) = vN  T (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  ϕ(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' x) = ϕN  T (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  x ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  where we choose vN  T and ϕN  T such that  (˜σN  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' vN  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' ϕN  T )† =  �  |n|≥N+1  ˜ah  nΦh  n  with ˜ah  nαn  1 := anP N(n) for all |n| ≥ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We write the solutions to the systems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='40) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='41)  respectively as  ˜σN(t, x) =  �  |n|≥N+1  anP N(n)e(¯uin−¯ω)(T −t)einx,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='42)  σN(t, x) =  �  |n|≥N+1  anP N(n)eνh  n(T −t)einx,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='43)  vN(t, x) =  �  |n|≥N+1  anP N(n)βn  1  αn  1  eνh  n(T −t)einx,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='44)  ϕN(t, x) =  �  |n|≥N+1  anP N(n) γn  1  αn  1  eνh  n(T −t)einx,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='45)  26  for all (t, x) ∈ [0, T ] × [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Similar to the barotropic case, we prove that the solution component σN  approximates the solution ˜σN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Indeed, we have  ��σN(·, x) − ˜σN(·, x)  ��2  L2(0,T ) ≤  �  |n|≥N+1  |an|2 ��P N(n)  ��2 ���eνh  n(T −t) − e(¯uin−¯ω)(T −t)���  2  L2(0,T )  ≤  �  |n|≥N+1  |an|2 ��P N(n)  ��2 ���e(¯uin−¯ω)(T −t)eO(|n|−1)(T −t) − 1  ���  2  L2(0,T )  ≤  C  |n|2  �  |n|≥N+1  |an|2 ��P N(n)  ��2 ,  for all x ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We also have for all x ∈ [0, 2π]  ��vN(·, x)  ��2  L2(0,T ) ≤  �  |n|≥N+1  |an|2 ��P N(n)  ��2 |βn  1 |2  |αn  1 |2  ���eνh  n(T −t)���  2  L2(0,T )  ≤ C  �  |n|≥N+1  |an|2 ��P N(n)  ��2  1  |n|2  ≤  C  |N|2  �  |n|≥N+1  |an|2 ��P N(n)  ��2  We suppose that the following observability inequality holds  � T  0  ��R¯θ¯uσN(t, 2π) + R¯θ¯ρvN(t, 2π)  ��2 dt ≥ C  ��(σN(0), vN(0), ϕN(0))†��2  (L2(0,2π))3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Then, we have  ��(σN(0), vN(0), ϕN(0))†��2  (L2(0,2π))3  ≤ C  � T  0  ��R¯θ¯uσN(t, 2π) + R¯θ¯ρvN(t, 2π)  ��2 dt  ≤ C  � T  0  �  ¯u2 ��(σN(t, 2π) − ˜σN(t, 2π))  ��2 + ¯u2 ��˜σN(t, 2π)  ��2 + ¯ρ2 ��vN(t, 2π)  ��2�  dt  ≤ C  N 2  �  |n|≥N+1  |an|2 ��P N(n)  ��2 ,  since ˜σN(t, 0) = 0 = ˜σN(t, 2π) for all t ∈ (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus, we get  ��σN(0)  ��2  L2(0,2π) ≤  ��(σN(0), vN(0), ϕN(0))†��2  (L2(0,2π))3  ≤ C  N 2  �  |n|≥N+1  |an|2 ��P N(n)  ��2 ≤ C  N 2  ��σN(0)  ��2  L2(0,2π) ,  since ℜ(νh  n) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Therefore, 1 ≤  C  N 2 for all N and hence the above inequality cannot hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' This  is a contradiction and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Further Comments and Conclusions  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Controllability Results Using Neumann Boundary Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We consider the system  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) with the initial state (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2) and the boundary conditions  ρ(t, 0) = ρ(t, 2π),  u(t, 0) = u(t, 2π),  ux(t, 0) = ux(t, 2π) + q1(t),  t ∈ (0, T ),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)  where q1 is a boundary control that acts on the velocity through Neumann conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since the obser-  vation terms satisfies similar estimates as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='20), following the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3 and Proposition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4, we can obtain the null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) at time T > 2π  ¯u in the space  ˙Hs  per(0, 2π) × ˙L2(0, 2π) for s ≥ 1, and the null controllability fails in the space Hs  per(0, 2π) × L2(0, 2π)  for 0 ≤ s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In this case also, null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) is inconclusive when  the time is small (0 < T ≤ 2π  ¯u ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  27  Similar to the barotropic case, we consider the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) with the initial state (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6) and the  boundary conditions  ρ(t, 0) = ρ(t, 2π),  u(t, 0) = u(t, 2π),  ux(t, 0) = ux(t, 2π) + q2(t),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)  θ(t, 0) = θ(t, 2π),  θx(t, 0) = θx(t, 2π),  t ∈ (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  In this case also, following the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='9 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='10, we get null controllability of  the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2) at time T > 2π  ¯u in the space ˙Hs  per(0, 2π) × ( ˙L2(0, 2π))2 for s ≥ 1, and null  controllability fails in the space Hs  per(0, 2π) × (L2(0, 2π))2 for 0 ≤ s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We next consider the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) with the initial state (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6) and the boundary conditions  ρ(t, 0) = ρ(t, 2π),  u(t, 0) = u(t, 2π),  ux(t, 0) = ux(t, 2π),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3)  θ(t, 0) = θ(t, 2π),  θx(t, 0) = θx(t, 2π) + q3(t),  t ∈ (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Similar to the previous case, following the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='11 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='12, we get null  controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) at time T > 2π  ¯u in the space ˙Hs  per(0, 2π) × ( ˙L2(0, 2π))2 for  s ≥ 1, and null controllability fails in the space Hs  per(0, 2π) × (L2(0, 2π))2 for 0 ≤ s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  For both systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='6)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3), null controllability is inconclusive for a small  time 0 < T ≤ 2π  ¯u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' More number of controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Adding controls in both velocity and temperature components through  periodic boundary conditions does not improve the null controllability result of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) with  respect to the regularity of the initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Estimates of the observation terms remain the same as in  the control acts in velocity or temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Controllability under Dirichlet Boundary Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Let us consider the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) in  the interval (0, 1) with the initial state (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2) and the following boundary conditions  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4)  ρ(t, 0) = p(t),  u(t, 0) = 0,  u(t, 1) = q(t),  t ∈ (0, T ),  where p and q are boundary controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' It is known in [3] that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4) (with q = 0) is  null controllability at a large time T using only one boundary control p ∈ L2(0, T ) provided the initial  states belong to the space Hs  per(0, 1) × L2(0, 1) with s > 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) using  a boundary control acts only in velocity through Dirichlet conditions (that is, p = 0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4)) is still an  open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  In the case of non-barotropic fluids, null controllability of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5) at large time T using only  one boundary control acts either in density, velocity or temperature through Dirichlet conditions is also  an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Proof of the Well-Posedness Results  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Existence of semigroup: proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The proof is divided into several parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' The operator A is dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' for all (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' ζ)† ∈ D(A)  ℜ ⟨AU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' U⟩(L2(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2π))3 = ℜ  �  \uf8eb  \uf8ec  \uf8ec  \uf8ed  −¯uξx − ¯ρηx  − R¯θ  ¯ρ ξx + λ0ηxx − ¯uηx − Rζx  − R¯θ  c0 ηx + κ0ζxx − ¯uζx  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8  �  (L2(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2π))3  = ℜ  �  −R¯θ¯u  � 2π  0  ¯ξξxdx − R¯θ¯ρ  � 2π  0  ¯ξηxdx − R¯θ¯ρ  � 2π  0  ξx¯ηdx + λ0¯ρ2  � 2π  0  ¯ηηxxdx − ¯ρ2¯u  � 2π  0  ¯ηηxdx  −R¯ρ2  � 2π  0  ¯ηζxdx − R¯ρ2  � 2π  0  ηx¯ζdx + κ0  ¯ρ2c0  ¯θ  � 2π  0  ¯ζζxxdx − ¯u ¯ρ2c0  ¯θ  � 2π  0  ¯ζζxdx  �  = −R¯θ¯u  2  � 2π  0  d  dx(|ξ|2)dx − λ0¯ρ2  � 2π  0  ¯ηxηxdx − ¯ρ2¯u  2  � 2π  0  d  dx(|η|2)dx − κ0  ¯ρ2c0  ¯θ  � 2π  0  ¯ζxζxdx  − ¯u  2  ¯ρ2c0  ¯θ  � 2π  0  d  dx(|ζ|2)dx − λ0¯ρ2  � 2π  0  |ux|2 dx − κ0  ¯ρ2c0  ¯θ  � 2π  0  |ζx|2 dx ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  28  Part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  The operator A is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  This is equivalent to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  For any ν > 0 and any  \uf8eb  \uf8ec  \uf8ed  f  g  h  \uf8f6  \uf8f7  \uf8f8 ∈ (L2(0, 2π))3, we can find a  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8 ∈ D(A) such that  (νI − A)  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  f  g  h  \uf8f6  \uf8f7  \uf8f8 ,  that is,  νξ + ¯uξx + ¯ρηx = f,  νη + R¯θ  ¯ρ ξx − λ0ηxx + ¯uηx + Rζx = g,  νζ + R¯θ  c0  ηx − κ0ζxx + ¯uζx = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Let ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Instead of solving the above problem, we will solve the following regularized problem  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1)  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  νξ + ¯uξx − ǫξxx + ¯ρηx = f,  νη + R¯θ  ¯ρ ξx − λ0ηxx + ¯uηx + Rζx = g,  νζ + R¯θ  c0 ηx − κ0ζxx + ¯uζx = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  with the following boundary conditions  ξ(0) = ξ(2π),  ξx(0) = ξx(2π),  η(0) = η(2π),  ηx(0) = ηx(2π),  ζ(0) = ζ(2π),  ζx(0) = ζx(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We now proceed through the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Using Lax-Milgram theorem, we first prove that the system (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='1) has a unique solution in  (H1  per(0, 2π))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Define the operator B : (H1  per(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π))3 × (H1  per(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π))3 → C by  B  \uf8eb  \uf8ec  \uf8ed  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  \uf8eb  \uf8ec  \uf8ed  ξ1  η1  ζ1  \uf8f6  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f8  = ǫ  � 2π  0  ξx( ¯ξ1)xdx + ¯ρ  � 2π  0  ηx ¯ξ1dx + ¯u  � 2π  0  ξx ¯ξ1dx + ν  � 2π  0  ξ ¯ξ1dx  + λ0  � 2π  0  ηx( ¯η1)xdx + ¯u  � 2π  0  ηx ¯η1dx + R¯θ  ¯ρ  � 1  0  ξx ¯η1dx + R  � 2π  0  ζx ¯η1dx + ν  � 2π  0  η ¯η1dx  + κ0  � 2π  0  ζx( ¯ζ1)xdx + ¯u  � 2π  0  ζx ¯ζ1dx + R¯θ  c0  � 2π  0  ηx ¯ζ1dx + ν  � 2π  0  ζ ¯ζ1dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  for all  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  \uf8eb  \uf8ec  \uf8ed  ξ1  η1  ζ1  \uf8f6  \uf8f7  \uf8f8 ∈ (H1  per(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Then, one can show that B is continuous and coercive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Thus, by  Lax-Milgram theorem, for every ǫ > 0, there exists a unique solution (ξǫ, ηǫ, ζǫ)† ∈ (H1  per(0, 2π))3 such  that  B  \uf8eb  \uf8ec  \uf8ed  \uf8eb  \uf8ec  \uf8ed  ξǫ  ηǫ  ζǫ  \uf8f6  \uf8f7  \uf8f8 ,  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f8 = F  \uf8eb  \uf8ec  \uf8ed  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f8 ,  ∀  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8 ∈ (H1  per(0, 2π))3,  where F : (H1  per(0, 2π))3 → C is the linear functional given by  F  \uf8eb  \uf8ec  \uf8ed  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f8 :=  � 2π  0  f ¯ξdx +  � 2π  0  g¯ηdx +  � 2π  0  h¯ζdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  29  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Observe that  ℜ  \uf8eb  \uf8ec  \uf8edB  \uf8eb  \uf8ec  \uf8ed  \uf8eb  \uf8ec  \uf8ed  ξǫ  ηǫ  ζǫ  \uf8f6  \uf8f7  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  \uf8eb  \uf8ec  \uf8ed  ξǫ  ηǫ  ζǫ  \uf8f6  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f8 ≤  � 2π  0  ��fξǫ�� dx +  � 2π  0  |gηǫ| dx +  � 2π  0  ��hζǫ�� dx  ≤ 1  2  � 2π  0  �  |f|2 + |g|2 + |h|2�  dx + 1  2  � 2π  0  ���ξǫ��2 + |ηǫ|2 +  ��ζǫ��2�  dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  which yields  ǫ  � 2π  0  |ξǫ  x|2+ν  2  � 2π  0  |ξǫ|2+λ0  � 2π  0  |ηǫ  x|2+ν  2  � 2π  0  |ηǫ|2+κ0  � 2π  0  |ζǫ  x|2+ν  2  � 2π  0  |ζǫ|2 ≤ 1  2  � 2π  0  (|f|2+|g|2+|h|2)  This shows that the sequences (ηǫ) and (ζǫ) are bounded in H1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π) and the sequences (ξǫ) and (√ǫξǫ  x)  are bounded in L2(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since the spaces H1(0, 2π) and L2(0, 2π) are reflexive, there exist subsequences,  still denoted by (ηǫ), (ζǫ), (ξǫ), and functions ξ ∈ L2(0, 2π) and η ∈ H1(0, 2π) such that  ηǫ ⇀ η in H1(0, 2π), and ξǫ ⇀ ξ in L2(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Furthermore, we have  � 2π  0  |ǫξǫ  x|2 = ǫ  � 1  0  ��√ǫξǫ��2 → 0, as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Now, since B  \uf8eb  \uf8ec  \uf8ed  \uf8eb  \uf8ec  \uf8ed  ξǫ  ηǫ  ζǫ  \uf8f6  \uf8f7  \uf8f8 ,  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f8 = F  \uf8eb  \uf8ec  \uf8ed  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f8, for all  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8 ∈ (H1  per(0, 2π))3, we may take  \uf8eb  \uf8ec  \uf8ed  ξ1  0  0  \uf8f6  \uf8f7  \uf8f8 ∈  (H1  per(0, 2π))3, so that we obtain  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2)  ǫ  � 2π  0  ξǫ  x( ¯ξ1)xdx + ¯ρ  � 2π  0  ηǫ  x ¯ξ1dx + ¯u  � 2π  0  ξǫ  x ¯ξ1dx + ν  � 2π  0  ξǫ ¯ξ1dx =  � 2π  0  f ¯ξ1dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Similarly, by taking  \uf8eb  \uf8ec  \uf8ed  0  η1  0  \uf8f6  \uf8f7  \uf8f8 ,  \uf8eb  \uf8ec  \uf8ed  0  0  ζ1  \uf8f6  \uf8f7  \uf8f8 ∈ (H1  per(0, 2π))3, we get  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) λ0  � 2π  0  ηǫ  x( ¯η1)xdx+ ¯u  � 2π  0  ηǫ  x ¯η1dx+ R¯θ  ¯ρ  � 1  0  ξǫ  x ¯η1dx+R  � 2π  0  ζǫ  x ¯η1dx+ν  � 2π  0  ηǫ ¯η1dx =  � 2π  0  g ¯η1dx,  and  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4)  κ0  � 2π  0  ζǫ  x( ¯ζ1)xdx + ¯u  � 2π  0  ζǫ  x ¯ζ1dx + R¯θ  c0  � 2π  0  ηǫ  x ¯ζ1dx + ν  � 2π  0  ζǫ ¯ζ1dx =  � 2π  0  h ¯ζ1dx  Integrating by parts, we get from equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='2) that,  ǫ  � 2π  0  ξǫ  x( ¯ξ1)xdx + ¯ρ  � 2π  0  ηǫ  x ¯ξ1dx − ¯u  � 2π  0  ξǫ( ¯ξ1)xdx + ν  � 2π  0  ξǫ ¯ξ1dx =  � 2π  0  f ¯ξ1dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Then, passing to the limit ǫ → 0, we obtain  ¯ρ  � 2π  0  ηx ¯ξ1dx + ¯u  � 2π  0  ξx ¯ξ1dx + ν  � 2π  0  ξ ¯ξ1dx =  � 2π  0  f ¯ξ1dx,  and the above relation is true ∀ξ1 ∈ C∞  c (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' As a consequence,  ¯ρηx + ¯uξx + νξ = f,  in the sense of distribution and therefore ¯uξx = f − ¯ρηx − νξ ∈ L2(0, 2π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' in other words, ξ ∈ H1(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We similarly have from identities (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='4)  νη + R¯θ  ¯ρ ξx − λ0ηxx + ¯uηx + Rζx = g,  νζ + R¯θ  c0  ηx − κ0ζxx + ¯uζx = h,  in the sense of distribution and therefore η, ζ ∈ H2(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  30  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' We now show η(0) = η(2π) and ηx(0) = ηx(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since the inclusion map i : H1(0, 2π) → C0( ¯  0, 2π)  is compact and ηǫ ⇀ η in H1(0, 2π), we obtain  ηǫ → η  in C0[0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Thus, (ηǫ(0), ηǫ(2π)) → (η(0), η(2π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Since ηǫ(0) = ηǫ(2π) for all ǫ > 0, we have  η(0) = η(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='3), we have after passing the limit as ǫ → 0  λ0  � 2π  0  ηx( ¯η1)xdx + ¯u  � 2π  0  ηx ¯η1dx + R¯θ  ¯ρ  � 1  0  ξx ¯η1dx + R  � 2π  0  ζx ¯η1dx + ν  � 2π  0  η ¯η1dx =  � 2π  0  g ¯η1dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Integrating by parts, we get  −λ0  � 2π  0  ηxx ¯η1dx + λ0(ηx(2π) ¯η1(2π) − ηx(0) ¯η1(0)) + ¯u  � 2π  0  ηx ¯η1dx + R¯θ  ¯ρ  � 1  0  ξx ¯η1dx  +R  � 2π  0  ζx ¯η1dx + ν  � 2π  0  η ¯η1dx =  � 2π  0  g ¯η1dx,  and therefore  ηx(2π) ¯η1(2π) − ηx(0) ¯η1(0) = 0  that is ηx(0) = ηx(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' In a similar way, we can obtain ζ(0) = ζ(2π) and ζx(0) = ζx(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  We now show ξ(0) = ξ(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Recall that we have after taking limit as ǫ → 0  ¯ρ  � 2π  0  ηx ¯ξ1dx − ¯u  � 2π  0  ξ( ¯ξ1)xdx + ν  � 2π  0  ξ ¯ξ1dx =  � 2π  0  f ¯ξ1dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  Integrating by parts, we get  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='5)  ¯ρ  � 2π  0  ηx ¯ξ1dx + ¯u  � 2π  0  ξx ¯ξ1dx − ¯u(ξ(2π)ξ1(2π) − ξ(0)ξ1(0)) + ν  � 2π  0  ξ ¯ξ1dx =  � 2π  0  f ¯ξ1dx,  and therefore  ξ(0) = ξ(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  So, we get  \uf8eb  \uf8ec  \uf8ed  ξ  η  ζ  \uf8f6  \uf8f7  \uf8f8 ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Hence, the operator A is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
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+page_content=', 21 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
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+page_content='  [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Martin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Rosier, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Rouchon, Null controllability of the structurally damped wave equation with moving  control, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content=' Control Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
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+page_content=' 660–684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
+page_content='  32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E2T4oBgHgl3EQfuggX/content/2301.04080v1.pdf'}
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+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+AMEDEO ALTAVILLA, EDOARDO BALLICO, AND MARIA CHIARA BRAMBILLA
+Abstract. We study surfaces of bidegree (1, d) contained in the flag threefold under
+the action of the twistor projection. First, we prove that there is no integral surfaces of
+bidegree (1, d) containing d + 2 twistor fibers such that three of them are not collinear.
+Then, fixed any union of 0 ≤ n ≤ d + 1 non-three-by-three collinear twistor fibers, we
+show that there is an integral (1, d)-surface containing them and no other twistor fibers.
+The result is also true for d + 2 twistor fibers with additional suitable hypotheses. Later,
+we focus on surfaces of low bidegrees and prove that, for any set of 0 ≤ n ≤ 3 (resp. n = 4)
+twistor fibers, there is a smooth (resp. integral) surface of bidegree (1, 2) containing them
+and no other twistor fiber. Finally, we prove that there is no integral (1, d)-surface, for
+d = 2, 3, containing d + 3 twistor fibers.
+Contents
+1.
+Introduction
+1
+2.
+Preliminaries and first results
+3
+2.1.
+Curves in F and smooth conics
+4
+2.2.
+Surfaces of bidegree (1, 0) and (0, 1)
+6
+2.3.
+Surfaces of bidegree (0, d) and (1, d)
+7
+2.4.
+Non-collinear smooth conics
+8
+3.
+Surfaces of bidegree (1, d)
+10
+4.
+Surfaces of bidegree (1, 2) and (1, 3)
+15
+4.1.
+Surfaces of bidegree (1, 2) containing 0 ≤ n ≤ 4 twistor fibers
+15
+4.2.
+Non existence results for surfaces of bidegree (1, 2) and (1, 3)
+22
+References
+24
+1. Introduction
+The flag threefold F can be seen as the twistor space of the complex projective plane
+P2 endowed with all its standard structures except for its orientation which, in this case,
+is the opposite of the usual one [6, 11],
+π : F → P2.
+The study of the twistor geometry of the flag is motivated by the search of Riemannian 4-
+manifolds admitting several integrable complex structures compatible with the prescribed
+metric (see e.g. [8, 12, 13]). In this context, a recent trend is that of study particular cases,
+in order to find explicit examples [1, 2, 5, 7, 9].
+In [4] we have started a detailed analysis of the geometry of the algebraic curves and
+surfaces contained in F, in relation with the twistor projection. In particular, twistor fibers
+are smooth integral curves of bidegree (1,1) (called smooth conics). In [3] we gave a first
+bound on the maximum number of smooth conics contained in a smooth surface S ⊂ F.
+2010 Mathematics Subject Classification. Primary: 32L25, 14M15; Secondary: 14D21, 14J26.
+Key words and phrases. flag threefold, twistor projection, twistor fiber, surfaces, bidegree.
+All the authors are partially supported by GNSAGA. The first named author is partially supported by
+the INdAM project ‘Teoria delle funzioni ipercomplesse e applicazioni’.
+1
+arXiv:2301.04874v1  [math.AG]  12 Jan 2023
+
+2
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+Here, by focusing our attention to a specific family of surfaces, we obtain more precise
+results. In fact we analyze the case of bidegree (1, d) surfaces in F and study the number
+and the arrangements of twistor fibers contained in them.
+In order to explain in details our results, we need to clarify some definitions. The flag
+threefold can be defined as
+F := {(p, ℓ) ∈ P2 × P2 | pℓ = 0},
+where p = [p0 : p1, p2], ℓ = [ℓ0 : ℓ1 : ℓ2] ∈ P2 and pℓ = p0ℓ0 + p1ℓ1 + p2ℓ2. The notation
+(p, ℓ), would recall a couple (point,line), and the condition pℓ = 0 translates as p belongs
+to ℓ. In order to simplify the notation, we identify the second factor P2∨ with P2.
+We have three projection maps: π1, π2, π : F → P2, defined as
+π1(p, ℓ) = p,
+π2(p, ℓ) = ℓ,
+π(p, ℓ) = p × ℓ = [p1ℓ2 − p2ℓ1, p2ℓ0 − p0ℓ2, p0ℓ1 + p1ℓ0],
+where the third one is the twistor map. The fibers of such three maps are the object of
+our investigation.
+Being embedded in P2 × P2, it is possible to define a natural notion of bidegree for
+surfaces and a (bit less natural) one for curves in F (see Definition 2.1). In particular, for
+any q ∈ P2, the three fibers π−1
+1 (q), π−1
+2 (q) and π−1(q) are curves of bidegree (0, 1), (1, 0)
+and (1, 1), respectively. While the fibers of π1 and π2 exhaust the family of bidegrees (1, 0)
+and (0, 1) curves, twistor fibers are only a (non-open Zariski dense) subset of those of
+bidegree (1, 1). Since the curves of bidegree (1, 1) are rational, we call them conics. There
+are only two types of bidegree (1, 1) curves: the reducible ones (union of a bidegree (1, 0)
+and of a bidegree (0, 1) curves intersecting at a point), or smooth conics. All of them can
+be described as
+Lq,m := {(p, ℓ) ∈ F | pm = 0, qℓ = 0},
+where q, m ∈ P2 and, the reducible and smooth cases are obtained for qm = 0 or qm ̸= 0,
+respectively.
+In twistor theory there is an antiholomorphic involution without fixed points that plays
+important roles in identifying twistor fibers [6, 11]. In our case, this map can be defined
+as j : F2 → F2, where
+j(p, ℓ) = (ℓ, p).
+A smooth conic Lq,m is a twistor fiber if and only if j(Lq,m) = Lq,m if and only if m = q.
+As explained in [6], the geometry of F as twistor space does not change if we consider a
+conformal copy of its base space P2. Therefore, it is natural to classify objects in F up to
+conformal transformations, which, in this case, means up to projective transformations of
+F coming from the lift via π of a conformal transformation of P2. It is possible to see that
+such transformations of F are exactly those which commute with j (see [4] for the case of
+the flag manifold). Hence, in particular, the number of twistor fibers contained in a given
+surface and their arrangement are conformal invariants.
+In order to state our main results we need some more notation. We denote by C(1) = C
+the set of smooth conics in F and by C(n), n ≥ 2, the set of n pairwise disjoint smooth
+conics. In an analogous way we define T (1) = T ⊂ C as the set of twistor fibers and
+T (n) ⊂ C(n), n ≥ 2, as the set of n pairwise disjoint twistor fibers.
+We will see in Remark 2.5 that for any couple of different smooth conics, there is a
+unique bidegree (1, 0) curve L = π−1
+2 (q2) and a unique bidegree (0, 1) curve R = π−1
+1 (q1)
+such that L and R intersect both smooth conics. In the case of a couple of twistor fibers
+we also have R = j(L). We say that three or more smooth conics are collinear if there is
+a (1, 0) curve L which intersects all of them. To be collinear, for three or more smooth
+conics, is a Zariski closed condition.
+To be more precise, in Definition 2.7, we define the set C∗(n) which parametrizes all
+A ∈ C(n) such that #(L ∩ A) ≤ 2 for all curves L of bidegree (1, 0). Clearly C∗(1) = C(1),
+and C∗(2) = C(2), while for n ≥ 3 the open set C∗(n) is given by the set of disjoint smooth
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+3
+conics such that no three of them are collinear.
+Moreover, we set T ∗(n) := T (n) ∩ C∗(n).
+In Theorem 2.19 we characterize the elements A ∈ C∗(d + 1) to be those which do not
+obstruct the linear system |IA(1, d)|.
+We now summarize the main results of the paper. In Section 3, we study surfaces of
+bidegree (1, d) containing a certain number of smooth conics or twistor fibers, and we
+prove the following two theorems.
+Theorem 1.1. For any d ∈ N and A ∈ T ∗(d + 2), there is no integral surfaces of bidegree
+(1, d) containing A.
+Theorem 1.2. Fix integer d ≥ 1 and 0 ≤ n ≤ d + 2. There is an integral S ∈ |OF(1, d)|
+containing exactly n twistor fibers.
+We also show, in Theorem 3.4, that the first result is sharp. Indeed, for any A ∈ T ∗(n),
+with 0 ≤ n ≤ d + 1, we are able to find an integral surface of bidegree (1, d) containing A
+and no other twistor fibers. This last issue requires some effort and the proof is divided
+into several particular case.
+Theorem 1.2 is a consequence of Theorem 3.4 and Theorem 3.8. More precisely, in
+Theorem 3.4, for 0 ≤ n ≤ d + 1, we prove that fixed any union A of 0 ≤ n ≤ d + 1
+non-three-by-three collinear twistor fibers, there is an integral (1, d)-surface containing A
+and no other twistor fibers. The extremal case n = d + 2 is considered in Theorem 3.8,
+where we prove that given d + 2 general collinear twistor fibers there is an integral surface
+of bidegree (1, d) containing them.
+In Section 4, we focus on surfaces of bidegree (1, 2) and (1, 3). The main results are
+summarized by the following statements:
+Theorem 1.3. Fix 0 ≤ n ≤ 3. There is a smooth S ∈ |OF(1, 2)| containing exactly n
+twistor fibers. Moreover, there exists a bidegree (1, 2) integral surface containing exactly 4
+twistor fibers.
+Theorem 1.4. There is no integral S ∈ |OF(1, 2)| containing at least 5 twistor fibers.
+Theorem 1.5. There is no integral S ∈ |OF(1, 3)| containing at least 6 twistor fibers.
+The first existence result (Theorem 1.3) follows from Theorem 4.6, for 0 ≤ n ≤ 3, and
+Theorem 4.10, for the case n = 4. In the extremal case n = 4, we will also show that the
+surfaces are singular along a line.
+The two non-existence results (Theorems 1.4 and 1.5) are proved in the last Section
+4.2.
+An essential tool is Lemma 4.11 which states that if a surface of bidegree (1, d)
+contains d + 3 or more collinear twistor fibers, then this surface is reducible and one of its
+components is a surface of bidegree (1, 1) containing 4 of the prescribed twistor fibers.
+We conclude here with a comparison with the other (smooth) algebraic twistor space of
+a Riemannian 4-manifold, which is the complex projective space. This is the twistor space
+of the standard 4-sphere [11]. In this case, surfaces of degree 2 and 3 were studied in some
+details. In particular, analogously to our case of surfaces of bidegree (1, 1), surfaces of
+degree 2 in P3 might contain 0, 1 or 2 twistor fibers. If a surface of degree 2 contains more
+than 2 twistor fibers, then it contains infinitely many of them [13]. For degree 3 surfaces,
+such a maximum is realized for 5 twistor fibers [5] which is more than our maximum of 4
+for surfaces of bidegree (1, 2). This difference could be explained by observing a certain
+unbalancedness of the case of “total degree” 3 in the flag threefold. On the other hand,
+this particular unbalancedness allows us to compute all the Betti numbers in the next
+section as well as the use of the geometry of the Hirzebruch surfaces.
+2. Preliminaries and first results
+In this section, we collect some known results about algebraic curves and surfaces in
+the flag. Then, we give first results on the space of bidegree (0, d) and (1, d) surfaces
+
+4
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+containing a certain amounts of twistor fibers. In particular, we introduce the concept of
+collinear smooth conics and give a topological characterization in terms of cohomology of
+certain ideal sheaves.
+For most of the known material about F, we refer mainly to [4] and to [3, Section 2].
+However, we recall here some basic notion and results in order to be as more self-contained
+as possible.
+Let us consider the multi projective space P2 × P2; an element (p, ℓ) ∈ P2 × P2 will
+be a couple written in the following form p = [p0 : p1 : p2], ℓ = [ℓ0 : ℓ1 : ℓ2]⊤, so that
+pℓ = p0ℓ0 + p1ℓ1 + p2ℓ2. Even if it is classically embedded in P2 × P2∨, we might see
+F := {(p, ℓ) ∈ P2 × P2 | pℓ = 0} as a hypersurface of bidegree (1, 1) of P2 × P2. We denote
+by Π1 and Π2 the two standard projections of P2×P2 and we will use small letters for their
+restrictions, i.e. πi = Πi|F, i = 1, 2. Thus, the two natural projections define a natural
+notion of bidegree for algebraic surfaces in F. Moreover, for all (a, b) ∈ Z2 we have the
+following natural exact sequence
+(1)
+0 → OP2×P2(a − 1, b − 1) → OP2×P2(a, b) → OF(a, b) → 0,
+and, for any (a, b) ∈ N2, we get (see e.g. [4, Lemma 2.3])
+(2)
+h0(OF(a, b)) = (a + 1)(b + 1)(a + b + 2)
+2
+and
+h1(OF(a, b)) = 0.
+It will be useful to recall from [4, Proposition 3.11] the multiplication rules in the Chow
+ring:
+OF(1, 0) · OF(1, 0) · OF(1, 0) = 0,
+OF(1, 0) · OF(0, 1) · OF(1, 0) = 1,
+OF(0, 1) · OF(1, 0) · OF(0, 1) = 1,
+OF(0, 1) · OF(0, 1) · OF(0, 1) = 0.
+(3)
+2.1. Curves in F and smooth conics. Let us recall a notion of bidegree for the family
+of algebraic curves in F already given in [4, 3].
+Definition 2.1. Let C ⊂ F be an integral algebraic curve. We define the bidegree of C
+as the couple of positive integer numbers (d1, d2), where di = 0 if πi(C) = {x}, otherwise
+di = deg(πi(C)) deg(πi|C).
+If a curve D has irreducible components C1, . . . , Cs then the bidegree of D is taken to
+be the sum of the bidegrees of C1, . . . , Cs.
+Recall from [3, Remark 2.4] that if a curve C is such that C · OF(1, 0) = d1 and
+C · OF(0, 1) = d2, then it has bidegree (d1, d2).
+From the previous table of multiplication, we can easily derive the following formula.
+Lemma 2.2. For any choice of non-negative integers a, b, c, d, the one-dimensional cycle
+OF(a, b) · OF(c, d) has bidegree
+(ad + b(c + d), a(c + d) + bc).
+Proof. We have OF(a, b) · OF(c, d) = acOF(1, 0) · OF(1, 0) + (ad + bc)OF(1, 0) · OF(0, 1) +
+bdOF(0, 1)·OF(0, 1). Hence the thesis is easily obtained by recalling that OF(1, 0)·OF(1, 0)
+(resp. OF(1, 0) · OF(0, 1), resp. OF(0, 1) · OF(0, 1)) is a one-dimensional cycle of bidegree
+(0, 1) (resp. bidegree (1, 1), resp. bidegree (1, 0)).
+□
+Remark 2.3. Notice that the fibers of π1 are algebraic curves of bidegree (0, 1), while
+those of π2 have bidegree (1, 0) (see, e.g. [4, Section 3]).
+Moreover, all bidegree (0, 1)
+curves can be seen as complete intersections between two different (1, 0) surfaces (and
+analogously for bidegree (1, 0) curves).
+Among all algebraic curves in F we focus our attention on the family of bidegree (1, 1)
+curves. These are geometrically described in [4, Section 3.1] and are parameterized by
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+5
+(q, m) ∈ P2 × P2. In fact, as anticipated in the introduction, any of these curves can be
+written as
+Lq,m := {(p, ℓ) ∈ F | p ∈ m, ℓ ∋ q} = {(p, ℓ) ∈ F | qℓ = 0, pm = 0}.
+There are two types of these curves: the smooth and irreducible ones (when qm ̸= 0) and
+the union of a (1, 0) and of a (0, 1) intersecting at a point (when qm = 0, i.e. (q, m) ∈ F).
+In any case, each bidegree (1, 1) curve can be seen as the complete intersection of a surface
+of bidegree (1, 0) with one of bidegree (0, 1). As already mentioned in the introduction,
+the 4-dimensional family of smooth irreducible (1, 1) curves will be denoted by C. The
+elements of C will be called smooth conics.
+Remark 2.4. From the very definition of smooth integral conics, it is clear that, for any
+C ∈ C we have that πi(C) is a line in P2.
+Remark 2.5. Notice that for any two different elements Lq,m, Lq′,m′ ∈ C there exist a
+unique curve L of bidegree (1, 0) and a unique R of bidegree (0, 1) such that L and R
+meets both Lq,m and Lq′,m′ at a point (see Figure 1).
+From the analysis made in [4,
+Section 3.1] it is easy to see that L = π−1
+2 (q × q′) and R = π−1
+1 (m × m′), where ×
+stands for the standard (formal) cross product. Equivalently, L = π−1
+2 (Sing(π2(A)) and
+R = π−1
+1 (Sing(π1(A)).
+We say that three disjoint smooth conics are collinear if they
+Figure 1. Any two smooth conics are connected by a curve of bidegree
+(1, 0) and by a curve of (0, 1).
+intersect the same (1, 0) curve L.
+The fibers of the twistor projection π : F → P2 (see [4, Section 5]) form a subset T
+of the family of conics C. The twistor fibers are also characterized to be the irreducible
+elements in C that are fixed by the anti-holomorphic involution j : F → F defined as
+j(p, ℓ) = (ℓ, p).
+Being the set of fixed point of j, a curve Lq,m belongs to T if and only if m = q. Moreover,
+the set T is a Zariski dense in C (see, e.g. [3, Section 4]).
+Remark 2.6. If L is the curve of bidegree (1, 0) connecting two different twistor fibers
+(see Remark 2.5), then the curve of bidegree (0, 1) connecting them is exactly R = j(L).
+Hence if three twistor fibers are collinear, then they intersect the same (1, 0) curve and
+the same (0, 1) curve.
+Recall from the introduction that, for any positive integer n, C(n) denotes the 4n-
+dimensional set of n pairwise disjoint elements of C and T (n) the set of n pairwise disjoint
+elements of T . As before, T (n) is a Zariski dense in C(n) (see again [3, Section 4]). We
+now introduce the following crucial definition.
+
+R
+L
+q,m
+q",m6
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+Definition 2.7. For any n ≥ 1 let C∗(n) be the set of all A ∈ C(n) such that for any curve
+L of bidegree (1, 0), it holds #(L ∩ D) ≤ 2. Set T ∗(n) := T (n) ∩ C∗(n).
+Clearly we have C∗(n) = C(n) and T ∗(n) = T (n), for n = 1, 2. For n ≥ 3 the set
+C(n) \ C∗(n) parametrizes unions of n disjoint smooth conics such that at least three of
+them are collinear, hence C∗(n) is an open Zariski dense in C(n), as well as T ∗(n) in T (n).
+Therefore, for any n ≥ 1, all the following inclusions are Zariski dense: T ∗(n) ⊂ T (n) ⊂
+C(n), T ∗(n) ⊂ C∗(n) ⊂ C(n).
+2.2. Surfaces of bidegree (1, 0) and (0, 1). We now turn our attention back to sur-
+faces. We recall from [4, Section 3.2] and [3, Section 2] that (1, 0) and (0, 1) surfaces are
+Hirzebruch surfaces of first type. In particular, a surface X of bidegree (1, 0) can be seen
+as the lift, via π1 of a line (and analogously for a bidegree (0, 1) surface Y ). Using this
+description, it is easy to see that any of these surfaces represent the blow up of P2 at a
+point. Let F1 be a Hirzebruch surface of type 1; we now describe the relation between the
+generators of the Picard group of F1 and the family of curves in F previously described.
+We recall that Pic(F1) = Zh ⊕ Zf, where
+h2 = −1,
+f2 = 0,
+hf = 1.
+Notation 2.8. For the following analysis and the rest of the paper X will denote a surface
+of bidegree (1, 0), while Y one of bidegree (0, 1).
+Identifying a surface X with F1 we obtain that OX(1, 0) ≃ OF1(f) which in turn corre-
+sponds to the set of curves in F of bidegree (0,1) contained in X. On the other hand we
+have that OX(0, 1) ≃ OF1(h + f) which corresponds to elements of C.
+Hence, for any a, b ∈ Z and for any α, β ∈ Z, we obtain the following two relations
+(4)
+OX(a, b) ∼= OF1(bh + (a + b)f),
+and
+OF1(αh + βf) ∼= OX(β − α, α).
+For a surface Y of bidegree (0, 1) we can derive similar formulæ:
+(5)
+OY (a, b) ∼= OF1(ah + (a + b)f),
+and
+OF1(αh + βf) ∼= OY (α, β − α),
+for any a, b ∈ Z and for any α, β ∈ Z.
+Remark 2.9. Let X be a surface of bidegree (1, 0). Then X does not contain any element
+of C(2). In fact, any element of C in X corresponds to an element of OF1(h + f) and any
+two elements of type h + f meets.
+The same holds for surfaces Y of bidegree (0, 1).
+In particular, for each bidegree (1, 0) or (0, 1) surface there is exactly one twistor fiber
+contained in it.
+We now recall from [3, Lemma 2.5] that, for any a, b ≥ 0, using the exact sequence
+0 → OF(a − 1, b) → OF(a, b) → OX(a, b) → 0,
+and its analogous for Y , we have that
+h0(OX(a, b)) = a(b + 1) +
+�b + 2
+2
+�
+,
+h0(OY (a, b)) = b(a + 1) +
+�a + 2
+2
+�
+,
+while
+h1(OX(a, b)) = h1(OY (a, b)) = 0.
+Moreover, if a > 0, b > 0, the line bundles OX(a, b) and OY (a, b) are very ample.
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+7
+2.3. Surfaces of bidegree (0, d) and (1, d). We now pass to study higher bidegree sur-
+faces. We start with some consideration about bidegree (0, d) surfaces.
+Remark 2.10. As described in [4, Section 3.3], any integral surface S of bidegree (0, d)
+is equal to π−1
+1 (C) for some degree d integral curve C. Therefore, for d ≥ 2, no integral
+S ∈ |OF(0, d)| contains a smooth conic. Otherwise, thanks to Remark 2.4, πi(S) would
+contain a line, but π1(S) is an integral curve of degree d.
+For n ≥ 2 we now compute how many (non integral) bidegree (0, d) surfaces contain a
+fixed element of C(n). First we set the following notation.
+Notation 2.11. We will denote by IU,V the ideal sheaf of a scheme U contained in a
+projective variety V ; whenever V = F we will omit it. So, in particular, if A ∈ C(n) we
+will write IA := IA,F.
+Lemma 2.12. Fix d ≥ 0, n ≥ 1, and A ∈ C(n). We have
+h0(IA(0, d)) = (d − n + 2)(d − n + 1)
+2
+and
+h1(IA(0, d)) =
+� n(n−1)
+2
+if n ≤ d + 1
+n(d + 1) − (d+2)(d+1)
+2
+if n ≥ d + 1.
+Proof. Recall that OF(0, d) = π∗
+1(OP2(d)), and IA(0, d) = π∗
+1(IT,P2(d)) where T = π1(A)
+is a union of n distinct lines in P2. In general, we have that:
+h0(OF(0, d)) =
+�d + 2
+2
+�
+,
+h0(IA(0, d)) =
+�d + 2 − n
+2
+�
+,
+h0(OA(0, d)) = n(d + 1),
+hence, using the exact sequence
+0 → IA(0, d) → OF(0, d) → OA(0, d) → 0,
+since h1(OF(0, d)) = 0 and
+�d+2−n
+2
+�
+= 0 if n ≥ d + 1, we get the result.
+□
+Remark 2.13. As a direct consequence of the previous lemma, we can state that, for any
+C ∈ C, there is only one surface Y in |IC(0, 1)| and, analogously, only one surface X in
+|IC(1, 0)|.
+Remark 2.14. If A ∈ C(n), we consider the following exact sequence:
+0 → IA(a, b) → OF(a, b) → OA(a, b) → 0.
+Since the conics in A are all disjoint, we have that for any (a, b) ∈ N2
+(6)
+h0(OA(a, b)) = n(a + b + 1),
+(see, e.g. [3, Sequence (7) and proof of Theorem 4.4]).
+Recall from Formula (2) that
+h0(OF(a, b)) = (a+1)(b+1)(a+b+2)
+2
+. Therefore, as h1(OA(a, b)) = h1(OF(a, b)) = 0, for any
+A ∈ C(n) and d ≥ 0, we have
+(7)
+χ(IA(1, d)) = h0(IA(1, d)) − h1(IA(1, d)) = (d + 1)(d + 3) − n(d + 2).
+Hence, if n = d + 2, we have h0(OA(1, d)) = (d + 2)2 and so
+(8)
+h0(OF(1, d)) = (d + 1)(d + 3) = (d + 2)2 − 1 = h0(OA(1, d)) − 1
+and, if n = d + 1:
+h0(OF(1, d)) = h0(OA(1, d)) + d + 1.
+In the following we will implicitly make use of the following simple observation.
+
+8
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+Remark 2.15. Since the elements of T (1) are the fibers of a map, the twistor map, each
+p ∈ F is contained in a unique element, C, of T . Since j(C) = C, j(p) ∈ C. Analogously,
+note that p is contained in a unique curve of bidegree (1, 0) and a unique curve of bidegree
+(0, 1), π−1
+2 (π2(p)) and π−1
+1 (π1(p)).
+The following remark will be used several times in the next pages.
+Remark 2.16. Fix a positive integer d and an integral S ∈ |OF(1, d)|.
+Since π1|X is
+birational onto its image, S is rational. By B´ezout Theorem, for any p ∈ P2, the bidegree
+(1, 0) curve π−1
+2 (p) either is contained in S, or intersects S in a single point (scheme-
+theoretically).
+We close this subsection with a technical result that will be used in the next pages.
+In [3, Proposition 4.1] we proved that there exists at most one surface of bidegree (a, b)
+containing a number equal or greater than a2 + ab + b2 of smooth conics. We will now
+generalize this result to a more general context.
+Proposition 2.17. Fix (a, b, c, d) ∈ N4 such that (a, b) ̸= (0, 0), c > 0, d > 0. Take
+A ∈ C(n). Assume the existence of an integral S ∈ |OF(a, b)| containing A and assume
+one of the following conditions:
+(1) ad + b(c + d) < n;
+(2) a(c + d) + bc < n;
+(3) ad + b(c + d) = a(c + d) + bc = n.
+Then each element of |IA(c, d)| contains S and in particular c ≥ a and d ≥ b.
+Proof. Assume by contradiction the existence of S′ ∈ |IA(c, d)| such that S′ ⊉ S. We have
+OF(a, b) · OF(c, d) =acOF(1, 0) · OF(1, 0) + (ad + bc)OF(1, 0) · OF(0, 1)
++ bdOF(0, 1) · OF(0, 1).
+Since S′ ⊉ S, the intersection S ∩ S′ is a curve of bidegree (ad + b(c + d), a(c + d) + bc)
+(see Lemma 2.2). Since c > 0 and d > 0, then OF(c, d) is ample. Moreover, since S is
+irreducible, A has bidegree (n, n) and A is not connected, then the intersection contains
+some more component in addition to A. Hence, either ad+bc+bd > n and ac+ad+bc ≥ n
+or ad + bc + bd ≥ n and ac + ad + bc > n.
+□
+2.4. Non-collinear smooth conics. We now want to characterize the conics in C∗(n) in
+terms of cohomology. We start showing that the vanishing of certain cohomology groups
+implies that an element A ∈ C(n) lies in C∗(n).
+Lemma 2.18. Fix d ≥ 0, 3 ≤ n ≤ d+1 and A ∈ C(n). If h1(IA(1, d)) = 0 then A ∈ C∗(n).
+Proof. Assume, by contradiction, that there exists a curve L of bidegree (1, 0) such that
+#(L ∩ A) ≥ 3 and consider the exact sequence defining IA:
+(9)
+0 → IA(1, d) → OF(1, d) → OA(1, d) → 0.
+We will prove that the restriction map H0(OF(1, d)) → H0(OA(1, d)) is not surjective,
+which implies that h1(IA(1, d)) > 0. In fact, assume that it is surjective. Then, consider
+the following diagram
+OF(1, d)
+OL(1, d)
+OA(1, d)
+OL∩A(1, d)
+......................................................................................
+....................................................................................... ............
+....................................................................... ............
+......................................................................................
+.
+As the vertical maps are surjective, then the map H0(OF(1, d)) → H0(OA∩L(1, d)), induced
+by the composition, is surjective too and hence of rank at least 3 (since L∩A has cardinality
+at least 3 and the irreducible components of A are pairwise disjoint). On the other hand the
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+9
+restriction map H0(OF(1, d)) → H0(OL(1, d)) has rank 2 and this gives a contradiction.
+□
+Before completing the characterization of the conics in C∗(d + 1), we expose a general
+construction that will be used in several following discussions. Let A ∈ C(n) and let C be
+any connected component of A. Set B := A \ C. Then, for any a, b ≥ 0, if Y ∈ |OC(0, 1)|,
+we have the following residual exact sequence
+0 → IResY (A)(a, b − 1) → IA(a, b) → IA∩Y,Y (a, b) → 0,
+but as ResY (A) = B and A ∩ Y = (B ∩ Y ) ∪ C, we have
+(10)
+0 → IB(a, b − 1) → IA(a, b) → I(B∩Y )∪C,Y (a, b) → 0.
+Clearly an analogous sequence can be written for X ∈ |OC(1, 0)|.
+Theorem 2.19. Fix d ≥ 0 and A ∈ C(d + 1). We have A ∈ C∗(d + 1) if and only if
+h1(IA(1, d)) = 0.
+Proof. By the previous lemma we need only to prove that A ∈ C∗(d+1) satisfies h1(IA(1, d)) =
+0. We use induction on d ≥ 0. The case d = 0 is true by Lemma 2.12. Therefore, we
+may assume d > 0 and use induction on d. Let C be a connected component of A, set
+B := A \ C and cal Y the only element of |IC(0, 1)|. Consider the residual exact sequence
+(10), with a = 1 and b = d. Since A ∈ C(d + 1), C ∩ B = ∅ and B ∩ Y is formed by d
+different points, up to the identification of D with F1 we have
+I(B∩Y )∪C,Y (1, d) ∼= I(B∩Y )∪C,F1(1, d)(h + (d + 1)f) ∼= IB∩Y,F1(df) .
+Using (10) and induction we are left to prove that h1(IB∩Y,F1(df)) = 0 if and only if
+A ∈ C∗(d + 1).
+Consider now the following exact sequence
+(11)
+0 → IB∩Y,F1(df) → OF1(df) → OB∩Y (df) → 0.
+Since h0(OF1(df)) = d + 1 and h0(OB∩Y (df)) = d, we have h1(IB∩Y,F1(df)) > 0 if
+and only if h0(IB∩Y,F1(df)) ≥ 2. The last inequality means that there exist at least two
+different sets of d fibers containing the set of d points B ∩Y . This is equivalent to the fact
+that there exists a fiber L ∈ |f| such that #(B ∩ L) ≥ 2. Since L is a curve of bidegree
+(1, 0) in F, then L · Y = 0 in the intersection ring of F, therefore L ⊂ Y and hence we get
+L ∩ C ̸= ∅. Thus #(L ∩ A) ≥ 3, which means that A ̸∈ C∗(d + 1).
+□
+Corollary 2.20. Fix d ≥ 0, 0 ≤ n ≤ d + 1 and A ∈ C∗(n). Then h1(IA(1, d)) = 0. In
+particular, for n = 0, 1, 2 and for any A ∈ C(n), we have
+h1(IA(1, 1)) = 0
+and
+h0(IA(1, 1)) = 8 − 3n.
+Proof. By using [3, Remark 4.3], we easily get the first part of the statement. The second
+one, follows from Formula (7).
+□
+We point out that since T (n) is a Zariski dense of C(n), then the characterization given
+by Theorem 2.19 also holds for the set T ∗(n).
+Lemma 2.21. Fix an integer d ≥ 0 and A ∈ C∗(d + 2). Then h1(IA(1, d)) ≤ 1.
+Proof. The lemma is true for d = 0, because h0(IA(1, 0)) = 0 (see Remark 2.9). We assume
+d > 0 and use induction on d. Let C be a connected component of A, set B := A \ C
+and call Y the only element of |IC(0, 1)|. Consider the residual exact sequence (10), with
+a = 1 and b = d. Since A ∈ C(d + 2), C ∩ B = ∅ and B ∩ Y is formed by d + 1 different
+points, up to the identification of D with F1 we have
+I(B∩Y )∪C,Y (1, d) ∼= I(B∩Y )∪C,F1(1, d)(h + (d + 1)f) ∼= IB∩Y,F1(df) .
+Using (10) and induction we are left to prove that h1(IB∩Y,F1(df)) = 0 if A ∈ C∗(d + 2).
+
+10
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+Consider now the following exact sequence
+(12)
+0 → IB∩Y,F1(df) → OF1(df) → OB∩Y (df) → 0 .
+Since h0(OF1(df)) = d + 1 and h0(OB∩Y (df)) = d + 1, we have h1(IB∩Y,F1) > 0 if and
+only if h0(IB∩Y,F1(df)) > 0. This is equivalent to the fact that there exists a fiber L ∈ |f|
+such that #(B ∩ L) ≥ 2. Since L is a curve of bidegree (1, 0) in F, then L · Y = 0 in the
+intersection ring of F, therefore L ⊂ Y and hence we get L ∩ C ̸= ∅. Thus #(L ∩ A) ≥ 3,
+which means that A ̸∈ C∗(d + 2).
+□
+As said in Remark 2.5, for any element A ∈ C(2), there is a unique curve L of bidegree
+(1, 0) and a unique R of bidegree (0, 1) such that both intersect the elements of A at a
+point. As described in the following result, it turns out that A ∪ L ∪ R is the base locus
+of |IA(1, 1)|.
+Proposition 2.22. For any A ∈ C(2), we have that
+(1) the general element in |IA(1, 1)| is integral;
+(2) the base locus B of |IA(1, 1)| is A∪L∪R, where L is and R are the curves described
+in Remark 2.5.
+Proof. Since A ∈ C(2), by Corollary 2.20 we have h1(IA(1, 1)) = 0 and h0(IA(1, 1)) = 2.
+Call C1 and C2 the connected components of A. Denote by Xi the only element of |OF(1, 0)|
+containing Ci and by Yi the only element of |OF(0, 1)| containing Ci. The surfaces X1 ∪Y2
+and X2 ∪ Y1 are the only reducible elements of |IA(1, 1)| and hence, the general element
+in |IA(1, 1)| is irreducible and (1) is proved.
+To prove (2) we analyze the base locus B of |IA(1, 1)|. If S, S′ ∈ |IA(1, 1)| are irreducible
+and S ̸= S′, then the one-dimensional cycle S ∩ S′ has bidegree (3, 3) and it contains A,
+which has bidegree (2, 2). Take R := X1 ∩ X2 and L := Y1 ∩ Y2, where Xi and Yi are the
+surfaces defined in the first part of this proof. The curves L and R are exactly the ones
+stated in Remark 2.5. In particular, #(L ∩ A) = #(R ∩ A) = 2 and hence, by B´ezout and
+Remark 2.16, L∪R ⊂ B. Moreover, recall that the reducible surfaces, X1 ∪Y2 and X2 ∪Y1
+belong to |IA(1, 1)| and their intersection (X1 ∪ Y2) ∩ (X2 ∪ Y1) is A ∪ L ∪ R. Hence the
+base locus of |IA(1, 1)| is exactly B = A ∪ L ∪ R.
+□
+Remark 2.23. By generalizing the proof of Proposition 2.22 we can say something about
+the base locus of IA(1, d), for A ∈ C(n). Fix integers d > 0 and n ≥ 2 and take any
+A ∈ C(n). Then, the base locus B of IA(1, d) contains all curves L of bidegree (1, 0) such
+that #(L ∩ A) ≥ 2. If A ∈ C∗(n), then there are exactly
+�n
+2
+�
+such curves L (the number of
+lines joining two points in a set of n general points). If A ∈ T (n), then #(j(L) ∩ A) ≥ 2
+for all L such that #(L ∩ A) ≥ 2, since j(A) = A.
+3. Surfaces of bidegree (1, d)
+In this section we prove Theorems 1.1 and 1.2. In particular, we give several results for
+the case of a surface of bidegree (1, d). Later, in the following section, we will specialize
+to the cases d = 2, 3. The case d = 1 was studied in many details in [4] and here we add
+a simple lemma useful for what follows.
+Lemma 3.1. For any A ∈ T ∗(3), there is no integral surfaces of bidegree (1, 1) containing
+A.
+Proof. Assume the existence of an integral surface M of bidegree (1, 1) containing A ∈
+T ∗(3).
+First of all, thanks to [4, Corollary 8.4], we have that M = j(M).
+Then, as
+explained at the beginning of Section 7.1 in [4], either M is smooth or reducible. But if
+M is a smooth j-invariant surface of bidegree (1, 1) containing 3 twistor fibers, then it
+contains infinitely many of them and these are parametrized by a circle (see [4, Theorem
+7.2]). However, smooth surfaces of bidegree (1, 1) can be seen as the blow-up of P2 at three
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+11
+points both via π1 and π2. In particular, up to unitary transformation it is possible to
+write M as the set {([p0 : p1 : p2], [ℓ0 : ℓ1 : ℓ2]) ∈ F | p1ℓ1 + λp2ℓ2}, with λ ∈ R \ {0, 1}. In
+these coordinates, M contains π−1
+µ ([1 : 0 : 0]), π−1
+µ ([0 : 1 : 0]), π−1
+µ ([0 : 0 : 1]), for µ = 1, 2
+and, the family of twistor fibers π−1([q0 : q1 : q2]) defined by:
+�
+�
+�
+�
+�
+q0 = 0 and |q1|2λ + |q2|2 = 0
+if λ < 0 ,
+q1 = 0 and |q2|2 − |q0|2(λ − 1) = 0
+if λ > 1 ,
+q2 = 0 and |q1|2λ + |q0|2(λ − 1) = 0
+if 0 < λ < 1 .
+Take for instance λ < 0, then any twistor fiber in M intersects the line L = π−1
+2 ([1 : 0 : 0])
+of bidegree (1, 0). An analogous consideration holds if 0 < λ < 1 or λ > 1. Hence we get
+a contradiction.
+□
+In the previous lemma we show that an integral (1, 1) surface cannot contain three
+general twistor fibers.
+On the other hand if M is a (1, 1) surface containing a given
+A ∈ T (3) \ T ∗(3), then by [4, Corollary 8.3] we have that M is j-invariant, hence either it
+is smooth or reducible. Moreover, if M contains infinitely many twistor fibers, then all of
+them intersect a bidegree (1, 0) curve L and its associated (0, 1) curve R = j(L).
+Remark 3.2. In [4, Section 8.1] we gave examples of bidegree (1, 1) smooth surfaces
+containing exactly 0, 1 or 2 twistor fibers.
+Remark 3.3. As a smooth surface of bidegree (1, 1) is a Del Pezzo surface of degree 6,
+then this is characterized either by the three bidegree (1, 0) curves that contains or by the
+three bidegree (0, 1) curves that contains. In fact, recall that these surfaces represent the
+blow-up of P2 at three points with respect to either π1 or π2. We notice that if a smooth
+surface of bidegree (1, 1) is j-invariant, then, this is uniquely determined by three twistor
+fibers contained in it and not by the curves L and R = j(L) (of bidegree (1, 0) and (0, 1),
+respectively), which intersect all the twistor fibers. In fact, in [4], we proved that these
+kind of surfaces are uniquely determined by the “circle” defined by the infinite family of
+twistor fibers contained in it and, such a “circle” is properly contained in in the (1, 0) line
+L which intersects all fibers.
+Thanks to these first considerations about bidegree (1, 1) surfaces, we are ready to give
+the proof of our first main theorem.
+Proof of Theorem 1.1: Thanks to Remark 2.9 and Lemma 3.1, the result is true for d = 0
+and d = 1.
+Assume now that d ≥ 2 and, by contradiction, that S is an integral (1, d) surface
+containing A ∈ T ∗(d+2). Call C a connected component of A and set B := A\C. Call Y
+the only (by Lemma 3.2) element of |IC(0, 1)| and consider the following exact sequence
+(which is a particular case of the one in Formula (10)):
+(13)
+0 → IB(1, d − 1) → IA(1, d) → I(B∩Y )∪C,Y (1, d) → 0.
+From Formulæ (6) and (8), we have h0(OA(1, d)) = (d+2)2 = h0(OF(1, d))+1. Clearly,
+we have that B ∩ Y is formed by d + 1 points, and, up to the identification of Y with F1
+given in Formula (5), as the curve C corresponds to an element of type h+f in F1, we can
+write I(B∩Y )∪C,Y (1, d) ∼= I(B∩Y )∪C,F1(h + (d + 1)f) ∼=IB∩Y,F1(df). Since A ∈ T ∗(d + 2)
+and every element of |f| meets C (indeed (h + f)f = 1), the restriction to B ∩ Y of the
+ruling morphism D → P1 associated to |f| is injective. Thus, h1(Y, IA∩Y,Y (1, d)) = 0 and
+the exact sequence (13) gives h1(IB(1, d−1)) ≥ h1(IA(1, d)) ≥ 2, where the last is greater
+or equal than 2 because χ(IA(1, d)) = −1 (see Formula (7)), and we are assuming that
+h0(IA(1, d)) ≥ 1. Thus, we also have h0(IB(1, d − 1)) > 0.
+Recall that B ∈ T ∗(d+1) and hence, by the inductive assumption, B is not contained in
+any integral E ∈ |OF(1, d − 1)|. Therefore, thanks to Remarks 2.9 and 2.10, there must be
+
+12
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+an integral M ∈ |OF(1, 1)| containing at least 3 connected components of B, say B′ ⊂ M
+with B′ of bidegree (3, 3). Hence, by Lemma 3.1, there is a curve L of bidegree (1, 0) such
+that #(L ∩ B′) = 3. Thus A /∈ T ∗(d + 2), a contradiction.
+□
+Having proved that an integral bidegree (1, d) surface cannot contains d + 2 (or more)
+non collinear twistor fibers, we now pass to prove that all the other cases can actually
+arise. In particular, we prove in the following result a stronger version of Theorem 1.2 in
+the case n ≤ d + 1.
+Theorem 3.4. Fix integers d ≥ 1 and 0 ≤ n ≤ d + 1. Then, for any A ∈ T ∗(n) there is
+an integral S ∈ |OF(1, d)| containing A. Moreover, the general S ∈ |IA(1, d)| contains no
+other twistor fibers.
+Proof. We use induction on the integer d. If d = 1 the statement is true by Remark 3.2.
+Assume now d ≥ 2 and take an element A ∈ T ∗(n). Since T ∗(n) is Zariski dense in
+C(n), A has the bigraded Hilbert function of a general element of C(n). Thus, thanks to
+Corollary 2.20, we have that h1(IA(1, d)) = 0 and, by Formula (7)
+h0(IA(1, d)) = (d + 1)(d + 3) − n(d + 2) =: Nn + 1.
+Fix a connected component C of A and set B := A \ C.
+If Y denotes be the only
+(0, 1) surface containing C, Corollary 2.20 entails that h1(IB(1, d − 1)) = 0 and, again by
+Formula 7
+h0(IB(1, d − 1)) = d(d + 2) − (n − 1)(d + 1).
+By the inductive assumption, we know that |IB(1, d − 1)| ̸= ∅ and a general W ∈
+|IB(1, d − 1)| is irreducible.
+Thus Y ∪ W ∈ |IA(1, d)| and Y ∪ W has 2 irreducible
+components, one of them having bidegree (1, d − 1).
+Let us denote by C1, . . . , Cn the connected components of A, by Bi := A \ Ci and by Yi
+the unique element in |ICi(0, 1)|. The set of all the reducible surfaces W ∪ Yi ∈ |IA(1, d)|,
+where W ∈ |IBi(1, d − 1)|, is the union of e projective spaces (one for each choice of
+Ci), each of them of codimension h0(IA(1, d)) − h0(IB(1, d − 1)) = d + 2 − n > 0 in
+|IA(1, d)|= PNn (in particular of codimension 1 if n = d+1). Therefore, they do not cover
+all |IA(1, d)|.
+We now want to exclude other possible splittings. In particular, we consider reducible
+surfaces of the form W1 ∪ D1 with W1 integral, D1 possible reducible of bidegree (0, x) for
+some x ≥ 2 and hence W1 of bidegree (1, d − x). Remark 2.10 shows that only irreducible
+components of D1 of bidegree (0, 1) may contain some component of A. We obtain that
+the surface is of the form W1 ∪D2 ∪D3 with W1 ∪D2 of bidegree (1, d−1), but, as showed
+before, these kind of surfaces do not cover all |IA(1, d)|, hence we have the thesis.
+Now we prove that a general S ∈ |IA(1, d)| contains no other twistor fibers. We start
+by analyzing the case d = 2 and discussing the cases n = 1, 2, 3 separately.
+Assume n = 1. Fix C ∈ T (1). Let CC(1) denote the set of all B ∈ C(1) such that B∩C =
+∅. Note that T (1) \ {C} = CC(1) ∩ T (1). For any B ∈ CC(1) we have h1(IC∪B(1, 2)) = 0
+and hence h0(IC∪B(1, 2)) = h0(IC(1, 2)) − 4. Let XC be the set of all smooth and integral
+surfaces of bidegree (1, 2) containing C. It is a non-empty Zariski open subset of |IC(1, 2)|.
+For any B ∈ CC(1) the set of all S ∈ XC containing B has complex codimension 4 and
+hence real codimension 8 as real manifolds. Since T (1) has real dimension 4, a general
+S ∈ XC contains no other twistor fiber.
+Let now n = 2. Fix A ∈ T (2) and let CA(1) denote the set of all B ∈ C(1) such that
+B∩A = ∅ and B∪A ∈ C∗(3). Note that h1(IA∪B(1, 2)) = 0. So as in the previous step, we
+get that a sufficiently general S ∈ |IA(1, 2)| contains no element of TA(1). Assume that S
+contains B ∈ T (1) such that there is a curve of bidegree (1, 0) intersecting each connected
+component of A ∪ B. Note that L is uniquely determined by A. Let C(A, L) denote the
+set of all B ∈ C(1) such that B ∩ A = ∅ and L meets B and set T (A, L) := C(A, L) ∩ T (1).
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+13
+For any o ∈ L \ (L ∩ C) the set of all B ∈ C(A, L) containing o is a non-empty family of
+complex dimension 1, while there is a unique twistor fiber containing o. The set C(A, L) is
+a complex manifold of dimension 2, while h1(IA∪B(1, 2)) = 1 and hence h0(IA∪B(1, 2)) =
+h0(IA(1, 2)) − 3. Since dim C(A, L) = 2, a general S ∈ |IA(1, 2)| contains no element of
+C(A, L). Hence, there is no twistor fiber B such that A ∪ B ∈ T ∗(3).
+Assume now that n = 3. Start by considering a general A ∈ T ∗(3). Then we have
+h1(IA(1, 2)) = 0 and h0(IA(1, 2)) = 3. For any x ∈ {0, 1, 2, 3} let C(A, x) denote the set of
+all C ∈ C(1) such that A∩C = ∅ and h0(IA∪C(1, 2)) = x and set T (A, x) := C(A, x)∩T (1).
+For a general D ∈ C(4) we have h0(ID(1, 1)) = 0 (but h1(ID(1, 1)) = 1). Fix C ∈ C(1)
+such that C ∩ A = ∅, call Y the only element of |IC(0, 1)|. Since C ∩ A = ∅, no connected
+component of A is contained in Y and Y ∩ A is formed by 3 points, all of them in Y \ C.
+We easily deal with the case x = 0 as any curve C ∈ C(A, 0) is not contained in any
+element of |IA(1, 2)|. We have OY (1, 2)(−C) ∼= OF1(2f) and hence C ∈ C(A, 0) if no curve
+of bidegree (1, 0) L ∈ |f| intersects 2 of the components of A (since h1(IA(1, 1)) = 1 the
+last statement is only an “if” and not an “if and only if”). A necessary condition for
+being C ∈ C(A, 2) is that L intersects all connected components of A, but this is excluded
+because A ∈ T ∗(3). Since A ∈ T ∗(3) there are exactly 3 curves L1, L2, L3 of bidegree (1, 0)
+intersecting 2 of the connected components of A. We claim that a general S ∈ |IA(1, 2)|
+contains no C ∈ C(1) such that C ∩ A = ∅. The family of smooth conics which intersect
+Li has complex dimension 2, while the family of twistor fibers intersecting Li has real
+dimension 2. As the general S ∈ |IA(1, 2)| has only finitely many conics, it only has a
+finite number of elements in C(A, 1) and, for the general, none of them is a twistor fiber.
+We now pass to analyze the case d ≥ 3. Assume the general surface of |IA(1, d)| contains
+the twistor fiber C ⊈ A. Thus A∩C = ∅. Set A′ := A∪C. Take Y ∈ |OY (0, 1)| containing
+C and consider the residual exact sequence
+(14)
+0 → IA(1, d − 1) → IA′(1, d) → I(Y ∩A)∪C,Y (1, d) → 0.
+We have IC,Y (1, d) ∼= OF1(df).
+Assume first n ≤ d. If A′ ∈ T ∗(n + 1), then h1(I′
+A(1, d)) = 0 and hence h0(I′
+A(1, d)) =
+h0(IA(1, d)) − d − 2. Since dim C(1) = 4, for d ≥ 3 the general S ∈ |IA(1, d)| contains no
+C such that A ∪ C ∈ T ∗(n + 1).
+Now assume A′ /∈ T ∗(n+1). Thus there are connected components C′ and C′′ of A such
+that C ∪ C′ ∪ C′′ /∈ T ∗(3), i.e. C intersects the unique line L meeting C′ and C′′. Since
+dim L = 1, to exclude this case it is sufficient to prove that h0(IB(1, d)) ≤ h0(IA(1, d))−2,
+i.e.
+h0(F1, IA∩Y (df)) ≤ d.
+But since h0(OF1(df)) = d + 1, then OF1(df) is globally
+generated and A ∩ Y ̸= ∅, therefore h0(F1, IA∩Y (df)) ≤ d.
+Assume now n = d+1 and that A′ ∈ C∗(d+2). By Lemma 2.21 we have h1(IA′(1, d)) ≤ 1
+and hence h0(IA′(1, d)) ≤ h0(IA(1, d)) − d − 1. Now assume A′ /∈ C∗(d + 2). We need
+h0(F1, IA∩Y (df)) ≤ d − 1. Let π : F1 → P1 denote the ruling of F1. Since OP1(d) is very
+ample, h0(F1, IA∩Y (df)) ≤ d − 1 if and only if #π(A ∩ Y ) ≥ 2, which is true because
+#(A∩Y ) = d+1 ≥ 3 and (since A ∈ C∗(d+1) no fiber F of π contains at least 3 points of
+A). The only remaining case is now d = 3 and n = 4, which can be dealt just by adapting
+the previous argument for d = 2 and n = 3.
+□
+Having proved Theorem 1.2 in the case n ≤ d + 1 for non collinear twistor fibers, we
+focus now to the case n = d + 2. In this setting we need some preliminary results.
+Lemma 3.5. Fix d > 0, n ≤ d+2 and consider a general A ∈ T (n). Then h1(IA(1, d)) =
+0.
+Proof. Since T (n) is Zariski dense in C(n), it is sufficient to prove the statement for a
+general A ∈ C(n). Clearly, it is sufficient to prove the case n = d + 2. Take a connected
+component C of A and set B := A \ C and Y ∈ |IC(0, 1)|. Consider the residual exact
+
+14
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+sequence of Y , as in Formula (13). Recall the correspondence Y ∼= F1 given in Formula (5)
+and let ρ : Y → P1 denote its ruling. Since A is general #ρ(A ∩ Y ) = d + 1 and hence
+hi(F1, IB∩Y (df)) = 0, for i = 0, 1. Therefore h1(IA(1, d)) = 0.
+□
+By Theorem 1.1, we already know that an integral (1, d) surface may contain a union of
+d + 2 twistor fibers only if it belongs to T (d + 2) \ T ∗(d + 2). Therefore, we introduce the
+following notation for sets of disjoint smooth conics which are collinear. Given a curve L
+of bidegree (1, 0) and an integer n > 0, let C(n, L) denote the set of all A ∈ C(n) such that
+each connected component of A meets L. The set C(n, L) is isomorphic (as real algebraic
+variety) to the set S(L, n) of all subsets of L with cardinality n and hence it is irreducible.
+An analogous definition and observation can be done for a curve R of bidegree (0, 1) and,
+of course, for the family T instead of C.
+Lemma 3.6. Fix integers n ≥ 3 and d ≥ 1 and take any A ∈ T (n, L). Then,
+h1(IA(1, d)) ≥ n − 2 + max{0, n − (d + 1)}.
+Proof. Call C1, . . . , Cn the connected component of A. Let Si ⊂ Ci be any union of d + 2
+distinct points on each conic and S := S1 ∪ · · · ∪ Sn. Since Ci is a smooth rational curve,
+the restriction map H0(OCi(1, d)) → H0(OSi(1, d)) is bijective. Thus, the restriction map
+H0(OA(1, d)) → H0(OS(1, d)) is bijective and χ(OA(1, d)) = χ(OS(1, d)). Thus we get
+χ(IA(1, d)) = χ(IS(1, d)).
+Since A ∈ T (n, L), we know that there exists L of bidegree (1, 0) which intersects each
+conic in A (and the (0, 1) curve j(L) does the same). Hence we can choose S such that n
+points are on L and n points are on j(L). In other words we assume (A∩L)∪(A∩j(L)) ⊆ S.
+By using B´ezout and the fact that the bidegree is (1, d), we get that (n − 2) + max{0, n −
+(d + 1)} of these points may be omitted without changing the set |IS(1, d)|, and hence
+H0(IS(1, d)). It follows that h1(IA(1, d)) ≥ n − 2 + max{0, n − (d + 1)}.
+□
+Remark 3.7. Thanks to the previous lemma, if A ∈ T (3, L), for some bidegree (1, 0)
+curve L, then h1(IA(1, 1)) ≥ 2, and hence h0(IA(1, 1)) ≥ 1. However, since no surface of
+bidegree (1, 0) or (0, 1) contains an element of T (2), then every S ∈ |IA(1, 1)| is irreducible.
+Therefore, Proposition 2.17 gives that |IA(1, 1)| = {S} and so, thanks to Formula (7),
+h1(IA(1, 1)) = 2.
+Finally, the following result completes the proof of Theorem 1.2, in the case of d + 2
+twistor fibers.
+Theorem 3.8. Fix an integer d ≥ 2 and take a general A ∈ T (d+2, L). Then h0(IA(1, d)) ≥
+d and the general S ∈ |IA(1, d)| is integral.
+Proof. By applying Lemma 3.6 with n = d + 2, we get h1(IA(1, d)) ≥ d + 1.
+Since
+χ(IA(1, d)) = (d+1)(d+3)−(d+2)2 = −1, we get h0(IA(1, d)) ≥ d and hence, |IA(1, d)| ̸=
+∅.
+We now prove that a general element in |IA(1, d)| is integral.
+Take S ∈ |IA(1, d)|.
+Every surface of bidegree (1, 0) or (0, 1) contains at most one connected component of A.
+Therefore, S cannot be the union of a surface of bidegree (1, 0) and d of bidegree (0, 1).
+No integral surface of bidegree (0, x), for x ≥ 2, contains a twistor fiber. If d = 2, then
+h0(IA(1, 2)) ≥ 2. However, thanks to Remark 3.7, for every choice of C ∈ A, there is only
+one M ∈ |I(A\C)(1, 1)|. Since we have considered all the possible reducible elements of
+|IA(1, 2)|, we get the thesis.
+Now assume d > 2. Since A has finitely many components, it is sufficient to prove that
+for any x ∈ {3, . . . , d−1}, any union E of x connected components of A and any connected
+component C of A \ E we have
+(15)
+h0(IE(1, x − 2)) < h0(IE∪C(1, x − 1)),
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+15
+and then proceed as in the proof of Theorem 3.4.
+Let Y be the only element of |OF(0, 1)| containing C. The exact sequence in Formula (10)
+gives h0(IE(1, x−2)) ≤ h0(IE∪C(1, x−1)) and equality holds if and only if Y is in the base
+locus B of |IE∪C(1, x − 1)|. Call C1, . . . , Cx the components of E and Yi the only surface
+of bidegree (0, 1) containing Ci. By Remark 2.13 the irreducible surfaces Y, Y1, . . . , Yx are
+all different one eachother. For a general A we get that each integer h0(IE(1, x−2)) is the
+same for all union of x connected components of A. Thus if the inequality is false, then B
+contains the surface Y ∪ Y1 ∪ · · · ∪ Yx of bidegree (0, x + 1), which is a contradiction.
+□
+4. Surfaces of bidegree (1, 2) and (1, 3)
+In this section we specialize our study to the case of surfaces of bidegree (1, 2) and (1, 3).
+In particular, we will prove Theorems 1.3, 1.4 and 1.5.
+Recall, from Formula (7), that for any A ∈ C(n), we have χ(IA(1, 2)) = 15 − 4n, and
+hence, if n ≤ 3, we get h0(IA(1, 2)) > 0. We also recall that a general S ∈ |OF(1, 2)|
+contains finitely many smooth conics and, thanks to Theorem 2.19, for every B ∈ C(2) we
+have h1(IB(1, 1)) = 0.
+4.1. Surfaces of bidegree (1, 2) containing 0 ≤ n ≤ 4 twistor fibers. In this section,
+we show the existence of a smooth surface of bidegree (1, 2) containing exactly 0, 1, 2, 3 or
+4 twistor fibers. In order to analyze the space |IA(1, 2)| when A is in C(n) (or in T (n)),
+for 0 ≤ n ≤ 4, we will need some preliminary results. Note that the extremal case, when
+n = 4, will be treated in a different way.
+We start by considering (1, 2)-surfaces containing three disjoint smooth conics.
+Proposition 4.1. Take A ∈ C(3) such that h0(IA(1, 1)) > 0. Then
+(1) there exists a curve L of bidegree (1, 0) and a curve R of bidegree (0, 1) such that
+A ∈ C(3, L) and A ∈ C(3, R)
+(2) there is an integral element in |IA(1, 2)|;
+(3) h1(IA(1, 2)) = 1;
+(4) the base locus B of |IA(1, 2)| is A ∪ L ∪ R, where L is and R are the curves defined
+in (1).
+Proof. We start arguing as in Remark 3.7. Thanks to Remark 2.9, each surface of bide-
+gree (1, 0) or (0, 1) does not contain any element of C(2), hence, as A ∈ C(3), any ele-
+ment in |IA(1, 1)| is irreducible. Proposition 2.17 (with a = b = c = d = 1) gives that
+h0(IA(1, 1)) = 1 and hence we set |IA(1, 1)| = {M} and by Formula (7) we compute
+h1(IA(1, 1)) = 2.
+We prove now the first statement. Let C be a connected component of A, set B := A\C
+and denote by X the only element of |IC(1, 0)|. Performing the same construction that
+leads to Formula (10), we have
+0 → IB(0, 1) → IA(1, 1) → IA∩X,X(1, 1) → 0.
+Since h0(IB(0, 1)) = 0 and h0(IA(1, 1)) = 1, the previous residual exact sequence gives
+(16)
+h0(IA∩X,X(1, 1)) ≥ 1.
+Thanks to Formula (4), we have that OX(1, 1) ≃ OF1(h + 2f); moreover, recall from
+Remark 2.9 that C is identified with an element of |OF1(h + f)|. Since A ∩ X is the union
+of C and the two points B ∩X, there is a fiber L ∈ |f| of the ruling of F1 containing B ∩X.
+Since f(h + f) = 1, we have that L meets C. Thus L meets each connected component of
+A. Taking instead of X the only element of |IC(0, 1)| we get the existence of R.
+We now prove (2).
+We will show that there is an integral element in |IA(1, 2)| by
+showing that the possible reducible cases do not cover the whole family. Remark 2.10
+shows that A is not contained in a surface of bidegree (1, 2) with an irreducible component
+
+16
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+of bidegree (0, 2). By Remark 2.9, a bidegree (0, 1) or (0, 1) surface does not contain any
+element of C(n), with n ≥ 2. Hence, there are only finitely many elements of |OF(1, 2)|
+with at least 3 irreducible components.
+Since h0(OF(0, 1)) = 3, the set of all reducible elements of |OF(1, 2)| with an irreducible
+component of bidegree (1, 1) containing A is isomorphic to P2. Thus, in order to prove the
+existence of an integral element in |IA(1, 2)|, it is sufficient to prove that h0(IA(1, 2)) ≥ 4.
+But, using the exact sequence (9), this is equivalent to prove that h1(IA(1, 2)) ≥ 1, and
+the last inequality is true, by Theorem 2.19 because #(L ∩ A) = 3.
+To prove (3), i.e. h1(IA(1, 2)) = 1, it is sufficient to prove that h1(IA(1, 2)) ≤ 1. As
+before, take a connected component C of A and set B = A\C. Let Y be the only element
+of |IC(0, 1)|. In the identification (5) of Y with F1 we have
+I(B∩Y )∪C,Y (1, 2) ∼= I(B∩Y )∪C,F1(h + 3f) ∼= O(B∩Y ),F1(2f).
+Since #(B ∩ Y ) = 2 and OF1(2f) is globally generated, h1(F1, IB∩Y,F1(2f)) ≤ 1. We have
+ResY (A) = B. Thanks to Theorem 2.19, we have h1(IB(1, 1)) = 0, and the residual exact
+sequence of Y
+0 → IB(1, 1) → IA(1, 2) → I(B∩Y )∪C,Y (1, 2) → 0,
+gives h1(IA(1, 2)) ≤ 1.
+Finally, we discuss the base locus of |IA(1, 2)| in order to prove (4). First of all, for any
+surface S ∈ |IA(1, 2)|, we clearly have A ⊂ S. Moreover, as #(L∩A) = 3 and #(R∩A) = 3,
+then by B´ezout, both curves are contained in S: in fact, thanks to Remark 2.3 and
+Formula (3), the general intersection between a curve of bidegree (1, 0) and S consists of
+one point while the intersection of a curve of bidegree (0, 1) and S consists of two points
+(see also Remark 2.16). Therefore, L ∪ R ⊂ S and A ∪ L ∪ R ⊂ B.
+We now prove that B ⊂ A∪L∪R. Fix p ∈ B\(A∪L∪R). Take a connected component Ci,
+i = 1, 2, 3, of A and set Bi := A\Ci. Let Yi be the only element of |OF(0, 1)| containing Ci.
+By Proposition 2.22 Bi ∪L∪R is the base locus of |IBi(1, 1)|. Thus there is Si ∈ |IBi(1, 1)|
+such that p /∈ Si. If p /∈ Yi, then p /∈ B. Since S1∩S2∩S3 = L∪R, we may take i ∈ {1, 2, 3}
+such that p /∈ Yi. Thus B = A ∪ L ∪ R.
+□
+The following remark shows that if A ∈ C(3) satisfies the condition (1) of Theorem 4.1,
+then the existence of a (1, 1)-surface containing A is granted. In particular, there exists a
+(1, 1)-surface containing any triplets of collinear twistor fibers.
+Remark 4.2. Take A ∈ C(3) and assume the existence of curves L of bidegree (1, 0) and
+R of bidegree (0, 1) intersecting each connected component of A. By adapting the proof of
+Lemma 3.6, since #(L ∩ A) = 3 and #(R ∩ A) = 3, we have that h1(IA(1, 1)) ≥ 2. Thus
+h0(IA(1, 1)) ≥ 1 and A satisfies the assumptions of Proposition 4.1.
+We can even be more specific and say that if A ∈ C(3) (with no assumption on L or R),
+then h0(IA(1, 1)) ≤ 1 and if |IA(1, 1)| ̸= ∅, then the only element of |IA(1, 1)| is integral.
+This is true because, thanks to Remark 2.9, any reducible element of |OF(1, 1)| contains
+at most 2 disjoint smooth conics.
+Note that if A ∈ T (3) and L exists, then we may take R := j(L). Thus if A ∈ T (3) to
+get h0(IA(1, 1)) > 0 it is sufficient to assume A /∈ T ∗(3).
+The following lemma is a sort of vice versa of the previous remark.
+Lemma 4.3. Take A ∈ C∗(3). Then h0(IA(1, 1)) = 0 and h1(IA(1, 1)) = 1.
+Proof. If A ∈ C(3), thanks to Formula (7), χ(IA(1, 1)) = −1. Hence h0(IA(1, 1)) = 0 if
+and only if h1(IA(1, 1)) = 1. We assume h0(IA(1, 1)) ̸= 0 and will prove that A /∈ C∗(3).
+Let B ⊂ A be the union of 2 connected components of A and set C := A \ B. Let L and
+R be the curves defined in Remark 2.5 for B ∈ C(2). Take any element D ∈ |IB(1, 1)|.
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+17
+Since #(B ∩ (L ∪ R)) = 2, B ⊂ D and D has bidegree (1, 1), then B´ezout theorem implies
+L ∪ R ⊂ D.
+By Theorem 2.19 and Proposition 2.22, we have h1(IB(1, 1)) = 0, h0(IB(1, 1)) = 2, and
+the general element M in |IB(1, 1)| is integral. Since h0(IB(1, 1)) = 2 and M is general,
+C ⊈ M. Consider the following residual exact sequence:
+(17)
+0 → IC → IA(1, 1) → IB∪(M∩C),M(1, 1) → 0
+Since M ∈ |IB(1, 1)| and h1(OF) = 0, the exact sequence
+0 → OF → IB(1, 1) → IB,M(1, 1) → 0
+gives h0(M, IB,M(1, 1)) = 1. Moreover, h0(IC) = 0, and so the sequence (17) and the
+assumption h0(IA(1, 1)) ≥ 1 imply h0(M, IB∪(M∩C),M(1, 1)) ≥ 1. By Proposition 2.22, the
+curve A∪L∪R is the base locus of |IB(1, 1)| and hence the base locus of H0(M, IB,M(1, 1))
+is the curve B ∪ L ∪ R. Since B ∩ C = ∅, the degree 2 scheme C ∩ M is contained in
+L ∪ R. To get A /∈ C∗(3) we need to prove that C ∩ L ̸= ∅. It is sufficient to observe that
+deg(C ∩ T) ≤ 1 for any curve T of bidegree (0, 1). Indeed, this is true by Remark 2.3 and
+the fact that C is the intersection of a surface of bidegree (1, 0) and a surface of bidegree
+(0, 1).
+□
+We now discuss the case of A ∈ C(2) contained in a smooth bidegree (1, 1) surface. In
+this case we will also prove smoothness for the general element in |IA(1, 2)|.
+Proposition 4.4. Take any A ∈ C(2) contained in a smooth element of |OF(1, 1)|. Then
+we have:
+(1) h1(IA(1, 2)) = 0 and h0(IA(1, 2)) = 7;
+(2) the set A ∪ L is contained in the base locus B of |IA(1, 2)|, where L is the bidegree
+(1, 0) curve described in Remark 2.5;
+(3) a general S ∈ |IA(1, 2)| is smooth.
+Proof. To prove (1) it is sufficient to apply Corollary 2.20 giving h1(IA(1, 2)) = 0 and
+Formula (7), which entails h0(IA(1, 2)) = 7.
+We now pass to point (2). Take L and R as in Remark 2.5. Since OF(0, 1) is globally
+generated, B ⊆ A ∪ L ∪ R; moreover, thanks to Remark 2.16 we also have A ∪ L ⊆ B.
+We are left to prove (3).
+By Bertini’s theorem Sing(S) ⊆ A ∪ L ∪ R for a general
+S ∈ |IA(1, 2)|. Fix a smooth M ∈ |IA(1, 1)|. Take a general Y ′ ∈ |OF(0, 1)|. Since Y ′
+is general L ∩ Y ′ = ∅ (and hence it is not singular at any p ∈ L). Thus, up to small
+deformation, we can say that S (which is general) is smooth in a neighborhood of L. We
+are left to exclude the case Sing(S) ⊆ A∪R. Fix p ∈ A∪R and let 2p be the 0-dimensional
+scheme of F defined by the ideal I2
+p,F. S is singular at p if and only if S ∈ |I2p∪A(1, 2)|.
+To conclude our proof we need to prove that
+h0(I2p∪A(1, 2)) = h0(IA(1, 2)) − 2,
+for all p ∈ (A ∪ R) \ A ∩ R and that, for p ∈ A ∩ R, h0(I2p∪A(1, 2)) < h0(IA(1, 2)). These
+two statements give the thesis because (A ∪ R) \ A ∩ R and A ∩ R are 1-dimensional and
+0-dimensional, respectively, and we are saying that the set of bidegree (1, 2) surfaces con-
+taining A and a singular points has codimension 2 in the first case and positive codimension
+in the second one.
+Let us start by taking p ∈ (A ∪ R) \ A ∩ R.
+Since p is a smooth point of A ∪ R,
+deg(2p ∩ (A ∪ R)) = 2. Consider the exact sequence
+(18)
+0 → I(A∪R)∪2p(1, 2) → IA∪R(1, 2) → IA∪R ⊗ O2p(1, 2) → 0.
+
+18
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+Since deg(2p) = 4 and A is smooth, we have h0(IA∪R ⊗ O2p(1, 2)) = 2 if p ∈ A ∪ R and
+h0(IA ⊗ O2p(1, 2)) = 4 if p ∈ R. Hence it is sufficient to prove that
+h1(I(A∪R)∪2p(1, 2)) = 0.
+First of all, assume that p ∈ A \ R. Let C be the connected component of A containing
+p. Set the following notation E := A \ C. As R is in the base locus of IA(1, 1) we have
+that h0(IA(1, 1)) = h0(IA∪R(1, 1)) (see [3, proof of Theorem 1.1]). Moreover, thanks to
+part (1) and to [3, Remark 4.3], we have h0(IA∪R(1, 1)) = h0(IA(1, 1)) = h0(IE(1, 1)) − 3.
+Thus p is not in the base locus of |IE(1, 1)|. Fix M ∈ |IE(1, 1)| such that p /∈ S. Let
+Y be the surface of |OF(0, 1)| containing C and consider the residual exact sequence with
+respect to Y :
+(19)
+0 → IE∪p(1, 1) → IA∪2p(1, 2) → I(E∩Y )∪C∪(2p∩Y ),Y (1, 2) → 0.
+Now we prove that
+(20)
+h1(IE∪p(1, 1)) = 0.
+Recall that A = E ∪ C and p ∈ C, hence we have the exact sequence
+0 → IA(1, 1) → IE∪p(1, 1) → Ip,C(2) → 0.
+Thanks to Theorem 2.19 we have that h1(IA(1, 1)) = 0; on the other hand, since C is a
+smooth rational curve, we have h1(Ip,C(2)) = h1(OC(1)) = 0 and this proves (20).
+In order to conclude it is sufficient to prove now that
+(21)
+h1(I(E∩Y )∪C∪(2p∩Y ),Y (1, 2)) = 0.
+Note that IC,Y (1, 2) ∼= OF1(2f) ∼= OY (0, 1), hence, by [3, Remark 2.11], we know that
+IC,Y (1, 2) is very ample. Therefore we get h1(IC∪(2p∩Y ),Y (1, 2)) = 0. Since E ∩ Y consists
+of a point we conclude that (21) holds.
+Thus the exact sequence (19) gives h1(IA∪2p,F(1, 2)) = 0, concluding the proof in the
+case p ∈ A \ R.
+Fix p ∈ R \ (A ∩ R) and ecall that we need to prove that h0(IA∪2p(1, 2)) = 5. Fix
+a general Y ′ ∈ |Ip(0, 1)|. Since Y ′ is general, R ⊈ Y ′ (and also L ⊈ Y ′). Since Y ′ is
+smooth, Y ′ ∩ 2p = (2p, Y ′) is a degree 3 scheme and ResY ′(2p) = {p}. As p ∈ R, we have
+that h0(IA∪{p}(1, 1)) = h0(IA(1, 1)) = 2. Thus, by the residual exact sequence of Y ′ it is
+sufficient to prove that
+h0(Y ′, I(A∩Y ′)∪(2p,Y ′)(1, 2)) ≤ 3.
+Since OY ′(1, 2) is very ample, we have h0(Y ′, I(2p,Y ′)(1, 2)) = h0(Y ′, OY ′(1, 2)) − 3 =
+4. Thus it is sufficient to prove that A ∩ Y ′ is not contained in the base locus, B′, of
+|O(2p,Y ′)(1, 2)|. In the identification between Y ′ and F1 we have OY ′(1, 2) ∼= OF1(h + 3f).
+Let N be the only element of |f| containing p. We have N ∼= P1 and h1(N, I2p∩N(1, 2)) = 0,
+but N ⊆ B′. Since OF1(h + 2f) is very ample, Ip(h + 2f) has only p in its base locus.
+Thus B′ = N and so, since R ⊈ Y ′ (and also L ⊈ Y ′), B′ cannot contain both points of
+A ∩ Y ′.
+The last case is p ∈ A ∩ R. To prove our claim, i.e. that h0(I2p∪A(1, 2)) < h0(IA(1, 2)),
+it is sufficient to use (18). Hence, a general S is smooth.
+□
+The following result is analogous to Proposition 4.1, when we choose the conics to be
+twistor fibers.
+Proposition 4.5. Take A ∈ T (3) such that h0(IA(1, 1)) = 0. Then we have the following:
+(1) h1(IA(1, 2)) = 0 and hence h0(IA(1, 2)) = 3;
+(2) there is an integral S ∈ |IA(1, 2)|;
+(3) the base locus of |IA(1, 2)| is contained in the union of A and 3 distinct curves of
+bidegree (1, 0);
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+19
+(4) for a sufficiently general A (contained in a dense euclidean open subset of T (3)),
+we may take a smooth S ∈ |IA(1, 2)|.
+Thanks to Lemma 3.1, the hypothesis A ∈ T (3) such that h0(IA(1, 1)) = 0 in the
+previous statement, implies that the conics in A do not belong to any infinite family of
+twistor fibers contained in a smooth j-invariant surface of bidegree (1, 1).
+Proof. We start by proving (1).
+Fix a connected component C of A and call D the
+only element of |IC(0, 1)|.
+Set B := A \ C.
+To get h1(IA(1, 2)) = 0 mimicking the
+proof of Proposition 4.1 it is sufficient to prove that h1(F1, IB∩D(2f)) = 0.
+Assume
+h1(F1, IB∩D(2f)) > 0, i.e. assume the existence of T ∈ |OF1(f)| containing the 2 points
+B ∩ D. Since C ∈ |OF1(h + f)|, C ∩ T ̸= ∅. Thus T meets each connected component of
+A. Remark 4.2 gives h0(IA(1, 1)) > 0, a contradiction.
+To prove (2) it is sufficient to show that the reducible cases do not cover the whole
+|IA(1, 2)|. In fact, reasoning as in the proof of Proposition 4.1, the only possible splitting
+are of the form (1, 0) + (0, 1) + (0, 1), which are in a finite number, or (1, 1) + (0, 1), where
+the bidegree (1, 1) component contains 2 connected components of A and the remainder
+bidegree (0, 1) part is uniquely determined.
+Now, h0(IB(1, 1)) = 2, so, the set of all
+reducible elements of |IA(1, 2)| with an irreducible component of bidegree (1, 1) does not
+cover |IA(1, 2)|.
+We now prove (3) and (4). Since h0(IA(1, 1)) = 0 and A is j-invariant, neither π1(A) nor
+π2(A) has a triple points (both have 3 double points). Set L1∪L2∪L3 := π−1
+2 (Sing(π2(A)))
+and R1 ∪ R2 ∪ R3 := π−1
+1 (Sing(π1(A))). Since #(Li ∩ A) = #(Ri ∩ A) = 2, L1 ∪ L2 ∪ L3
+are in the base locus of |IA(1, 2)| and each Li and each Ri meets exactly 2 connected
+components of A.
+To prove the existence of a smooth element, it is sufficient to reason as in the proof of
+Proposition 4.4 case (3).
+□
+We are now ready to prove the first part of Theorem 1.3.
+Theorem 4.6. Fix n ∈ {0, 1, 2, 3}. There is a smooth S ∈ |OF(1, 2)| containing exactly n
+twistor fibers.
+Proof. A general S ∈ |OF(1, 2)| contains only finitely many smooth conics. Since the set
+of all twistor fibers has real codimension 4 in the space of all smooth conics, a general
+S ∈ |OF(1, 2)| contains no twistor fiber.
+Now we prove the case n = 1. Fix a twistor fiber C and take a general S ∈ |IC(1, 2)|. As-
+sume that S contains another twistor fiber, E. We have h1(IC(1, 2)) = h1(IC∪E(1, 2)) = 0
+(Theorem 2.19 and Remark 2.14). Thus |IC∪E(1, 2)| is a 4-codimensional complex projec-
+tive subspace of |IC(1, 2)| (this is explained by the equality h0(IC∪E(1, 2)) = h0(IC(1, 2))−
+4 contained in [3, proof of Theorem 1.1]). However T (1) is a real 4-dimensional space. So
+a general S ∈ |IC(1, 2)| contains no other twistor fiber.
+Note that C is the base locus of |IC(1, 2)|. By Bertini theorem a general S ∈ |IC(1, 2)|
+is smooth outside C. Fix p ∈ C and let 2p the closed subscheme of F with (Ip)2 as its
+ideal sheaf. Recall that 2p ⊂ S if and only if p ∈ Sing(S). Since dim C = 1 to get that S
+is smooth it is sufficient to prove that h0(I2p∪C(1, 2)) ≤ h0(IC(1, 2)) − 2 = 9. This follows
+from the proof of Proposition 4.4 case (3).
+The case n = 2 is true by Proposition 4.4 with T (2) instead of C(2).
+The case n = 3 is true by Proposition 4.5.
+□
+In the remainder of the section, we will construct a smooth (1, 2)-surface containing 4
+twistor fibers. The following lemma, in the case d = 2, says that if an integral (1, 2)-surface
+contains 4 disjoint smooth conics, then these conics are not general, because three of them
+must be collinear.
+
+20
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+Lemma 4.7. Let d ≥ 2 and A ∈ C(d + 2). If there is an integral S ∈ |OF(1, d)|, then
+A /∈ C∗(d + 2)
+Proof. We prove the lemma by induction on d. We start with the case d = 2. Assume
+that A ∈ C∗(4), i.e. there is no union B of 3 of the connected components of A such that
+#(L ∩ B) = 3 for some curve L of bidegree (1, 0). Fix a connected component C of A and
+set B := A \ C. Call Y the only element of |IC(0, 1)|. Remark 2.9 gives ResY (A) = B. By
+assumption and Lemma 4.3, h0(IB(1, 1)) = 0. Since h0(IA(1, 2)) ̸= 0, the residual exact
+sequence
+0 → IB(1, 1) → IA(1, 2) → IA∩Y,Y (1, 2) → 0 ,
+gives h0(Y, IA∩Y,Y (1, 2)) > 0 (otherwise |IA(1, 2)| = ∅). The scheme A∩Y is the union of C
+and the 3 points B ∩Y . In the identification of Y with F1 the line bundle OY (1, 2) goes to
+the line bundle OF1(h+3f) and C goes to an element of |h+f|. Thus h0(F1, IB∩Y,F1(2f)) >
+0. Hence at least 2 of the 3 points B ∩ Y are in the same fiber ˆL of the ruling |f| of F1.
+Since ˆL ∩ C ̸= ∅, ˆL is a curve of bidegree (1, 0) meeting at least 3 connected components
+of A. Call B′ a union of 3 components of A intersecting ˆL. The curves B′ and ˆL give a
+contradiction.
+Assume now that the result is true for d+1. Notice that, as a byproduct of the previous
+part, if B ∈ C∗(d + 1), then h0(IB(1, d − 1)) = 0.
+Assume A ∈ C∗(d + 2) and that there is an integral S ∈ |OF(1, d)|. Fix a connected
+component C of A and set B := A \ C. Take a surface Y of bidegree (0, 1) containing
+C. By means of the sequence in Formula (10), we either have h0(IB(1, d − 1)) > 0 or
+h0(Y, IA∩Y,Y (1, d)) > 0. Since A ∈ C∗(d + 2), B ∈ C∗(d + 1) and hence, thanks to the
+inductive assumption, we have h0(IB(1, d − 1)) = 0. The scheme A ∩ Y is the union of
+C and the scheme B ∩ Y with A ∩ B ∩ Y = ∅. Up to the identification of Y and F1 we
+have OY (1, d)(−C) ∼= OF1(df). Since Y has bidegree (0, 1) each connected component of
+B is either contained in Y or it intersects transversely Y at a unique point. By Remark
+2.9, the set B ∩ Y is formed by d + 1 points. Thus IA∩Y,Y (1, d) ∼= IB∩Y (df). We saw
+h0(Y, IA∩Y,Y (1, d)) > 0 and this is true if and only if there are u1, . . . ud+1 ∈ B ∩ Y and
+F ∈ |f| such that that ui ̸= uj, for i ̸= j and {u1, . . . , ud+1} ⊂ F. The set F ∩ C is a
+unique point, o, and o /∈ {u1, . . . , ud+1}, because B ∩ C = ∅. The curve F has bidegree
+(0, 1) and hence A /∈ C∗(d + 2), a contradiction.
+□
+Thanks to the previous result, if an integral (1, 2)-surface contains 4 disjoint smooth
+conics, then these are in special position. We now show that if these 4 conics are twistor
+fibers, then their position is very special. We begin by introducing the following notation.
+For n ≥ 4, we denote by C(n)− the set of elements A ∈ C(n) for which there exists a
+bidegree (1, 0) curve L such that A ∈ C(n, L). The set C(n)− parametrizes the families of
+n collinear disjoint smooth conics. For n ≥ 4 we also write T (n)− := T (n) ∩ C(n)−. The
+families C(n)− and T (n)− are Zariski closed in C(n) and T (n), respectively.
+The following lemma shows that if an integral (1, 2)-surface contains 4 twistor fibers,
+then they are all collinear.
+Lemma 4.8. Take an integral S ∈ |OF(1, 2)| containing A ∈ T (4). Then A ∈ T (4)−
+Proof. Assume the existence of an integral S ∈ |OF(1, 2)| containing A ∈ T (4). By Lemma
+4.7 there is a union B of 3 of the connected components of A such that B ∈ T (3) \ T ∗(3),
+i.e., there exists a bidegree (1, 0) curve L, such that B ∈ T (3, L) and hence, thanks to
+Remark 3.7 h0(IB(1, 1)) > 0. However, the same remark tells us that h0(IB(1, 1)) = 1,
+h1(IB(1, 1)) = 2 and that the only element M of |IB(1, 1)| is integral.
+As usual, set C := A \ B.
+As in Remark 4.2 since L of bidegree (1, 0) meets each
+connected components of B, then R := j(L), of bidegree (0, 1), do the same.
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+21
+Thanks to Remark 2.16 we get B ∪ L ∪ R ⊂ M. Since S and M are integral, thanks
+to Lemma 2.2, the one-dimensional scheme S ∩ M has bidegree (5, 4). Since B ∪ L ∪ R
+has bidegree (4, 4), then C ⊈ M. Let Y be only element of |IC(0, 1)|. Since B ⊂ M ∪ Y ,
+then M ∪ Y ∈ |IA(1, 2)|. Moreover, as S is irreducible, then S ̸= M ∪ Y , and hence
+h0(IA(1, 2)) ≥ 2, i.e. h1(IA(1, 2)) ≥ 3. Since h1(IB(1, 1)) = 2, the residual exact sequence
+0 → IB(1, 1) → IA(1, 2) → IA∩Y,Y (1, 2) → 0,
+gives h1(Y, IA∩Y,Y (1, 2)) > 0.
+As in the proof of Lemma 4.7 we obtain the following
+inequality h1(F1, IB∩Y (2f)) > 0, i.e. there is a curve ˆL ∈ |f| of bidegree (1, 0) intersecting
+at least 2 of the connected components of B.
+Call B′ the union of 2 of the connected components of B intersecting ˆL. Since ˆL∩C ̸= ∅
+and B′ ∪ C is j-invariant, each connected component of B′ ∪ C meets j(ˆL). Remark 4.2,
+Proposition 4.1 and B´ezout imply the existence of an integral surface M′ of bidegree (1, 1)
+containing B′ ∪ C ∪ ˆL ∪ j(ˆL). Since B′ ⊂ M′, ˆL and j(ˆL) contain at least 2 points of
+M′, then B′ ∪ ˆL ∪ j(ˆL) ⊂ M. But by Remark 2.5 there is a unique curve of bidegree
+(1, 0) intersecting two different smooth conics, hence ˆL = L and both L and j(L) intersect
+each connected component of B. Thus L intersects each connected component of A, i.e.
+A ∈ C(4)−.
+□
+As a byproduct of the proof of the previous result, we get the following lemma. It
+essentially says that there are infinitely many integral (1, 2)-surfaces containing 4 collinear
+twistor fibers.
+Lemma 4.9. Take A ∈ T (4)− and assume that A is not contained in a surface of bidegree
+(1, 1).
+Then dim |IA(1, 2)| = 1 and |IA(1, 2)| contains exactly 4 reducible elements of
+|OF(1, 2)|.
+Proof. Call L the curve of bidegree (1, 0) intersecting each connected component of A.
+Since each connected component of A is j-invariant, j(L) intersects each connected com-
+ponent of A. In the proof of Lemma 4.8 we showed that h0(IA(1, 2)) ≥ 2. From the lines of
+that proof, it is possible to derive that only 4 elements in |IA(1, 2)| are reducible and they
+are all obtained fixing a connected component C of A and taking the union of the unique
+surface MC of bidegree (1, 1) containing A\C and the unique surface YC of bidegree (0, 1)
+containing C. To conclude the proof it is sufficient to prove that h0(IA(1, 2)) ≤ 2. Take a
+connected component C of A and consider the residual exact sequence
+(22)
+0 → IC(0, 1) → IA(1, 2) → IMC∩A,MC(1, 2) → 0.
+We have h0(IC(0, 1)) = 1, because the intersection of 2 different elements of |OY (0, 1)| is
+a curve of bidegree (1, 0). Thus by (22) to conclude the proof it is sufficient to prove that
+the image V of H0(IA(1, 2)) in H0(MC, IMC∩A,MC(1, 2)) has dimension at most 1. Bez´out
+gives that A ∪ L ∪ j(L) is contained in the base locus of |IB,MC(1, 2)|. Every D ∈ |V|
+has bidegree (5, 4) as a curve of F and hence a general D ∈ |V| is the union (counting
+multiplicities as divisors of the smooth surface MC) of A ∪ L ∪ j(L) and a curve E of
+bidegree (1, 0)) as a curve of F. Recall that MC is the blow up of P2 at 3 non collinear
+points and that these 3 exceptional divisors are the only curve of MC with bidegree (1, 0).
+Since MC has only finitely many curves of bidegree (1, 0), D is the same for all non-zero
+elements of V and hence dim V = 1.
+□
+The following result completes the proof of Theorem 1.3.
+Theorem 4.10. There are integral S ∈ |OF(1, 2)| containing exactly 4 twistor fibers and
+for any such S and A ∈ T (4) with A ⊂ S, there is a curve L of bidegree (1, 0) intersecting
+each connected component of A. Moreover, h0(IA(1, 2)) = 2 and each S ∈ |IA(1, 2)| is
+singular along L.
+
+22
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+Proof. By Theorem 1.4 no integral surface of bidegree (1, 2) contains at least 5 twistor
+fiber. The curve L exists by Lemma 4.8. Now we reverse the construction. We start with
+A ∈ T (4, L)−. Let 2L denote the closed subscheme of the “double line”. To prove that
+each S ∈ |IA(1, 2)| is singular at each point of L it is sufficient to prove that h0(IA(1, 2)) =
+h0(IA∪2L(1, 2)). Lemma 4.9 gives h0(IA(1, 2)) = 2. Hence, it is sufficient to prove that
+h0(IA∪2L(1, 2)) > 1. For any connected component C of A let MC the only surface of
+bidegree (1, 1) containing A \ C and let YC the only surface of bidegree (0, 1).
+Since
+C ∩ L ̸= ∅, L ∩ YC ̸= ∅. Since YC has bidegree (0, 1) and L bidegree (1, 0), we get L ⊂ YC.
+Thus L ⊆ MC ∩ YC and hence |IA∪2L(1, 2)| contains at least the 4 reducible elements of
+|IA(1, 2)|. Hence h0(IA∪2L(1, 2)) > 1.
+□
+4.2. Non existence results for surfaces of bidegree (1, 2) and (1, 3). In this last
+part, we prove our two last main results, i.e. Theorems 1.4 and 1.5.
+For any A ∈ T (n)−, n ≥ 4, let us call L and R := j(L) the curves of bidegree (1, 0) e
+(0, 1), respectively, intersecting all the connected components of A.
+In view of our goal, we need to discuss the reducibility of some surfaces containing a
+certain amount of twistor fibers. First of all, fix an integer n ≥ 2, take B ∈ T (4) such that
+h0(IB(1, 1)) > 0 and call M the unique (see e.g. Remark 4.2) surface of bidegree (1, 1)
+containing B. Since each element of C(1) is contained in an element of |OF(0, 1)| for each
+E ∈ T (n − 1) there is a reducible element W ∈ |OF(1, k)|, union of M and n − 1 surfaces
+of bidegree (0, 1) such that B ∪ E ⊂ W. The following lemma is a sort of viceversa of this
+remark. Moreover, it will be a key tool in the last two proofs.
+Lemma 4.11. If d ≥ 2 and A ∈ T (d + 3)− is such that h0(IA(1, d)) > 0, then each
+element of |IA(1, d)| has an irreducible component M of bidegree (1, 1) containing at least
+4 connected components of A.
+In particular, for any n ≥ d + 3, there is no integral
+S ∈ |OF(1, d)| containing A ∈ T (n)−.
+Proof. In order to prove the last statement, it is sufficient to do the case n = d + 3 and
+thus it is sufficient to prove the first assertion.
+We use induction on d ≥ 2. Let us assume first d = 2. Take A ∈ T (5)− and let L and
+j(L) be the curves of bidegree (1, 0) and (0, 1) intersecting all the connected components
+of A. Fix a connected component C of A and set B := A \ C. Since C ∩ L ̸= ∅, the curve
+C ∪ L is a connected and nodal curve of bidegree (2, 1) with arithmetic genus 0. Hence
+h0(OC∪L(0, 1)) = 2. Thus there is Y ∈ |IC∪L(0, 1)| and such a Y is unique. Since any two
+smooth conics of Y meet, no component of B is contained in Y . Hence B ∩ Y is formed
+by 4 points of L \ (L ∩ C). Recall that OY (1, 2) ∼= OF1(h + 3f) and that C ∈ |OF1(h + f)|
+and thus IA∩Y,Y ∼= IB∩L,Y (2f). Since each element of |f| contains a unique point of L we
+have that h0(D, IA∩Y,Y (1, 2)) = 0. The residual exact sequence of Y
+0 → IB(1, 1) → IA(1, 2) → IA∩Y,Y (1, 2) → 0,
+gives an isomorphism ϕ : H0(IB(1, 1)) → H0(IA(1, 2)). If h0(IB(1, 1)) = 0, then h0(IA(1, 2)) =
+0. Now assume h0(IB(1, 1)) ̸= 0. The isomorphism ϕ says that each W ∈ |IA(1, 2)| has Y
+as an irreducible component, say W = Y ∪ W1 with W1 ∈ |IB(1, 1)|, and hence we have
+the thesis.
+Assume now d ≥ 3 and use induction on d. By reasoning as in the base case, take
+A ∈ T (d + 3)− and use the exact sequence
+0 → IB(1, d − 1) → IA(1, d) → IA∩Y,Y (1, d) → 0,
+to prove that h0(Y, IA∩Y,Y (1, d)) = 0 and hence that there is an isomorphism ϕ : H0(IB(1, d−
+1)) → H0(IA(1, d)). Now, again, if h0(IB(1, d − 1)) = 0, then h0(IA(1, d)) = 0. Hence,
+assume h0(IB(1, d − 1)) ̸= 0. The isomorphism ϕ says that each S ∈ |IA(1, d)| has Y
+as an irreducible component, i.e. S = D ∪ S1 with S1 ∈ |IB(1, d − 1)|. The inductive
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+23
+assumption says that S1 has an irreducible component M of bidegree (1, 1) containing at
+least 4 components of B.
+□
+We now have all the ingredients to prove Theorems 1.4 and 1.5. First we prove that no
+integral surface of bidegree (1, 2) contains 5 twistor fibers.
+Proof of Theorem 1.4. Assume the existence of an integral S ∈ |OF(1, 2)| containing A ∈
+T (5). Lemma 4.8 shows that for any union A′ ⊂ A of 4 components of A there is a union
+A′′ ⊂ A′ of 3 connected components intersecting some L of bidegree (1, 0). Let L be a
+curve of bidegree (1, 0) intersecting the maximal number, z, of components of A. Clearly
+z ≥ 3. By Lemma 4.11 to get a contradiction it is sufficient to prove that z ≥ 5.
+Assume z ∈ {3, 4}. Take any ordering C1, . . . , C5 of the connected components of A
+and set Bi := π1(Ci), 1 ≤ i ≤ 5. Each Bi is a line of P2. Since any two conics contained
+in an element of |OF(1, 0)| meet, B1, . . . , B5 are 5 different lines of P2. For any i < j < h,
+there is a curve T of bidegree (1, 0) intersecting Ci, Cj and Ch if and only if Bh contains
+the point Bi ∩ Bj and in this case L = π−1
+1 (Ci ∩ Cj). With no loss of generality we may
+assume that L meets C1, . . . , Cz.
+(a) Assume z = 3 and hence B1 ∩B2 ∈ B3. Applying Lemma 4.8 to C1 ∪C2 ∪C4 ∪C5
+we have one of the following mutually esclusive relations:
+B1 ∩ B2 ∩ B4 ̸= ∅,
+B1 ∩ B2 ∩ B5 ̸= ∅,
+B1 ∩ B4 ∩ B5 ̸= ∅,
+B2 ∩ B4 ∩ B5 ̸= ∅.
+Since B1 ∩ B2 ∈ B3 and z = 3, we can exclude the first two cases, i.e. we have
+B1 ∩ B2 ∩ B4 = B1 ∩ B2 ∩ B5 = ∅.
+Thus, either B1 ∩ B4 ∩ B5 ̸= ∅ or B2 ∩ B4 ∩ B5 ̸= ∅. Exchanging if necessary C1 and C2
+we may assume B1 ∩ B4 ∩ B5 ̸= ∅, i.e. B4 ∩ B5 ∈ B1, and hence B2 ∩ B4 ∩ B5 = ∅. Since
+B2 ∩ B4 ∩ B5 = ∅, applying Lemma 4.8 to C2 ∪ C3 ∪ C4 ∪ C5 we have one of the following
+mutually esclusive relations
+B2 ∩ B3 ∩ B4 ̸= ∅,
+B2 ∩ B3 ∩ B5 ̸= ∅,
+B3 ∩ B4 ∩ B5 ̸= ∅.
+Since B4 ∩ B5 ∈ B1 and z = 3, B3 ∩ B4 ∩ B5 = ∅. Since B2 ∩ B3 ∈ B1 and z = 3,
+B2 ∩ B3 ∩ B5 = B2 ∩ B3 ∩ B4 = ∅, a contradiction.
+(b) Assume z = 4. Since C1 ∩ L ̸= ∅, the curve C1 ∪ L is a connected and nodal
+curve of arithmetic genus 0 and bidegree (2, 1). Thus h0(OC1∪L(0, 1)) = 2. Thus there is
+Y ∈ |IC1∪L(0, 1)|. Since S is irreducible, Y is not an irreducible component of S. Thus
+the residual exact sequence of Y
+0 → IB(1, 1) → IA(1, 2) → IA∩Y,Y (1, 2) → 0,
+gives h0(Y, IA∩Y,Y (1, 2)) ̸= 0.
+Up to the isomorphism of Y and F1 we have L = h,
+C1 ∈ |OF1(h + f)| and OY (1, 2) ∼= OF1(h + 3f). Since (A \ C1) ∩ C1 = ∅, IA∩Y,Y (1, 3) ∼=
+I(A\C1)∩D,D(2f). Since L meets C1, . . . , C4, (A \ C1) ∩ D contains a set F ⊂ L such that
+#F = 3. Since each element of |f| contains a unique point of L, h0(Y, IA\C)∩Y,Y (2f)) = 0,
+a contradiction.
+□
+We now conclude our paper with the proof of Theorem 1.5 which concerns surfaces of
+bidegree (1, 3).
+Proof of Theorem 1.5: Assume the existence of A ∈ T (6) and of an integral S ∈ |OF(1, 3)|
+containing A. By Lemma 4.11 to get a contradiction it is sufficient to prove the existence
+of a curve L of bidegree (1, 0) such that all the components of A intersects L. By Lemma
+4.7 for any union A′ ⊂ A of 5 components of A there is a union A′′ ⊂ A′ of 3 connected
+components intersecting some L of bidegree (1, 0). Let L be a curve of bidegree (1, 0)
+intersecting the maximal number, z, of components of A. We have that z ≥ 3. Hence, by
+
+24
+A. ALTAVILLA, E. BALLICO, AND M. C. BRAMBILLA
+Lemma 4.11 it is sufficient to prove that z ≥ 6. Assume then that z ≤ 5. We will now
+exclude all the cases z = 3, 4, 5.
+For any connected component C of A, Lemma 4.7 tells us that there is a curve L of
+bidegree (1, 0) intersecting at least 3 connected components of A \ C. In particular there
+is an integral M ∈ |OF(1, 1)| containing at least 3 components of A (see Remark 4.2).
+Note that j(M) = M. We may take M with the additional condition that it contains
+the maximal number e of components of A. Let E be the union of the components of A
+contained in M. Thus 3 ≤ e ≤ z ≤ 5.
+Since each twistor fiber is j-invariant, j(L) meets each connected component of E.
+B´ezout gives L ∪ j(L) ⊂ M and L ⊂ S. If e ≥ 4 B´ezout gives j(L) ⊂ S. However,
+the one-dimensional cycle M ∩ S has bidegree (7, 5) and thus e ≤ 4. Set Σ := S ∩ M
+(as a scheme-theoretic intersection). Since the one-dimensional scheme Σ is the complete
+intersection of F with 2 very ample divisors, h0(OΣ) = 1. Set F := A \ E.
+(a) Assume e = 4.
+Hence E ∪ L ∪ j(L) ⊂ Σ.
+Since E ∪ L ∪ j(L) has bidegree
+(5, 5) and h0(OΣ) = 1, Σ is the union of E ∪ L ∪ j(L) and a multiple structure on L.
+Note that Σ ∈ |OF(1, 3)| and that Σ contains E ∪ j(L) with multiplicity 1 and L with
+multiplicity 3 (as divisors of the smooth surface M).
+Since Σ has multidegree (7, 5),
+Σ = 3L ∪ j(L) ∪ E. Note that Σ contains the degree 4 zero-dimensional scheme F ∩ M.
+Since F ∩ E = ∅, F ∩ (j(L) ∪ L) ̸= ∅. Thus at least one irreducible component, T, of
+F meets L ∪ j(L). Since j(T) = T, T ∩ L ̸= ∅. Thus z = 5. Let C be a component
+of E. Since C ∩ L ̸= ∅, C ∪ L is a connected and nodal curve of bidegree (2, 1) with
+arithmetic genus 0.
+Thus h0(OC∪L(0, 1)) = 2.
+Thus there is Y ∈ |IC∪L(0, 1)| ̸= 0.
+Since S is irreducible, Y is not an irreducible component of S. Thus the residual exact
+sequence of Y gives h0(Y, IA∩Y,Y (1, 3)) ̸= 0.
+Up to the isomorphism of Y and F1 we
+have L = h, C ∈ |OF1(h + f)| and OY (1, 3) ∼= OF1(h + 4f). Since (A \ C) ∩ C = ∅,
+IA∩Y,Y (1, 3) ∼= I(A\C)∩Y,Y (3f). Since z = 5, (A \ C) ∩ Y contains a set H ⊂ L such that
+#H = 4. Since each element of |f| contains a unique point of L, h0(Y, IA\C)∩Y,Y (3f)) = 0,
+a contradiction.
+(b) Assume e = 3. Fix a connected component C of E and set B := A \ C. Set
+{Y } := |IC(0, 1)|. As in step (a) we have that L ⊂ Y . The following exact sequence
+(23)
+0 → IB(1, 2) → IA(1, 3) → IC∪(B∩Y ),Y (1, 3) → 0
+is the residual exact sequence of Y . Since Y is not an irreducible component of S, we have
+h0(Y, IC∪(B∩Y ),Y (1, 3)) > 0. As in step (a) we have IC∪(B∩Y ),Y (1, 3) ∼= IB∩Y,F1(3f). We
+now have two possibilities: either h0(IB(1, 2)) = 0 or h0(IB(1, 2)) > 0.
+(b1) Assume for the moment h0(IB(1, 2)) = 0. Thus h0(Y, IC∪(B∩Y ),Y (1, 3)) ≥ 2 and
+h1(Y, IC∪(B∩Y ),Y (1, 3)) ≥ 3. Up to the identification of Y and F1 we have IC,Y (1, 3) ∼=
+OF1(3f).
+Hence the 5 points B ∩ Y give at most one condition to the linear system
+|OF1(3f)|. Thus there is J ∈ |OF1(f)| such that B ∩ Y ⊂ J. Note that J is a curve of
+bidegree (1, 0). The maximality of the integer e gives a contradiction.
+(b2) Assume that h0(IB(1, 2)) > 0. By Theorem 1.4 any surface containing B is re-
+ducible, say M1∪Y with M1 irreducible of bidegree (1, 1) containing at least 4 components
+of B. Thus e ≥ 4, a contradiction.
+□
+References
+[1] A. Altavilla, E. Ballico. Twistor lines on algebraic surfaces. Ann. Global Anal. Geom. 55 (2019), no.
+3, 555–573.
+[2] A. Altavilla, G. Sarfatti. Slice-polynomial functions and twistor geometry of ruled surfaces in CP3.
+Math. Z. 291 (2019), no. 3-4, 1059–1092.
+[3] A. Altavilla, E. Ballico, M. C. Brambilla. Surfaces in the flag threefold containing smooth conics and
+twistor fibers. Mediterr. J. Math. 19, 281 (2022). https://doi.org/10.1007/s00009-022-02202-3
+
+TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD
+25
+[4] A. Altavilla, E. Ballico, M. C. Brambilla, S. Salamon. Twistor geometry of the Flag manifold. Math.
+Z. 303, 24 (2023). https://doi.org/10.1007/s00209-022-03161-x
+[5] J. Armstrong, M. Povero, S. Salamon. Twistor lines on cubic surfaces. Rend. Semin. Mat. Univ.
+Politec. Torino 71 (2013), no. 3-4, 317–338.
+[6] M. F. Atiyah, N. J. Hitchin, I. M. Singer. Self-duality in four-dimensional Riemannian geometry. Proc.
+Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461.
+[7] E. M. Chirka. Orthogonal complex structures in R4. (Russian) Uspekhi Mat. Nauk 73 (2018), no.
+1(439), 99–172; translation in Russian Math. Surveys 73 (2018), no. 1, 91–159
+[8] A. Fujiki, M. Pontecorvo. Twistors and bi-Hermitian surfaces of non-K¨ahler type. SIGMA Symmetry
+Integrability Geom. Methods Appl. 10 (2014), Paper 042, 13 pp.
+[9] G. Gentili, S. Salamon, C. Stoppato. Twistor transforms of quaternionic functions and orthogonal
+complex structures. J. Eur. Math. Soc. (JEMS) 16 (2014), no. 11, 2323–2353.
+[10] R. Hartshorne, Algebraic Geometry, Springer-Verlag, Berlin–Heidelberg–New York, 1977.
+[11] N. J. Hitchin. K¨ahlerian twistor spaces. Proc. London Math. Soc. (3) 43 (1981), no. 1, 133–150.
+[12] M. Pontecorvo. Complex structures on Riemannian four-manifolds. Math. Ann. 309 (1997), no. 1,
+159–177.
+[13] S. Salamon, J. Viaclovsky. Orthogonal complex structures on domains in R4. Math. Ann. 343 (2009),
+no. 4, 853–899.
+Dipartimento di Matematica, Universit`a degli Studi di Bari ‘Aldo Moro’, via Edoardo
+Orabona, 4, 70125, Bari, Italia
+Email address: amedeo.altavilla@uniba.it
+Dipartimento Di Matematica, Universit`a di Trento, Via Sommarive 14, 38123, Povo, Trento,
+Italia
+Email address: edoardo.ballico@unitn.it
+Universit`a Politecnica delle Marche, via Brecce Bianche, I-60131 Ancona, Italia
+Email address: brambilla@dipmat.univpm.it
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf,len=1580
+page_content='TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  AMEDEO ALTAVILLA, EDOARDO BALLICO, AND MARIA CHIARA BRAMBILLA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We study surfaces of bidegree (1, d) contained in the flag threefold under  the action of the twistor projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' First, we prove that there is no integral surfaces of  bidegree (1, d) containing d + 2 twistor fibers such that three of them are not collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Then, fixed any union of 0 ≤ n ≤ d + 1 non-three-by-three collinear twistor fibers, we  show that there is an integral (1, d)-surface containing them and no other twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The result is also true for d + 2 twistor fibers with additional suitable hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Later,  we focus on surfaces of low bidegrees and prove that, for any set of 0 ≤ n ≤ 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' n = 4)  twistor fibers, there is a smooth (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' integral) surface of bidegree (1, 2) containing them  and no other twistor fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Finally, we prove that there is no integral (1, d)-surface, for  d = 2, 3, containing d + 3 twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Preliminaries and first results  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Curves in F and smooth conics  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Surfaces of bidegree (1, 0) and (0, 1)  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Surfaces of bidegree (0, d) and (1, d)  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Non-collinear smooth conics  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Surfaces of bidegree (1, d)  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Surfaces of bidegree (1, 2) and (1, 3)  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Surfaces of bidegree (1, 2) containing 0 ≤ n ≤ 4 twistor fibers  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Non existence results for surfaces of bidegree (1, 2) and (1, 3)  22  References  24  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Introduction  The flag threefold F can be seen as the twistor space of the complex projective plane  P2 endowed with all its standard structures except for its orientation which, in this case,  is the opposite of the usual one [6, 11],  π : F → P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The study of the twistor geometry of the flag is motivated by the search of Riemannian 4-  manifolds admitting several integrable complex structures compatible with the prescribed  metric (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' [8, 12, 13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In this context, a recent trend is that of study particular cases,  in order to find explicit examples [1, 2, 5, 7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In [4] we have started a detailed analysis of the geometry of the algebraic curves and  surfaces contained in F, in relation with the twistor projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, twistor fibers  are smooth integral curves of bidegree (1,1) (called smooth conics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In [3] we gave a first  bound on the maximum number of smooth conics contained in a smooth surface S ⊂ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Primary: 32L25, 14M15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Secondary: 14D21, 14J26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' flag threefold, twistor projection, twistor fiber, surfaces, bidegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  All the authors are partially supported by GNSAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The first named author is partially supported by  the INdAM project ‘Teoria delle funzioni ipercomplesse e applicazioni’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='04874v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='AG]  12 Jan 2023  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  Here, by focusing our attention to a specific family of surfaces, we obtain more precise  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In fact we analyze the case of bidegree (1, d) surfaces in F and study the number  and the arrangements of twistor fibers contained in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In order to explain in details our results, we need to clarify some definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The flag  threefold can be defined as  F := {(p, ℓ) ∈ P2 × P2 | pℓ = 0},  where p = [p0 : p1, p2], ℓ = [ℓ0 : ℓ1 : ℓ2] ∈ P2 and pℓ = p0ℓ0 + p1ℓ1 + p2ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The notation  (p, ℓ), would recall a couple (point,line), and the condition pℓ = 0 translates as p belongs  to ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In order to simplify the notation, we identify the second factor P2∨ with P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We have three projection maps: π1, π2, π : F → P2, defined as  π1(p, ℓ) = p,  π2(p, ℓ) = ℓ,  π(p, ℓ) = p × ℓ = [p1ℓ2 − p2ℓ1, p2ℓ0 − p0ℓ2, p0ℓ1 + p1ℓ0],  where the third one is the twistor map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The fibers of such three maps are the object of  our investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Being embedded in P2 × P2, it is possible to define a natural notion of bidegree for  surfaces and a (bit less natural) one for curves in F (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, for  any q ∈ P2, the three fibers π−1  1 (q), π−1  2 (q) and π−1(q) are curves of bidegree (0, 1), (1, 0)  and (1, 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' While the fibers of π1 and π2 exhaust the family of bidegrees (1, 0)  and (0, 1) curves, twistor fibers are only a (non-open Zariski dense) subset of those of  bidegree (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since the curves of bidegree (1, 1) are rational, we call them conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' There  are only two types of bidegree (1, 1) curves: the reducible ones (union of a bidegree (1, 0)  and of a bidegree (0, 1) curves intersecting at a point), or smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' All of them can  be described as  Lq,m := {(p, ℓ) ∈ F | pm = 0, qℓ = 0},  where q, m ∈ P2 and, the reducible and smooth cases are obtained for qm = 0 or qm ̸= 0,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In twistor theory there is an antiholomorphic involution without fixed points that plays  important roles in identifying twistor fibers [6, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In our case, this map can be defined  as j : F2 → F2, where  j(p, ℓ) = (ℓ, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  A smooth conic Lq,m is a twistor fiber if and only if j(Lq,m) = Lq,m if and only if m = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  As explained in [6], the geometry of F as twistor space does not change if we consider a  conformal copy of its base space P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore, it is natural to classify objects in F up to  conformal transformations, which, in this case, means up to projective transformations of  F coming from the lift via π of a conformal transformation of P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' It is possible to see that  such transformations of F are exactly those which commute with j (see [4] for the case of  the flag manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence, in particular, the number of twistor fibers contained in a given  surface and their arrangement are conformal invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In order to state our main results we need some more notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We denote by C(1) = C  the set of smooth conics in F and by C(n), n ≥ 2, the set of n pairwise disjoint smooth  conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In an analogous way we define T (1) = T ⊂ C as the set of twistor fibers and  T (n) ⊂ C(n), n ≥ 2, as the set of n pairwise disjoint twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We will see in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5 that for any couple of different smooth conics, there is a  unique bidegree (1, 0) curve L = π−1  2 (q2) and a unique bidegree (0, 1) curve R = π−1  1 (q1)  such that L and R intersect both smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In the case of a couple of twistor fibers  we also have R = j(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We say that three or more smooth conics are collinear if there is  a (1, 0) curve L which intersects all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' To be collinear, for three or more smooth  conics, is a Zariski closed condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  To be more precise, in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7, we define the set C∗(n) which parametrizes all  A ∈ C(n) such that #(L ∩ A) ≤ 2 for all curves L of bidegree (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Clearly C∗(1) = C(1),  and C∗(2) = C(2), while for n ≥ 3 the open set C∗(n) is given by the set of disjoint smooth  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  3  conics such that no three of them are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Moreover, we set T ∗(n) := T (n) ∩ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='19 we characterize the elements A ∈ C∗(d + 1) to be those which do not  obstruct the linear system |IA(1, d)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We now summarize the main results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In Section 3, we study surfaces of  bidegree (1, d) containing a certain number of smooth conics or twistor fibers, and we  prove the following two theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For any d ∈ N and A ∈ T ∗(d + 2), there is no integral surfaces of bidegree  (1, d) containing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix integer d ≥ 1 and 0 ≤ n ≤ d + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' There is an integral S ∈ |OF(1, d)|  containing exactly n twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We also show, in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4, that the first result is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Indeed, for any A ∈ T ∗(n),  with 0 ≤ n ≤ d + 1, we are able to find an integral surface of bidegree (1, d) containing A  and no other twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' This last issue requires some effort and the proof is divided  into several particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2 is a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' More precisely, in  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4, for 0 ≤ n ≤ d + 1, we prove that fixed any union A of 0 ≤ n ≤ d + 1  non-three-by-three collinear twistor fibers, there is an integral (1, d)-surface containing A  and no other twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The extremal case n = d + 2 is considered in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8,  where we prove that given d + 2 general collinear twistor fibers there is an integral surface  of bidegree (1, d) containing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In Section 4, we focus on surfaces of bidegree (1, 2) and (1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The main results are  summarized by the following statements:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix 0 ≤ n ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' There is a smooth S ∈ |OF(1, 2)| containing exactly n  twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, there exists a bidegree (1, 2) integral surface containing exactly 4  twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' There is no integral S ∈ |OF(1, 2)| containing at least 5 twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' There is no integral S ∈ |OF(1, 3)| containing at least 6 twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The first existence result (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3) follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='6, for 0 ≤ n ≤ 3, and  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='10, for the case n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In the extremal case n = 4, we will also show that the  surfaces are singular along a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The two non-existence results (Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5) are proved in the last Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  An essential tool is Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='11 which states that if a surface of bidegree (1, d)  contains d + 3 or more collinear twistor fibers, then this surface is reducible and one of its  components is a surface of bidegree (1, 1) containing 4 of the prescribed twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We conclude here with a comparison with the other (smooth) algebraic twistor space of  a Riemannian 4-manifold, which is the complex projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' This is the twistor space  of the standard 4-sphere [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In this case, surfaces of degree 2 and 3 were studied in some  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, analogously to our case of surfaces of bidegree (1, 1), surfaces of  degree 2 in P3 might contain 0, 1 or 2 twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If a surface of degree 2 contains more  than 2 twistor fibers, then it contains infinitely many of them [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For degree 3 surfaces,  such a maximum is realized for 5 twistor fibers [5] which is more than our maximum of 4  for surfaces of bidegree (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' This difference could be explained by observing a certain  unbalancedness of the case of “total degree” 3 in the flag threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' On the other hand,  this particular unbalancedness allows us to compute all the Betti numbers in the next  section as well as the use of the geometry of the Hirzebruch surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Preliminaries and first results  In this section, we collect some known results about algebraic curves and surfaces in  the flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then, we give first results on the space of bidegree (0, d) and (1, d) surfaces  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  containing a certain amounts of twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, we introduce the concept of  collinear smooth conics and give a topological characterization in terms of cohomology of  certain ideal sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  For most of the known material about F, we refer mainly to [4] and to [3, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  However, we recall here some basic notion and results in order to be as more self-contained  as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Let us consider the multi projective space P2 × P2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' an element (p, ℓ) ∈ P2 × P2 will  be a couple written in the following form p = [p0 : p1 : p2], ℓ = [ℓ0 : ℓ1 : ℓ2]⊤, so that  pℓ = p0ℓ0 + p1ℓ1 + p2ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Even if it is classically embedded in P2 × P2∨, we might see  F := {(p, ℓ) ∈ P2 × P2 | pℓ = 0} as a hypersurface of bidegree (1, 1) of P2 × P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We denote  by Π1 and Π2 the two standard projections of P2×P2 and we will use small letters for their  restrictions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' πi = Πi|F, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus, the two natural projections define a natural  notion of bidegree for algebraic surfaces in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, for all (a, b) ∈ Z2 we have the  following natural exact sequence  (1)  0 → OP2×P2(a − 1, b − 1) → OP2×P2(a, b) → OF(a, b) → 0,  and, for any (a, b) ∈ N2, we get (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' [4, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3])  (2)  h0(OF(a, b)) = (a + 1)(b + 1)(a + b + 2)  2  and  h1(OF(a, b)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  It will be useful to recall from [4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='11] the multiplication rules in the Chow  ring:  OF(1, 0) · OF(1, 0) · OF(1, 0) = 0,  OF(1, 0) · OF(0, 1) · OF(1, 0) = 1,  OF(0, 1) · OF(1, 0) · OF(0, 1) = 1,  OF(0, 1) · OF(0, 1) · OF(0, 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (3)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Curves in F and smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let us recall a notion of bidegree for the family  of algebraic curves in F already given in [4, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let C ⊂ F be an integral algebraic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We define the bidegree of C  as the couple of positive integer numbers (d1, d2), where di = 0 if πi(C) = {x}, otherwise  di = deg(πi(C)) deg(πi|C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  If a curve D has irreducible components C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , Cs then the bidegree of D is taken to  be the sum of the bidegrees of C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Recall from [3, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4] that if a curve C is such that C · OF(1, 0) = d1 and  C · OF(0, 1) = d2, then it has bidegree (d1, d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  From the previous table of multiplication, we can easily derive the following formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For any choice of non-negative integers a, b, c, d, the one-dimensional cycle  OF(a, b) · OF(c, d) has bidegree  (ad + b(c + d), a(c + d) + bc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We have OF(a, b) · OF(c, d) = acOF(1, 0) · OF(1, 0) + (ad + bc)OF(1, 0) · OF(0, 1) +  bdOF(0, 1)·OF(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence the thesis is easily obtained by recalling that OF(1, 0)·OF(1, 0)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' OF(1, 0) · OF(0, 1), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' OF(0, 1) · OF(0, 1)) is a one-dimensional cycle of bidegree  (0, 1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' bidegree (1, 1), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' bidegree (1, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Notice that the fibers of π1 are algebraic curves of bidegree (0, 1), while  those of π2 have bidegree (1, 0) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' [4, Section 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Moreover, all bidegree (0, 1)  curves can be seen as complete intersections between two different (1, 0) surfaces (and  analogously for bidegree (1, 0) curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Among all algebraic curves in F we focus our attention on the family of bidegree (1, 1)  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' These are geometrically described in [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1] and are parameterized by  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  5  (q, m) ∈ P2 × P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In fact, as anticipated in the introduction, any of these curves can be  written as  Lq,m := {(p, ℓ) ∈ F | p ∈ m, ℓ ∋ q} = {(p, ℓ) ∈ F | qℓ = 0, pm = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  There are two types of these curves: the smooth and irreducible ones (when qm ̸= 0) and  the union of a (1, 0) and of a (0, 1) intersecting at a point (when qm = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' (q, m) ∈ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In any case, each bidegree (1, 1) curve can be seen as the complete intersection of a surface  of bidegree (1, 0) with one of bidegree (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As already mentioned in the introduction,  the 4-dimensional family of smooth irreducible (1, 1) curves will be denoted by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The  elements of C will be called smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' From the very definition of smooth integral conics, it is clear that, for any  C ∈ C we have that πi(C) is a line in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Notice that for any two different elements Lq,m, Lq′,m′ ∈ C there exist a  unique curve L of bidegree (1, 0) and a unique R of bidegree (0, 1) such that L and R  meets both Lq,m and Lq′,m′ at a point (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  From the analysis made in [4,  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1] it is easy to see that L = π−1  2 (q × q′) and R = π−1  1 (m × m′), where ×  stands for the standard (formal) cross product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Equivalently, L = π−1  2 (Sing(π2(A)) and  R = π−1  1 (Sing(π1(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We say that three disjoint smooth conics are collinear if they  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Any two smooth conics are connected by a curve of bidegree  (1, 0) and by a curve of (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  intersect the same (1, 0) curve L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The fibers of the twistor projection π : F → P2 (see [4, Section 5]) form a subset T  of the family of conics C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The twistor fibers are also characterized to be the irreducible  elements in C that are fixed by the anti-holomorphic involution j : F → F defined as  j(p, ℓ) = (ℓ, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Being the set of fixed point of j, a curve Lq,m belongs to T if and only if m = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover,  the set T is a Zariski dense in C (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' [3, Section 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If L is the curve of bidegree (1, 0) connecting two different twistor fibers  (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5), then the curve of bidegree (0, 1) connecting them is exactly R = j(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Hence if three twistor fibers are collinear, then they intersect the same (1, 0) curve and  the same (0, 1) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Recall from the introduction that, for any positive integer n, C(n) denotes the 4n-  dimensional set of n pairwise disjoint elements of C and T (n) the set of n pairwise disjoint  elements of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As before, T (n) is a Zariski dense in C(n) (see again [3, Section 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We  now introduce the following crucial definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  R  L  q,m  q",m6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For any n ≥ 1 let C∗(n) be the set of all A ∈ C(n) such that for any curve  L of bidegree (1, 0), it holds #(L ∩ D) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Set T ∗(n) := T (n) ∩ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Clearly we have C∗(n) = C(n) and T ∗(n) = T (n), for n = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For n ≥ 3 the set  C(n) \\ C∗(n) parametrizes unions of n disjoint smooth conics such that at least three of  them are collinear, hence C∗(n) is an open Zariski dense in C(n), as well as T ∗(n) in T (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Therefore, for any n ≥ 1, all the following inclusions are Zariski dense: T ∗(n) ⊂ T (n) ⊂  C(n), T ∗(n) ⊂ C∗(n) ⊂ C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Surfaces of bidegree (1, 0) and (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We now turn our attention back to sur-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We recall from [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2] and [3, Section 2] that (1, 0) and (0, 1) surfaces are  Hirzebruch surfaces of first type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, a surface X of bidegree (1, 0) can be seen  as the lift, via π1 of a line (and analogously for a bidegree (0, 1) surface Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Using this  description, it is easy to see that any of these surfaces represent the blow up of P2 at a  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let F1 be a Hirzebruch surface of type 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' we now describe the relation between the  generators of the Picard group of F1 and the family of curves in F previously described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We recall that Pic(F1) = Zh ⊕ Zf, where  h2 = −1,  f2 = 0,  hf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For the following analysis and the rest of the paper X will denote a surface  of bidegree (1, 0), while Y one of bidegree (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Identifying a surface X with F1 we obtain that OX(1, 0) ≃ OF1(f) which in turn corre-  sponds to the set of curves in F of bidegree (0,1) contained in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' On the other hand we  have that OX(0, 1) ≃ OF1(h + f) which corresponds to elements of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Hence, for any a, b ∈ Z and for any α, β ∈ Z, we obtain the following two relations  (4)  OX(a, b) ∼= OF1(bh + (a + b)f),  and  OF1(αh + βf) ∼= OX(β − α, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  For a surface Y of bidegree (0, 1) we can derive similar formulæ:  (5)  OY (a, b) ∼= OF1(ah + (a + b)f),  and  OF1(αh + βf) ∼= OY (α, β − α),  for any a, b ∈ Z and for any α, β ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let X be a surface of bidegree (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then X does not contain any element  of C(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In fact, any element of C in X corresponds to an element of OF1(h + f) and any  two elements of type h + f meets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The same holds for surfaces Y of bidegree (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In particular, for each bidegree (1, 0) or (0, 1) surface there is exactly one twistor fiber  contained in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We now recall from [3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5] that, for any a, b ≥ 0, using the exact sequence  0 → OF(a − 1, b) → OF(a, b) → OX(a, b) → 0,  and its analogous for Y , we have that  h0(OX(a, b)) = a(b + 1) +  �b + 2  2  �  ,  h0(OY (a, b)) = b(a + 1) +  �a + 2  2  �  ,  while  h1(OX(a, b)) = h1(OY (a, b)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Moreover, if a > 0, b > 0, the line bundles OX(a, b) and OY (a, b) are very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Surfaces of bidegree (0, d) and (1, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We now pass to study higher bidegree sur-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We start with some consideration about bidegree (0, d) surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As described in [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3], any integral surface S of bidegree (0, d)  is equal to π−1  1 (C) for some degree d integral curve C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore, for d ≥ 2, no integral  S ∈ |OF(0, d)| contains a smooth conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Otherwise, thanks to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4, πi(S) would  contain a line, but π1(S) is an integral curve of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  For n ≥ 2 we now compute how many (non integral) bidegree (0, d) surfaces contain a  fixed element of C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' First we set the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We will denote by IU,V the ideal sheaf of a scheme U contained in a  projective variety V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' whenever V = F we will omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' So, in particular, if A ∈ C(n) we  will write IA := IA,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix d ≥ 0, n ≥ 1, and A ∈ C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We have  h0(IA(0, d)) = (d − n + 2)(d − n + 1)  2  and  h1(IA(0, d)) =  � n(n−1)  2  if n ≤ d + 1  n(d + 1) − (d+2)(d+1)  2  if n ≥ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Recall that OF(0, d) = π∗  1(OP2(d)), and IA(0, d) = π∗  1(IT,P2(d)) where T = π1(A)  is a union of n distinct lines in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In general, we have that:  h0(OF(0, d)) =  �d + 2  2  �  ,  h0(IA(0, d)) =  �d + 2 − n  2  �  ,  h0(OA(0, d)) = n(d + 1),  hence, using the exact sequence  0 → IA(0, d) → OF(0, d) → OA(0, d) → 0,  since h1(OF(0, d)) = 0 and  �d+2−n  2  �  = 0 if n ≥ d + 1, we get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As a direct consequence of the previous lemma, we can state that, for any  C ∈ C, there is only one surface Y in |IC(0, 1)| and, analogously, only one surface X in  |IC(1, 0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If A ∈ C(n), we consider the following exact sequence:  0 → IA(a, b) → OF(a, b) → OA(a, b) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since the conics in A are all disjoint, we have that for any (a, b) ∈ N2  (6)  h0(OA(a, b)) = n(a + b + 1),  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' [3, Sequence (7) and proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Recall from Formula (2) that  h0(OF(a, b)) = (a+1)(b+1)(a+b+2)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore, as h1(OA(a, b)) = h1(OF(a, b)) = 0, for any  A ∈ C(n) and d ≥ 0, we have  (7)  χ(IA(1, d)) = h0(IA(1, d)) − h1(IA(1, d)) = (d + 1)(d + 3) − n(d + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Hence, if n = d + 2, we have h0(OA(1, d)) = (d + 2)2 and so  (8)  h0(OF(1, d)) = (d + 1)(d + 3) = (d + 2)2 − 1 = h0(OA(1, d)) − 1  and, if n = d + 1:  h0(OF(1, d)) = h0(OA(1, d)) + d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In the following we will implicitly make use of the following simple observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since the elements of T (1) are the fibers of a map, the twistor map, each  p ∈ F is contained in a unique element, C, of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since j(C) = C, j(p) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Analogously,  note that p is contained in a unique curve of bidegree (1, 0) and a unique curve of bidegree  (0, 1), π−1  2 (π2(p)) and π−1  1 (π1(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The following remark will be used several times in the next pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix a positive integer d and an integral S ∈ |OF(1, d)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since π1|X is  birational onto its image, S is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By B´ezout Theorem, for any p ∈ P2, the bidegree  (1, 0) curve π−1  2 (p) either is contained in S, or intersects S in a single point (scheme-  theoretically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We close this subsection with a technical result that will be used in the next pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In [3, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1] we proved that there exists at most one surface of bidegree (a, b)  containing a number equal or greater than a2 + ab + b2 of smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We will now  generalize this result to a more general context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix (a, b, c, d) ∈ N4 such that (a, b) ̸= (0, 0), c > 0, d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take  A ∈ C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume the existence of an integral S ∈ |OF(a, b)| containing A and assume  one of the following conditions:  (1) ad + b(c + d) < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (2) a(c + d) + bc < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (3) ad + b(c + d) = a(c + d) + bc = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Then each element of |IA(c, d)| contains S and in particular c ≥ a and d ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume by contradiction the existence of S′ ∈ |IA(c, d)| such that S′ ⊉ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We have  OF(a, b) · OF(c, d) =acOF(1, 0) · OF(1, 0) + (ad + bc)OF(1, 0) · OF(0, 1)  + bdOF(0, 1) · OF(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since S′ ⊉ S, the intersection S ∩ S′ is a curve of bidegree (ad + b(c + d), a(c + d) + bc)  (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since c > 0 and d > 0, then OF(c, d) is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, since S is  irreducible, A has bidegree (n, n) and A is not connected, then the intersection contains  some more component in addition to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence, either ad+bc+bd > n and ac+ad+bc ≥ n  or ad + bc + bd ≥ n and ac + ad + bc > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Non-collinear smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We now want to characterize the conics in C∗(n) in  terms of cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We start showing that the vanishing of certain cohomology groups  implies that an element A ∈ C(n) lies in C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix d ≥ 0, 3 ≤ n ≤ d+1 and A ∈ C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If h1(IA(1, d)) = 0 then A ∈ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume, by contradiction, that there exists a curve L of bidegree (1, 0) such that  #(L ∩ A) ≥ 3 and consider the exact sequence defining IA:  (9)  0 → IA(1, d) → OF(1, d) → OA(1, d) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We will prove that the restriction map H0(OF(1, d)) → H0(OA(1, d)) is not surjective,  which implies that h1(IA(1, d)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In fact, assume that it is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then, consider  the following diagram  OF(1, d)  OL(1, d)  OA(1, d)  OL∩A(1, d)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
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+page_content='.  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  As the vertical maps are surjective, then the map H0(OF(1, d)) → H0(OA∩L(1, d)), induced  by the composition, is surjective too and hence of rank at least 3 (since L∩A has cardinality  at least 3 and the irreducible components of A are pairwise disjoint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' On the other hand the  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  9  restriction map H0(OF(1, d)) → H0(OL(1, d)) has rank 2 and this gives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  Before completing the characterization of the conics in C∗(d + 1), we expose a general  construction that will be used in several following discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let A ∈ C(n) and let C be  any connected component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Set B := A \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then, for any a, b ≥ 0, if Y ∈ |OC(0, 1)|,  we have the following residual exact sequence  0 → IResY (A)(a, b − 1) → IA(a, b) → IA∩Y,Y (a, b) → 0,  but as ResY (A) = B and A ∩ Y = (B ∩ Y ) ∪ C, we have  (10)  0 → IB(a, b − 1) → IA(a, b) → I(B∩Y )∪C,Y (a, b) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Clearly an analogous sequence can be written for X ∈ |OC(1, 0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix d ≥ 0 and A ∈ C(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We have A ∈ C∗(d + 1) if and only if  h1(IA(1, d)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By the previous lemma we need only to prove that A ∈ C∗(d+1) satisfies h1(IA(1, d)) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We use induction on d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The case d = 0 is true by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore, we  may assume d > 0 and use induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let C be a connected component of A, set  B := A \\ C and cal Y the only element of |IC(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Consider the residual exact sequence  (10), with a = 1 and b = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since A ∈ C(d + 1), C ∩ B = ∅ and B ∩ Y is formed by d  different points, up to the identification of D with F1 we have  I(B∩Y )∪C,Y (1, d) ∼= I(B∩Y )∪C,F1(1, d)(h + (d + 1)f) ∼= IB∩Y,F1(df) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Using (10) and induction we are left to prove that h1(IB∩Y,F1(df)) = 0 if and only if  A ∈ C∗(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Consider now the following exact sequence  (11)  0 → IB∩Y,F1(df) → OF1(df) → OB∩Y (df) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since h0(OF1(df)) = d + 1 and h0(OB∩Y (df)) = d, we have h1(IB∩Y,F1(df)) > 0 if  and only if h0(IB∩Y,F1(df)) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The last inequality means that there exist at least two  different sets of d fibers containing the set of d points B ∩Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' This is equivalent to the fact  that there exists a fiber L ∈ |f| such that #(B ∩ L) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since L is a curve of bidegree  (1, 0) in F, then L · Y = 0 in the intersection ring of F, therefore L ⊂ Y and hence we get  L ∩ C ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus #(L ∩ A) ≥ 3, which means that A ̸∈ C∗(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix d ≥ 0, 0 ≤ n ≤ d + 1 and A ∈ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then h1(IA(1, d)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In  particular, for n = 0, 1, 2 and for any A ∈ C(n), we have  h1(IA(1, 1)) = 0  and  h0(IA(1, 1)) = 8 − 3n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By using [3, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3], we easily get the first part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The second  one, follows from Formula (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  We point out that since T (n) is a Zariski dense of C(n), then the characterization given  by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='19 also holds for the set T ∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix an integer d ≥ 0 and A ∈ C∗(d + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then h1(IA(1, d)) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The lemma is true for d = 0, because h0(IA(1, 0)) = 0 (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We assume  d > 0 and use induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let C be a connected component of A, set B := A \\ C  and call Y the only element of |IC(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Consider the residual exact sequence (10), with  a = 1 and b = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since A ∈ C(d + 2), C ∩ B = ∅ and B ∩ Y is formed by d + 1 different  points, up to the identification of D with F1 we have  I(B∩Y )∪C,Y (1, d) ∼= I(B∩Y )∪C,F1(1, d)(h + (d + 1)f) ∼= IB∩Y,F1(df) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Using (10) and induction we are left to prove that h1(IB∩Y,F1(df)) = 0 if A ∈ C∗(d + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  Consider now the following exact sequence  (12)  0 → IB∩Y,F1(df) → OF1(df) → OB∩Y (df) → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since h0(OF1(df)) = d + 1 and h0(OB∩Y (df)) = d + 1, we have h1(IB∩Y,F1) > 0 if and  only if h0(IB∩Y,F1(df)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' This is equivalent to the fact that there exists a fiber L ∈ |f|  such that #(B ∩ L) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since L is a curve of bidegree (1, 0) in F, then L · Y = 0 in the  intersection ring of F, therefore L ⊂ Y and hence we get L ∩ C ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus #(L ∩ A) ≥ 3,  which means that A ̸∈ C∗(d + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  As said in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5, for any element A ∈ C(2), there is a unique curve L of bidegree  (1, 0) and a unique R of bidegree (0, 1) such that both intersect the elements of A at a  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As described in the following result, it turns out that A ∪ L ∪ R is the base locus  of |IA(1, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For any A ∈ C(2), we have that  (1) the general element in |IA(1, 1)| is integral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (2) the base locus B of |IA(1, 1)| is A∪L∪R, where L is and R are the curves described  in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since A ∈ C(2), by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='20 we have h1(IA(1, 1)) = 0 and h0(IA(1, 1)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Call C1 and C2 the connected components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Denote by Xi the only element of |OF(1, 0)|  containing Ci and by Yi the only element of |OF(0, 1)| containing Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The surfaces X1 ∪Y2  and X2 ∪ Y1 are the only reducible elements of |IA(1, 1)| and hence, the general element  in |IA(1, 1)| is irreducible and (1) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  To prove (2) we analyze the base locus B of |IA(1, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If S, S′ ∈ |IA(1, 1)| are irreducible  and S ̸= S′, then the one-dimensional cycle S ∩ S′ has bidegree (3, 3) and it contains A,  which has bidegree (2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take R := X1 ∩ X2 and L := Y1 ∩ Y2, where Xi and Yi are the  surfaces defined in the first part of this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The curves L and R are exactly the ones  stated in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, #(L ∩ A) = #(R ∩ A) = 2 and hence, by B´ezout and  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='16, L∪R ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, recall that the reducible surfaces, X1 ∪Y2 and X2 ∪Y1  belong to |IA(1, 1)| and their intersection (X1 ∪ Y2) ∩ (X2 ∪ Y1) is A ∪ L ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence the  base locus of |IA(1, 1)| is exactly B = A ∪ L ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By generalizing the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='22 we can say something about  the base locus of IA(1, d), for A ∈ C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix integers d > 0 and n ≥ 2 and take any  A ∈ C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then, the base locus B of IA(1, d) contains all curves L of bidegree (1, 0) such  that #(L ∩ A) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If A ∈ C∗(n), then there are exactly  �n  2  �  such curves L (the number of  lines joining two points in a set of n general points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If A ∈ T (n), then #(j(L) ∩ A) ≥ 2  for all L such that #(L ∩ A) ≥ 2, since j(A) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Surfaces of bidegree (1, d)  In this section we prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, we give several results for  the case of a surface of bidegree (1, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Later, in the following section, we will specialize  to the cases d = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The case d = 1 was studied in many details in [4] and here we add  a simple lemma useful for what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For any A ∈ T ∗(3), there is no integral surfaces of bidegree (1, 1) containing  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume the existence of an integral surface M of bidegree (1, 1) containing A ∈  T ∗(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  First of all, thanks to [4, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4], we have that M = j(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Then, as  explained at the beginning of Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1 in [4], either M is smooth or reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' But if  M is a smooth j-invariant surface of bidegree (1, 1) containing 3 twistor fibers, then it  contains infinitely many of them and these are parametrized by a circle (see [4, Theorem  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' However, smooth surfaces of bidegree (1, 1) can be seen as the blow-up of P2 at three  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  11  points both via π1 and π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, up to unitary transformation it is possible to  write M as the set {([p0 : p1 : p2], [ℓ0 : ℓ1 : ℓ2]) ∈ F | p1ℓ1 + λp2ℓ2}, with λ ∈ R \\ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In  these coordinates, M contains π−1  µ ([1 : 0 : 0]), π−1  µ ([0 : 1 : 0]), π−1  µ ([0 : 0 : 1]), for µ = 1, 2  and, the family of twistor fibers π−1([q0 : q1 : q2]) defined by:  �  �  �  �  �  q0 = 0 and |q1|2λ + |q2|2 = 0  if λ < 0 ,  q1 = 0 and |q2|2 − |q0|2(λ − 1) = 0  if λ > 1 ,  q2 = 0 and |q1|2λ + |q0|2(λ − 1) = 0  if 0 < λ < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Take for instance λ < 0, then any twistor fiber in M intersects the line L = π−1  2 ([1 : 0 : 0])  of bidegree (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' An analogous consideration holds if 0 < λ < 1 or λ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence we get  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  In the previous lemma we show that an integral (1, 1) surface cannot contain three  general twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  On the other hand if M is a (1, 1) surface containing a given  A ∈ T (3) \\ T ∗(3), then by [4, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3] we have that M is j-invariant, hence either it  is smooth or reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, if M contains infinitely many twistor fibers, then all of  them intersect a bidegree (1, 0) curve L and its associated (0, 1) curve R = j(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In [4, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1] we gave examples of bidegree (1, 1) smooth surfaces  containing exactly 0, 1 or 2 twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As a smooth surface of bidegree (1, 1) is a Del Pezzo surface of degree 6,  then this is characterized either by the three bidegree (1, 0) curves that contains or by the  three bidegree (0, 1) curves that contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In fact, recall that these surfaces represent the  blow-up of P2 at three points with respect to either π1 or π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We notice that if a smooth  surface of bidegree (1, 1) is j-invariant, then, this is uniquely determined by three twistor  fibers contained in it and not by the curves L and R = j(L) (of bidegree (1, 0) and (0, 1),  respectively), which intersect all the twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In fact, in [4], we proved that these  kind of surfaces are uniquely determined by the “circle” defined by the infinite family of  twistor fibers contained in it and, such a “circle” is properly contained in in the (1, 0) line  L which intersects all fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thanks to these first considerations about bidegree (1, 1) surfaces, we are ready to give  the proof of our first main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1: Thanks to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1, the result is true for d = 0  and d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume now that d ≥ 2 and, by contradiction, that S is an integral (1, d) surface  containing A ∈ T ∗(d+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Call C a connected component of A and set B := A\\C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Call Y  the only (by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2) element of |IC(0, 1)| and consider the following exact sequence  (which is a particular case of the one in Formula (10)):  (13)  0 → IB(1, d − 1) → IA(1, d) → I(B∩Y )∪C,Y (1, d) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  From Formulæ (6) and (8), we have h0(OA(1, d)) = (d+2)2 = h0(OF(1, d))+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Clearly,  we have that B ∩ Y is formed by d + 1 points, and, up to the identification of Y with F1  given in Formula (5), as the curve C corresponds to an element of type h+f in F1, we can  write I(B∩Y )∪C,Y (1, d) ∼= I(B∩Y )∪C,F1(h + (d + 1)f) ∼=IB∩Y,F1(df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since A ∈ T ∗(d + 2)  and every element of |f| meets C (indeed (h + f)f = 1), the restriction to B ∩ Y of the  ruling morphism D → P1 associated to |f| is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus, h1(Y, IA∩Y,Y (1, d)) = 0 and  the exact sequence (13) gives h1(IB(1, d−1)) ≥ h1(IA(1, d)) ≥ 2, where the last is greater  or equal than 2 because χ(IA(1, d)) = −1 (see Formula (7)), and we are assuming that  h0(IA(1, d)) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus, we also have h0(IB(1, d − 1)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Recall that B ∈ T ∗(d+1) and hence, by the inductive assumption, B is not contained in  any integral E ∈ |OF(1, d − 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore, thanks to Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='10, there must be  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  an integral M ∈ |OF(1, 1)| containing at least 3 connected components of B, say B′ ⊂ M  with B′ of bidegree (3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1, there is a curve L of bidegree (1, 0) such  that #(L ∩ B′) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus A /∈ T ∗(d + 2), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  Having proved that an integral bidegree (1, d) surface cannot contains d + 2 (or more)  non collinear twistor fibers, we now pass to prove that all the other cases can actually  arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, we prove in the following result a stronger version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2 in  the case n ≤ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix integers d ≥ 1 and 0 ≤ n ≤ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then, for any A ∈ T ∗(n) there is  an integral S ∈ |OF(1, d)| containing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, the general S ∈ |IA(1, d)| contains no  other twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We use induction on the integer d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If d = 1 the statement is true by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume now d ≥ 2 and take an element A ∈ T ∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since T ∗(n) is Zariski dense in  C(n), A has the bigraded Hilbert function of a general element of C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus, thanks to  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='20, we have that h1(IA(1, d)) = 0 and, by Formula (7)  h0(IA(1, d)) = (d + 1)(d + 3) − n(d + 2) =: Nn + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Fix a connected component C of A and set B := A \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  If Y denotes be the only  (0, 1) surface containing C, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='20 entails that h1(IB(1, d − 1)) = 0 and, again by  Formula 7  h0(IB(1, d − 1)) = d(d + 2) − (n − 1)(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  By the inductive assumption, we know that |IB(1, d − 1)| ̸= ∅ and a general W ∈  |IB(1, d − 1)| is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thus Y ∪ W ∈ |IA(1, d)| and Y ∪ W has 2 irreducible  components, one of them having bidegree (1, d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Let us denote by C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , Cn the connected components of A, by Bi := A \\ Ci and by Yi  the unique element in |ICi(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The set of all the reducible surfaces W ∪ Yi ∈ |IA(1, d)|,  where W ∈ |IBi(1, d − 1)|, is the union of e projective spaces (one for each choice of  Ci), each of them of codimension h0(IA(1, d)) − h0(IB(1, d − 1)) = d + 2 − n > 0 in  |IA(1, d)|= PNn (in particular of codimension 1 if n = d+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore, they do not cover  all |IA(1, d)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We now want to exclude other possible splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, we consider reducible  surfaces of the form W1 ∪ D1 with W1 integral, D1 possible reducible of bidegree (0, x) for  some x ≥ 2 and hence W1 of bidegree (1, d − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='10 shows that only irreducible  components of D1 of bidegree (0, 1) may contain some component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We obtain that  the surface is of the form W1 ∪D2 ∪D3 with W1 ∪D2 of bidegree (1, d−1), but, as showed  before, these kind of surfaces do not cover all |IA(1, d)|, hence we have the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Now we prove that a general S ∈ |IA(1, d)| contains no other twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We start  by analyzing the case d = 2 and discussing the cases n = 1, 2, 3 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix C ∈ T (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let CC(1) denote the set of all B ∈ C(1) such that B∩C =  ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Note that T (1) \\ {C} = CC(1) ∩ T (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For any B ∈ CC(1) we have h1(IC∪B(1, 2)) = 0  and hence h0(IC∪B(1, 2)) = h0(IC(1, 2)) − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let XC be the set of all smooth and integral  surfaces of bidegree (1, 2) containing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' It is a non-empty Zariski open subset of |IC(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  For any B ∈ CC(1) the set of all S ∈ XC containing B has complex codimension 4 and  hence real codimension 8 as real manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since T (1) has real dimension 4, a general  S ∈ XC contains no other twistor fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Let now n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix A ∈ T (2) and let CA(1) denote the set of all B ∈ C(1) such that  B∩A = ∅ and B∪A ∈ C∗(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Note that h1(IA∪B(1, 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' So as in the previous step, we  get that a sufficiently general S ∈ |IA(1, 2)| contains no element of TA(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume that S  contains B ∈ T (1) such that there is a curve of bidegree (1, 0) intersecting each connected  component of A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Note that L is uniquely determined by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let C(A, L) denote the  set of all B ∈ C(1) such that B ∩ A = ∅ and L meets B and set T (A, L) := C(A, L) ∩ T (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  13  For any o ∈ L \\ (L ∩ C) the set of all B ∈ C(A, L) containing o is a non-empty family of  complex dimension 1, while there is a unique twistor fiber containing o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The set C(A, L) is  a complex manifold of dimension 2, while h1(IA∪B(1, 2)) = 1 and hence h0(IA∪B(1, 2)) =  h0(IA(1, 2)) − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since dim C(A, L) = 2, a general S ∈ |IA(1, 2)| contains no element of  C(A, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence, there is no twistor fiber B such that A ∪ B ∈ T ∗(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume now that n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Start by considering a general A ∈ T ∗(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then we have  h1(IA(1, 2)) = 0 and h0(IA(1, 2)) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For any x ∈ {0, 1, 2, 3} let C(A, x) denote the set of  all C ∈ C(1) such that A∩C = ∅ and h0(IA∪C(1, 2)) = x and set T (A, x) := C(A, x)∩T (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  For a general D ∈ C(4) we have h0(ID(1, 1)) = 0 (but h1(ID(1, 1)) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix C ∈ C(1)  such that C ∩ A = ∅, call Y the only element of |IC(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since C ∩ A = ∅, no connected  component of A is contained in Y and Y ∩ A is formed by 3 points, all of them in Y \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We easily deal with the case x = 0 as any curve C ∈ C(A, 0) is not contained in any  element of |IA(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We have OY (1, 2)(−C) ∼= OF1(2f) and hence C ∈ C(A, 0) if no curve  of bidegree (1, 0) L ∈ |f| intersects 2 of the components of A (since h1(IA(1, 1)) = 1 the  last statement is only an “if” and not an “if and only if”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' A necessary condition for  being C ∈ C(A, 2) is that L intersects all connected components of A, but this is excluded  because A ∈ T ∗(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since A ∈ T ∗(3) there are exactly 3 curves L1, L2, L3 of bidegree (1, 0)  intersecting 2 of the connected components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We claim that a general S ∈ |IA(1, 2)|  contains no C ∈ C(1) such that C ∩ A = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The family of smooth conics which intersect  Li has complex dimension 2, while the family of twistor fibers intersecting Li has real  dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As the general S ∈ |IA(1, 2)| has only finitely many conics, it only has a  finite number of elements in C(A, 1) and, for the general, none of them is a twistor fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We now pass to analyze the case d ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume the general surface of |IA(1, d)| contains  the twistor fiber C ⊈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus A∩C = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Set A′ := A∪C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take Y ∈ |OY (0, 1)| containing  C and consider the residual exact sequence  (14)  0 → IA(1, d − 1) → IA′(1, d) → I(Y ∩A)∪C,Y (1, d) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We have IC,Y (1, d) ∼= OF1(df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume first n ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If A′ ∈ T ∗(n + 1), then h1(I′  A(1, d)) = 0 and hence h0(I′  A(1, d)) =  h0(IA(1, d)) − d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since dim C(1) = 4, for d ≥ 3 the general S ∈ |IA(1, d)| contains no  C such that A ∪ C ∈ T ∗(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Now assume A′ /∈ T ∗(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus there are connected components C′ and C′′ of A such  that C ∪ C′ ∪ C′′ /∈ T ∗(3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C intersects the unique line L meeting C′ and C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since  dim L = 1, to exclude this case it is sufficient to prove that h0(IB(1, d)) ≤ h0(IA(1, d))−2,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  h0(F1, IA∩Y (df)) ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  But since h0(OF1(df)) = d + 1, then OF1(df) is globally  generated and A ∩ Y ̸= ∅, therefore h0(F1, IA∩Y (df)) ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume now n = d+1 and that A′ ∈ C∗(d+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='21 we have h1(IA′(1, d)) ≤ 1  and hence h0(IA′(1, d)) ≤ h0(IA(1, d)) − d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Now assume A′ /∈ C∗(d + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We need  h0(F1, IA∩Y (df)) ≤ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let π : F1 → P1 denote the ruling of F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since OP1(d) is very  ample, h0(F1, IA∩Y (df)) ≤ d − 1 if and only if #π(A ∩ Y ) ≥ 2, which is true because  #(A∩Y ) = d+1 ≥ 3 and (since A ∈ C∗(d+1) no fiber F of π contains at least 3 points of  A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The only remaining case is now d = 3 and n = 4, which can be dealt just by adapting  the previous argument for d = 2 and n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  Having proved Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2 in the case n ≤ d + 1 for non collinear twistor fibers, we  focus now to the case n = d + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In this setting we need some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix d > 0, n ≤ d+2 and consider a general A ∈ T (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then h1(IA(1, d)) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since T (n) is Zariski dense in C(n), it is sufficient to prove the statement for a  general A ∈ C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Clearly, it is sufficient to prove the case n = d + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take a connected  component C of A and set B := A \\ C and Y ∈ |IC(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Consider the residual exact  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  sequence of Y , as in Formula (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Recall the correspondence Y ∼= F1 given in Formula (5)  and let ρ : Y → P1 denote its ruling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since A is general #ρ(A ∩ Y ) = d + 1 and hence  hi(F1, IB∩Y (df)) = 0, for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore h1(IA(1, d)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1, we already know that an integral (1, d) surface may contain a union of  d + 2 twistor fibers only if it belongs to T (d + 2) \\ T ∗(d + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore, we introduce the  following notation for sets of disjoint smooth conics which are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Given a curve L  of bidegree (1, 0) and an integer n > 0, let C(n, L) denote the set of all A ∈ C(n) such that  each connected component of A meets L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The set C(n, L) is isomorphic (as real algebraic  variety) to the set S(L, n) of all subsets of L with cardinality n and hence it is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  An analogous definition and observation can be done for a curve R of bidegree (0, 1) and,  of course, for the family T instead of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix integers n ≥ 3 and d ≥ 1 and take any A ∈ T (n, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then,  h1(IA(1, d)) ≥ n − 2 + max{0, n − (d + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Call C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , Cn the connected component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let Si ⊂ Ci be any union of d + 2  distinct points on each conic and S := S1 ∪ · · · ∪ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since Ci is a smooth rational curve,  the restriction map H0(OCi(1, d)) → H0(OSi(1, d)) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus, the restriction map  H0(OA(1, d)) → H0(OS(1, d)) is bijective and χ(OA(1, d)) = χ(OS(1, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus we get  χ(IA(1, d)) = χ(IS(1, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since A ∈ T (n, L), we know that there exists L of bidegree (1, 0) which intersects each  conic in A (and the (0, 1) curve j(L) does the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence we can choose S such that n  points are on L and n points are on j(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In other words we assume (A∩L)∪(A∩j(L)) ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  By using B´ezout and the fact that the bidegree is (1, d), we get that (n − 2) + max{0, n −  (d + 1)} of these points may be omitted without changing the set |IS(1, d)|, and hence  H0(IS(1, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' It follows that h1(IA(1, d)) ≥ n − 2 + max{0, n − (d + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thanks to the previous lemma, if A ∈ T (3, L), for some bidegree (1, 0)  curve L, then h1(IA(1, 1)) ≥ 2, and hence h0(IA(1, 1)) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' However, since no surface of  bidegree (1, 0) or (0, 1) contains an element of T (2), then every S ∈ |IA(1, 1)| is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Therefore, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='17 gives that |IA(1, 1)| = {S} and so, thanks to Formula (7),  h1(IA(1, 1)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Finally, the following result completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2, in the case of d + 2  twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix an integer d ≥ 2 and take a general A ∈ T (d+2, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then h0(IA(1, d)) ≥  d and the general S ∈ |IA(1, d)| is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='6 with n = d + 2, we get h1(IA(1, d)) ≥ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since  χ(IA(1, d)) = (d+1)(d+3)−(d+2)2 = −1, we get h0(IA(1, d)) ≥ d and hence, |IA(1, d)| ̸=  ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We now prove that a general element in |IA(1, d)| is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Take S ∈ |IA(1, d)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Every surface of bidegree (1, 0) or (0, 1) contains at most one connected component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Therefore, S cannot be the union of a surface of bidegree (1, 0) and d of bidegree (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  No integral surface of bidegree (0, x), for x ≥ 2, contains a twistor fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If d = 2, then  h0(IA(1, 2)) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' However, thanks to Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7, for every choice of C ∈ A, there is only  one M ∈ |I(A\\C)(1, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since we have considered all the possible reducible elements of  |IA(1, 2)|, we get the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Now assume d > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since A has finitely many components, it is sufficient to prove that  for any x ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , d−1}, any union E of x connected components of A and any connected  component C of A \\ E we have  (15)  h0(IE(1, x − 2)) < h0(IE∪C(1, x − 1)),  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  15  and then proceed as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Let Y be the only element of |OF(0, 1)| containing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The exact sequence in Formula (10)  gives h0(IE(1, x−2)) ≤ h0(IE∪C(1, x−1)) and equality holds if and only if Y is in the base  locus B of |IE∪C(1, x − 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Call C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , Cx the components of E and Yi the only surface  of bidegree (0, 1) containing Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='13 the irreducible surfaces Y, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , Yx are  all different one eachother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For a general A we get that each integer h0(IE(1, x−2)) is the  same for all union of x connected components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus if the inequality is false, then B  contains the surface Y ∪ Y1 ∪ · · · ∪ Yx of bidegree (0, x + 1), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Surfaces of bidegree (1, 2) and (1, 3)  In this section we specialize our study to the case of surfaces of bidegree (1, 2) and (1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In particular, we will prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Recall, from Formula (7), that for any A ∈ C(n), we have χ(IA(1, 2)) = 15 − 4n, and  hence, if n ≤ 3, we get h0(IA(1, 2)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We also recall that a general S ∈ |OF(1, 2)|  contains finitely many smooth conics and, thanks to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='19, for every B ∈ C(2) we  have h1(IB(1, 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Surfaces of bidegree (1, 2) containing 0 ≤ n ≤ 4 twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In this section,  we show the existence of a smooth surface of bidegree (1, 2) containing exactly 0, 1, 2, 3 or  4 twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In order to analyze the space |IA(1, 2)| when A is in C(n) (or in T (n)),  for 0 ≤ n ≤ 4, we will need some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Note that the extremal case, when  n = 4, will be treated in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We start by considering (1, 2)-surfaces containing three disjoint smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take A ∈ C(3) such that h0(IA(1, 1)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then  (1) there exists a curve L of bidegree (1, 0) and a curve R of bidegree (0, 1) such that  A ∈ C(3, L) and A ∈ C(3, R)  (2) there is an integral element in |IA(1, 2)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (3) h1(IA(1, 2)) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (4) the base locus B of |IA(1, 2)| is A ∪ L ∪ R, where L is and R are the curves defined  in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We start arguing as in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thanks to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9, each surface of bide-  gree (1, 0) or (0, 1) does not contain any element of C(2), hence, as A ∈ C(3), any ele-  ment in |IA(1, 1)| is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='17 (with a = b = c = d = 1) gives that  h0(IA(1, 1)) = 1 and hence we set |IA(1, 1)| = {M} and by Formula (7) we compute  h1(IA(1, 1)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We prove now the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let C be a connected component of A, set B := A\\C  and denote by X the only element of |IC(1, 0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Performing the same construction that  leads to Formula (10), we have  0 → IB(0, 1) → IA(1, 1) → IA∩X,X(1, 1) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since h0(IB(0, 1)) = 0 and h0(IA(1, 1)) = 1, the previous residual exact sequence gives  (16)  h0(IA∩X,X(1, 1)) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thanks to Formula (4), we have that OX(1, 1) ≃ OF1(h + 2f);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' moreover, recall from  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9 that C is identified with an element of |OF1(h + f)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since A ∩ X is the union  of C and the two points B ∩X, there is a fiber L ∈ |f| of the ruling of F1 containing B ∩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since f(h + f) = 1, we have that L meets C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus L meets each connected component of  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Taking instead of X the only element of |IC(0, 1)| we get the existence of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We now prove (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We will show that there is an integral element in |IA(1, 2)| by  showing that the possible reducible cases do not cover the whole family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='10  shows that A is not contained in a surface of bidegree (1, 2) with an irreducible component  16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  of bidegree (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9, a bidegree (0, 1) or (0, 1) surface does not contain any  element of C(n), with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence, there are only finitely many elements of |OF(1, 2)|  with at least 3 irreducible components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since h0(OF(0, 1)) = 3, the set of all reducible elements of |OF(1, 2)| with an irreducible  component of bidegree (1, 1) containing A is isomorphic to P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus, in order to prove the  existence of an integral element in |IA(1, 2)|, it is sufficient to prove that h0(IA(1, 2)) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  But, using the exact sequence (9), this is equivalent to prove that h1(IA(1, 2)) ≥ 1, and  the last inequality is true, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='19 because #(L ∩ A) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  To prove (3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' h1(IA(1, 2)) = 1, it is sufficient to prove that h1(IA(1, 2)) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As  before, take a connected component C of A and set B = A\\C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let Y be the only element  of |IC(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In the identification (5) of Y with F1 we have  I(B∩Y )∪C,Y (1, 2) ∼= I(B∩Y )∪C,F1(h + 3f) ∼= O(B∩Y ),F1(2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since #(B ∩ Y ) = 2 and OF1(2f) is globally generated, h1(F1, IB∩Y,F1(2f)) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We have  ResY (A) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thanks to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='19, we have h1(IB(1, 1)) = 0, and the residual exact  sequence of Y  0 → IB(1, 1) → IA(1, 2) → I(B∩Y )∪C,Y (1, 2) → 0,  gives h1(IA(1, 2)) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Finally, we discuss the base locus of |IA(1, 2)| in order to prove (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' First of all, for any  surface S ∈ |IA(1, 2)|, we clearly have A ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, as #(L∩A) = 3 and #(R∩A) = 3,  then by B´ezout, both curves are contained in S: in fact, thanks to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3 and  Formula (3), the general intersection between a curve of bidegree (1, 0) and S consists of  one point while the intersection of a curve of bidegree (0, 1) and S consists of two points  (see also Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore, L ∪ R ⊂ S and A ∪ L ∪ R ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We now prove that B ⊂ A∪L∪R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix p ∈ B\\(A∪L∪R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take a connected component Ci,  i = 1, 2, 3, of A and set Bi := A\\Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let Yi be the only element of |OF(0, 1)| containing Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='22 Bi ∪L∪R is the base locus of |IBi(1, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus there is Si ∈ |IBi(1, 1)|  such that p /∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If p /∈ Yi, then p /∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since S1∩S2∩S3 = L∪R, we may take i ∈ {1, 2, 3}  such that p /∈ Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus B = A ∪ L ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  The following remark shows that if A ∈ C(3) satisfies the condition (1) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1,  then the existence of a (1, 1)-surface containing A is granted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular, there exists a  (1, 1)-surface containing any triplets of collinear twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take A ∈ C(3) and assume the existence of curves L of bidegree (1, 0) and  R of bidegree (0, 1) intersecting each connected component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By adapting the proof of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='6, since #(L ∩ A) = 3 and #(R ∩ A) = 3, we have that h1(IA(1, 1)) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus  h0(IA(1, 1)) ≥ 1 and A satisfies the assumptions of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We can even be more specific and say that if A ∈ C(3) (with no assumption on L or R),  then h0(IA(1, 1)) ≤ 1 and if |IA(1, 1)| ̸= ∅, then the only element of |IA(1, 1)| is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  This is true because, thanks to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9, any reducible element of |OF(1, 1)| contains  at most 2 disjoint smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Note that if A ∈ T (3) and L exists, then we may take R := j(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus if A ∈ T (3) to  get h0(IA(1, 1)) > 0 it is sufficient to assume A /∈ T ∗(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The following lemma is a sort of vice versa of the previous remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take A ∈ C∗(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then h0(IA(1, 1)) = 0 and h1(IA(1, 1)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If A ∈ C(3), thanks to Formula (7), χ(IA(1, 1)) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence h0(IA(1, 1)) = 0 if  and only if h1(IA(1, 1)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We assume h0(IA(1, 1)) ̸= 0 and will prove that A /∈ C∗(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Let B ⊂ A be the union of 2 connected components of A and set C := A \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let L and  R be the curves defined in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5 for B ∈ C(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take any element D ∈ |IB(1, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  17  Since #(B ∩ (L ∪ R)) = 2, B ⊂ D and D has bidegree (1, 1), then B´ezout theorem implies  L ∪ R ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='19 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='22, we have h1(IB(1, 1)) = 0, h0(IB(1, 1)) = 2, and  the general element M in |IB(1, 1)| is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since h0(IB(1, 1)) = 2 and M is general,  C ⊈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Consider the following residual exact sequence:  (17)  0 → IC → IA(1, 1) → IB∪(M∩C),M(1, 1) → 0  Since M ∈ |IB(1, 1)| and h1(OF) = 0, the exact sequence  0 → OF → IB(1, 1) → IB,M(1, 1) → 0  gives h0(M, IB,M(1, 1)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, h0(IC) = 0, and so the sequence (17) and the  assumption h0(IA(1, 1)) ≥ 1 imply h0(M, IB∪(M∩C),M(1, 1)) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='22, the  curve A∪L∪R is the base locus of |IB(1, 1)| and hence the base locus of H0(M, IB,M(1, 1))  is the curve B ∪ L ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since B ∩ C = ∅, the degree 2 scheme C ∩ M is contained in  L ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' To get A /∈ C∗(3) we need to prove that C ∩ L ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' It is sufficient to observe that  deg(C ∩ T) ≤ 1 for any curve T of bidegree (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Indeed, this is true by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3 and  the fact that C is the intersection of a surface of bidegree (1, 0) and a surface of bidegree  (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  We now discuss the case of A ∈ C(2) contained in a smooth bidegree (1, 1) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In  this case we will also prove smoothness for the general element in |IA(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take any A ∈ C(2) contained in a smooth element of |OF(1, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then  we have:  (1) h1(IA(1, 2)) = 0 and h0(IA(1, 2)) = 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (2) the set A ∪ L is contained in the base locus B of |IA(1, 2)|, where L is the bidegree  (1, 0) curve described in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (3) a general S ∈ |IA(1, 2)| is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' To prove (1) it is sufficient to apply Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='20 giving h1(IA(1, 2)) = 0 and  Formula (7), which entails h0(IA(1, 2)) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We now pass to point (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take L and R as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since OF(0, 1) is globally  generated, B ⊆ A ∪ L ∪ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' moreover, thanks to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='16 we also have A ∪ L ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We are left to prove (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  By Bertini’s theorem Sing(S) ⊆ A ∪ L ∪ R for a general  S ∈ |IA(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix a smooth M ∈ |IA(1, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take a general Y ′ ∈ |OF(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since Y ′  is general L ∩ Y ′ = ∅ (and hence it is not singular at any p ∈ L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus, up to small  deformation, we can say that S (which is general) is smooth in a neighborhood of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We  are left to exclude the case Sing(S) ⊆ A∪R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix p ∈ A∪R and let 2p be the 0-dimensional  scheme of F defined by the ideal I2  p,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' S is singular at p if and only if S ∈ |I2p∪A(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  To conclude our proof we need to prove that  h0(I2p∪A(1, 2)) = h0(IA(1, 2)) − 2,  for all p ∈ (A ∪ R) \\ A ∩ R and that, for p ∈ A ∩ R, h0(I2p∪A(1, 2)) < h0(IA(1, 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' These  two statements give the thesis because (A ∪ R) \\ A ∩ R and A ∩ R are 1-dimensional and  0-dimensional, respectively, and we are saying that the set of bidegree (1, 2) surfaces con-  taining A and a singular points has codimension 2 in the first case and positive codimension  in the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Let us start by taking p ∈ (A ∪ R) \\ A ∩ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since p is a smooth point of A ∪ R,  deg(2p ∩ (A ∪ R)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Consider the exact sequence  (18)  0 → I(A∪R)∪2p(1, 2) → IA∪R(1, 2) → IA∪R ⊗ O2p(1, 2) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  Since deg(2p) = 4 and A is smooth, we have h0(IA∪R ⊗ O2p(1, 2)) = 2 if p ∈ A ∪ R and  h0(IA ⊗ O2p(1, 2)) = 4 if p ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence it is sufficient to prove that  h1(I(A∪R)∪2p(1, 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  First of all, assume that p ∈ A \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let C be the connected component of A containing  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Set the following notation E := A \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As R is in the base locus of IA(1, 1) we have  that h0(IA(1, 1)) = h0(IA∪R(1, 1)) (see [3, proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, thanks to  part (1) and to [3, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3], we have h0(IA∪R(1, 1)) = h0(IA(1, 1)) = h0(IE(1, 1)) − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thus p is not in the base locus of |IE(1, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix M ∈ |IE(1, 1)| such that p /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let  Y be the surface of |OF(0, 1)| containing C and consider the residual exact sequence with  respect to Y :  (19)  0 → IE∪p(1, 1) → IA∪2p(1, 2) → I(E∩Y )∪C∪(2p∩Y ),Y (1, 2) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Now we prove that  (20)  h1(IE∪p(1, 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Recall that A = E ∪ C and p ∈ C, hence we have the exact sequence  0 → IA(1, 1) → IE∪p(1, 1) → Ip,C(2) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thanks to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='19 we have that h1(IA(1, 1)) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' on the other hand, since C is a  smooth rational curve, we have h1(Ip,C(2)) = h1(OC(1)) = 0 and this proves (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In order to conclude it is sufficient to prove now that  (21)  h1(I(E∩Y )∪C∪(2p∩Y ),Y (1, 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Note that IC,Y (1, 2) ∼= OF1(2f) ∼= OY (0, 1), hence, by [3, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='11], we know that  IC,Y (1, 2) is very ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Therefore we get h1(IC∪(2p∩Y ),Y (1, 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since E ∩ Y consists  of a point we conclude that (21) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thus the exact sequence (19) gives h1(IA∪2p,F(1, 2)) = 0, concluding the proof in the  case p ∈ A \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Fix p ∈ R \\ (A ∩ R) and ecall that we need to prove that h0(IA∪2p(1, 2)) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix  a general Y ′ ∈ |Ip(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since Y ′ is general, R ⊈ Y ′ (and also L ⊈ Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since Y ′ is  smooth, Y ′ ∩ 2p = (2p, Y ′) is a degree 3 scheme and ResY ′(2p) = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As p ∈ R, we have  that h0(IA∪{p}(1, 1)) = h0(IA(1, 1)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus, by the residual exact sequence of Y ′ it is  sufficient to prove that  h0(Y ′, I(A∩Y ′)∪(2p,Y ′)(1, 2)) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since OY ′(1, 2) is very ample, we have h0(Y ′, I(2p,Y ′)(1, 2)) = h0(Y ′, OY ′(1, 2)) − 3 =  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus it is sufficient to prove that A ∩ Y ′ is not contained in the base locus, B′, of  |O(2p,Y ′)(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In the identification between Y ′ and F1 we have OY ′(1, 2) ∼= OF1(h + 3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Let N be the only element of |f| containing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We have N ∼= P1 and h1(N, I2p∩N(1, 2)) = 0,  but N ⊆ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since OF1(h + 2f) is very ample, Ip(h + 2f) has only p in its base locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thus B′ = N and so, since R ⊈ Y ′ (and also L ⊈ Y ′), B′ cannot contain both points of  A ∩ Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The last case is p ∈ A ∩ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' To prove our claim, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' that h0(I2p∪A(1, 2)) < h0(IA(1, 2)),  it is sufficient to use (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence, a general S is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  The following result is analogous to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1, when we choose the conics to be  twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take A ∈ T (3) such that h0(IA(1, 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then we have the following:  (1) h1(IA(1, 2)) = 0 and hence h0(IA(1, 2)) = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (2) there is an integral S ∈ |IA(1, 2)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (3) the base locus of |IA(1, 2)| is contained in the union of A and 3 distinct curves of  bidegree (1, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  19  (4) for a sufficiently general A (contained in a dense euclidean open subset of T (3)),  we may take a smooth S ∈ |IA(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thanks to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1, the hypothesis A ∈ T (3) such that h0(IA(1, 1)) = 0 in the  previous statement, implies that the conics in A do not belong to any infinite family of  twistor fibers contained in a smooth j-invariant surface of bidegree (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We start by proving (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Fix a connected component C of A and call D the  only element of |IC(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Set B := A \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  To get h1(IA(1, 2)) = 0 mimicking the  proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1 it is sufficient to prove that h1(F1, IB∩D(2f)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume  h1(F1, IB∩D(2f)) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' assume the existence of T ∈ |OF1(f)| containing the 2 points  B ∩ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since C ∈ |OF1(h + f)|, C ∩ T ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus T meets each connected component of  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2 gives h0(IA(1, 1)) > 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  To prove (2) it is sufficient to show that the reducible cases do not cover the whole  |IA(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In fact, reasoning as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1, the only possible splitting  are of the form (1, 0) + (0, 1) + (0, 1), which are in a finite number, or (1, 1) + (0, 1), where  the bidegree (1, 1) component contains 2 connected components of A and the remainder  bidegree (0, 1) part is uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Now, h0(IB(1, 1)) = 2, so, the set of all  reducible elements of |IA(1, 2)| with an irreducible component of bidegree (1, 1) does not  cover |IA(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We now prove (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since h0(IA(1, 1)) = 0 and A is j-invariant, neither π1(A) nor  π2(A) has a triple points (both have 3 double points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Set L1∪L2∪L3 := π−1  2 (Sing(π2(A)))  and R1 ∪ R2 ∪ R3 := π−1  1 (Sing(π1(A))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since #(Li ∩ A) = #(Ri ∩ A) = 2, L1 ∪ L2 ∪ L3  are in the base locus of |IA(1, 2)| and each Li and each Ri meets exactly 2 connected  components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  To prove the existence of a smooth element, it is sufficient to reason as in the proof of  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 case (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  We are now ready to prove the first part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix n ∈ {0, 1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' There is a smooth S ∈ |OF(1, 2)| containing exactly n  twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' A general S ∈ |OF(1, 2)| contains only finitely many smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since the set  of all twistor fibers has real codimension 4 in the space of all smooth conics, a general  S ∈ |OF(1, 2)| contains no twistor fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Now we prove the case n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix a twistor fiber C and take a general S ∈ |IC(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As-  sume that S contains another twistor fiber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We have h1(IC(1, 2)) = h1(IC∪E(1, 2)) = 0  (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='19 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus |IC∪E(1, 2)| is a 4-codimensional complex projec-  tive subspace of |IC(1, 2)| (this is explained by the equality h0(IC∪E(1, 2)) = h0(IC(1, 2))−  4 contained in [3, proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' However T (1) is a real 4-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' So  a general S ∈ |IC(1, 2)| contains no other twistor fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Note that C is the base locus of |IC(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Bertini theorem a general S ∈ |IC(1, 2)|  is smooth outside C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix p ∈ C and let 2p the closed subscheme of F with (Ip)2 as its  ideal sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Recall that 2p ⊂ S if and only if p ∈ Sing(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since dim C = 1 to get that S  is smooth it is sufficient to prove that h0(I2p∪C(1, 2)) ≤ h0(IC(1, 2)) − 2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' This follows  from the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 case (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The case n = 2 is true by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 with T (2) instead of C(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The case n = 3 is true by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  In the remainder of the section, we will construct a smooth (1, 2)-surface containing 4  twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The following lemma, in the case d = 2, says that if an integral (1, 2)-surface  contains 4 disjoint smooth conics, then these conics are not general, because three of them  must be collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  20  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let d ≥ 2 and A ∈ C(d + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If there is an integral S ∈ |OF(1, d)|, then  A /∈ C∗(d + 2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We prove the lemma by induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We start with the case d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume  that A ∈ C∗(4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' there is no union B of 3 of the connected components of A such that  #(L ∩ B) = 3 for some curve L of bidegree (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix a connected component C of A and  set B := A \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Call Y the only element of |IC(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9 gives ResY (A) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By  assumption and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3, h0(IB(1, 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since h0(IA(1, 2)) ̸= 0, the residual exact  sequence  0 → IB(1, 1) → IA(1, 2) → IA∩Y,Y (1, 2) → 0 ,  gives h0(Y, IA∩Y,Y (1, 2)) > 0 (otherwise |IA(1, 2)| = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The scheme A∩Y is the union of C  and the 3 points B ∩Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In the identification of Y with F1 the line bundle OY (1, 2) goes to  the line bundle OF1(h+3f) and C goes to an element of |h+f|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus h0(F1, IB∩Y,F1(2f)) >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence at least 2 of the 3 points B ∩ Y are in the same fiber ˆL of the ruling |f| of F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since ˆL ∩ C ̸= ∅, ˆL is a curve of bidegree (1, 0) meeting at least 3 connected components  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Call B′ a union of 3 components of A intersecting ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The curves B′ and ˆL give a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume now that the result is true for d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Notice that, as a byproduct of the previous  part, if B ∈ C∗(d + 1), then h0(IB(1, d − 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume A ∈ C∗(d + 2) and that there is an integral S ∈ |OF(1, d)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix a connected  component C of A and set B := A \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take a surface Y of bidegree (0, 1) containing  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By means of the sequence in Formula (10), we either have h0(IB(1, d − 1)) > 0 or  h0(Y, IA∩Y,Y (1, d)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since A ∈ C∗(d + 2), B ∈ C∗(d + 1) and hence, thanks to the  inductive assumption, we have h0(IB(1, d − 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The scheme A ∩ Y is the union of  C and the scheme B ∩ Y with A ∩ B ∩ Y = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Up to the identification of Y and F1 we  have OY (1, d)(−C) ∼= OF1(df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since Y has bidegree (0, 1) each connected component of  B is either contained in Y or it intersects transversely Y at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Remark  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9, the set B ∩ Y is formed by d + 1 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus IA∩Y,Y (1, d) ∼= IB∩Y (df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We saw  h0(Y, IA∩Y,Y (1, d)) > 0 and this is true if and only if there are u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ud+1 ∈ B ∩ Y and  F ∈ |f| such that that ui ̸= uj, for i ̸= j and {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , ud+1} ⊂ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The set F ∩ C is a  unique point, o, and o /∈ {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , ud+1}, because B ∩ C = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The curve F has bidegree  (0, 1) and hence A /∈ C∗(d + 2), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  Thanks to the previous result, if an integral (1, 2)-surface contains 4 disjoint smooth  conics, then these are in special position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We now show that if these 4 conics are twistor  fibers, then their position is very special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We begin by introducing the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  For n ≥ 4, we denote by C(n)− the set of elements A ∈ C(n) for which there exists a  bidegree (1, 0) curve L such that A ∈ C(n, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The set C(n)− parametrizes the families of  n collinear disjoint smooth conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For n ≥ 4 we also write T (n)− := T (n) ∩ C(n)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The  families C(n)− and T (n)− are Zariski closed in C(n) and T (n), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  The following lemma shows that if an integral (1, 2)-surface contains 4 twistor fibers,  then they are all collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take an integral S ∈ |OF(1, 2)| containing A ∈ T (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Then A ∈ T (4)−  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume the existence of an integral S ∈ |OF(1, 2)| containing A ∈ T (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7 there is a union B of 3 of the connected components of A such that B ∈ T (3) \\ T ∗(3),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=', there exists a bidegree (1, 0) curve L, such that B ∈ T (3, L) and hence, thanks to  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7 h0(IB(1, 1)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' However, the same remark tells us that h0(IB(1, 1)) = 1,  h1(IB(1, 1)) = 2 and that the only element M of |IB(1, 1)| is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  As usual, set C := A \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  As in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2 since L of bidegree (1, 0) meets each  connected components of B, then R := j(L), of bidegree (0, 1), do the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  21  Thanks to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='16 we get B ∪ L ∪ R ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since S and M are integral, thanks  to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2, the one-dimensional scheme S ∩ M has bidegree (5, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since B ∪ L ∪ R  has bidegree (4, 4), then C ⊈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let Y be only element of |IC(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since B ⊂ M ∪ Y ,  then M ∪ Y ∈ |IA(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, as S is irreducible, then S ̸= M ∪ Y , and hence  h0(IA(1, 2)) ≥ 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' h1(IA(1, 2)) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since h1(IB(1, 1)) = 2, the residual exact sequence  0 → IB(1, 1) → IA(1, 2) → IA∩Y,Y (1, 2) → 0,  gives h1(Y, IA∩Y,Y (1, 2)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  As in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7 we obtain the following  inequality h1(F1, IB∩Y (2f)) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' there is a curve ˆL ∈ |f| of bidegree (1, 0) intersecting  at least 2 of the connected components of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Call B′ the union of 2 of the connected components of B intersecting ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since ˆL∩C ̸= ∅  and B′ ∪ C is j-invariant, each connected component of B′ ∪ C meets j(ˆL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2,  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1 and B´ezout imply the existence of an integral surface M′ of bidegree (1, 1)  containing B′ ∪ C ∪ ˆL ∪ j(ˆL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since B′ ⊂ M′, ˆL and j(ˆL) contain at least 2 points of  M′, then B′ ∪ ˆL ∪ j(ˆL) ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' But by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5 there is a unique curve of bidegree  (1, 0) intersecting two different smooth conics, hence ˆL = L and both L and j(L) intersect  each connected component of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus L intersects each connected component of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  A ∈ C(4)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  As a byproduct of the proof of the previous result, we get the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' It  essentially says that there are infinitely many integral (1, 2)-surfaces containing 4 collinear  twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take A ∈ T (4)− and assume that A is not contained in a surface of bidegree  (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Then dim |IA(1, 2)| = 1 and |IA(1, 2)| contains exactly 4 reducible elements of  |OF(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Call L the curve of bidegree (1, 0) intersecting each connected component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since each connected component of A is j-invariant, j(L) intersects each connected com-  ponent of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8 we showed that h0(IA(1, 2)) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' From the lines of  that proof, it is possible to derive that only 4 elements in |IA(1, 2)| are reducible and they  are all obtained fixing a connected component C of A and taking the union of the unique  surface MC of bidegree (1, 1) containing A\\C and the unique surface YC of bidegree (0, 1)  containing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' To conclude the proof it is sufficient to prove that h0(IA(1, 2)) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take a  connected component C of A and consider the residual exact sequence  (22)  0 → IC(0, 1) → IA(1, 2) → IMC∩A,MC(1, 2) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We have h0(IC(0, 1)) = 1, because the intersection of 2 different elements of |OY (0, 1)| is  a curve of bidegree (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus by (22) to conclude the proof it is sufficient to prove that  the image V of H0(IA(1, 2)) in H0(MC, IMC∩A,MC(1, 2)) has dimension at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Bez´out  gives that A ∪ L ∪ j(L) is contained in the base locus of |IB,MC(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Every D ∈ |V|  has bidegree (5, 4) as a curve of F and hence a general D ∈ |V| is the union (counting  multiplicities as divisors of the smooth surface MC) of A ∪ L ∪ j(L) and a curve E of  bidegree (1, 0)) as a curve of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Recall that MC is the blow up of P2 at 3 non collinear  points and that these 3 exceptional divisors are the only curve of MC with bidegree (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since MC has only finitely many curves of bidegree (1, 0), D is the same for all non-zero  elements of V and hence dim V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  The following result completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' There are integral S ∈ |OF(1, 2)| containing exactly 4 twistor fibers and  for any such S and A ∈ T (4) with A ⊂ S, there is a curve L of bidegree (1, 0) intersecting  each connected component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, h0(IA(1, 2)) = 2 and each S ∈ |IA(1, 2)| is  singular along L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  22  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 no integral surface of bidegree (1, 2) contains at least 5 twistor  fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The curve L exists by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Now we reverse the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We start with  A ∈ T (4, L)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let 2L denote the closed subscheme of the “double line”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' To prove that  each S ∈ |IA(1, 2)| is singular at each point of L it is sufficient to prove that h0(IA(1, 2)) =  h0(IA∪2L(1, 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='9 gives h0(IA(1, 2)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence, it is sufficient to prove that  h0(IA∪2L(1, 2)) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For any connected component C of A let MC the only surface of  bidegree (1, 1) containing A \\ C and let YC the only surface of bidegree (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since  C ∩ L ̸= ∅, L ∩ YC ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since YC has bidegree (0, 1) and L bidegree (1, 0), we get L ⊂ YC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thus L ⊆ MC ∩ YC and hence |IA∪2L(1, 2)| contains at least the 4 reducible elements of  |IA(1, 2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence h0(IA∪2L(1, 2)) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Non existence results for surfaces of bidegree (1, 2) and (1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In this last  part, we prove our two last main results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  For any A ∈ T (n)−, n ≥ 4, let us call L and R := j(L) the curves of bidegree (1, 0) e  (0, 1), respectively, intersecting all the connected components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In view of our goal, we need to discuss the reducibility of some surfaces containing a  certain amount of twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' First of all, fix an integer n ≥ 2, take B ∈ T (4) such that  h0(IB(1, 1)) > 0 and call M the unique (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2) surface of bidegree (1, 1)  containing B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since each element of C(1) is contained in an element of |OF(0, 1)| for each  E ∈ T (n − 1) there is a reducible element W ∈ |OF(1, k)|, union of M and n − 1 surfaces  of bidegree (0, 1) such that B ∪ E ⊂ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The following lemma is a sort of viceversa of this  remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Moreover, it will be a key tool in the last two proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If d ≥ 2 and A ∈ T (d + 3)− is such that h0(IA(1, d)) > 0, then each  element of |IA(1, d)| has an irreducible component M of bidegree (1, 1) containing at least  4 connected components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  In particular, for any n ≥ d + 3, there is no integral  S ∈ |OF(1, d)| containing A ∈ T (n)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In order to prove the last statement, it is sufficient to do the case n = d + 3 and  thus it is sufficient to prove the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  We use induction on d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let us assume first d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take A ∈ T (5)− and let L and  j(L) be the curves of bidegree (1, 0) and (0, 1) intersecting all the connected components  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix a connected component C of A and set B := A \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since C ∩ L ̸= ∅, the curve  C ∪ L is a connected and nodal curve of bidegree (2, 1) with arithmetic genus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence  h0(OC∪L(0, 1)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus there is Y ∈ |IC∪L(0, 1)| and such a Y is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since any two  smooth conics of Y meet, no component of B is contained in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence B ∩ Y is formed  by 4 points of L \\ (L ∩ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Recall that OY (1, 2) ∼= OF1(h + 3f) and that C ∈ |OF1(h + f)|  and thus IA∩Y,Y ∼= IB∩L,Y (2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since each element of |f| contains a unique point of L we  have that h0(D, IA∩Y,Y (1, 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The residual exact sequence of Y  0 → IB(1, 1) → IA(1, 2) → IA∩Y,Y (1, 2) → 0,  gives an isomorphism ϕ : H0(IB(1, 1)) → H0(IA(1, 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If h0(IB(1, 1)) = 0, then h0(IA(1, 2)) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Now assume h0(IB(1, 1)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The isomorphism ϕ says that each W ∈ |IA(1, 2)| has Y  as an irreducible component, say W = Y ∪ W1 with W1 ∈ |IB(1, 1)|, and hence we have  the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume now d ≥ 3 and use induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By reasoning as in the base case, take  A ∈ T (d + 3)− and use the exact sequence  0 → IB(1, d − 1) → IA(1, d) → IA∩Y,Y (1, d) → 0,  to prove that h0(Y, IA∩Y,Y (1, d)) = 0 and hence that there is an isomorphism ϕ : H0(IB(1, d−  1)) → H0(IA(1, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Now, again, if h0(IB(1, d − 1)) = 0, then h0(IA(1, d)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence,  assume h0(IB(1, d − 1)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The isomorphism ϕ says that each S ∈ |IA(1, d)| has Y  as an irreducible component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' S = D ∪ S1 with S1 ∈ |IB(1, d − 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The inductive  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  23  assumption says that S1 has an irreducible component M of bidegree (1, 1) containing at  least 4 components of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  We now have all the ingredients to prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' First we prove that no  integral surface of bidegree (1, 2) contains 5 twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume the existence of an integral S ∈ |OF(1, 2)| containing A ∈  T (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8 shows that for any union A′ ⊂ A of 4 components of A there is a union  A′′ ⊂ A′ of 3 connected components intersecting some L of bidegree (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let L be a  curve of bidegree (1, 0) intersecting the maximal number, z, of components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Clearly  z ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='11 to get a contradiction it is sufficient to prove that z ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Assume z ∈ {3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Take any ordering C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , C5 of the connected components of A  and set Bi := π1(Ci), 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Each Bi is a line of P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since any two conics contained  in an element of |OF(1, 0)| meet, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , B5 are 5 different lines of P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' For any i < j < h,  there is a curve T of bidegree (1, 0) intersecting Ci, Cj and Ch if and only if Bh contains  the point Bi ∩ Bj and in this case L = π−1  1 (Ci ∩ Cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' With no loss of generality we may  assume that L meets C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , Cz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (a) Assume z = 3 and hence B1 ∩B2 ∈ B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8 to C1 ∪C2 ∪C4 ∪C5  we have one of the following mutually esclusive relations:  B1 ∩ B2 ∩ B4 ̸= ∅,  B1 ∩ B2 ∩ B5 ̸= ∅,  B1 ∩ B4 ∩ B5 ̸= ∅,  B2 ∩ B4 ∩ B5 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since B1 ∩ B2 ∈ B3 and z = 3, we can exclude the first two cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' we have  B1 ∩ B2 ∩ B4 = B1 ∩ B2 ∩ B5 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thus, either B1 ∩ B4 ∩ B5 ̸= ∅ or B2 ∩ B4 ∩ B5 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Exchanging if necessary C1 and C2  we may assume B1 ∩ B4 ∩ B5 ̸= ∅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' B4 ∩ B5 ∈ B1, and hence B2 ∩ B4 ∩ B5 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since  B2 ∩ B4 ∩ B5 = ∅, applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='8 to C2 ∪ C3 ∪ C4 ∪ C5 we have one of the following  mutually esclusive relations  B2 ∩ B3 ∩ B4 ̸= ∅,  B2 ∩ B3 ∩ B5 ̸= ∅,  B3 ∩ B4 ∩ B5 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since B4 ∩ B5 ∈ B1 and z = 3, B3 ∩ B4 ∩ B5 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since B2 ∩ B3 ∈ B1 and z = 3,  B2 ∩ B3 ∩ B5 = B2 ∩ B3 ∩ B4 = ∅, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (b) Assume z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since C1 ∩ L ̸= ∅, the curve C1 ∪ L is a connected and nodal  curve of arithmetic genus 0 and bidegree (2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus h0(OC1∪L(0, 1)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus there is  Y ∈ |IC1∪L(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since S is irreducible, Y is not an irreducible component of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus  the residual exact sequence of Y  0 → IB(1, 1) → IA(1, 2) → IA∩Y,Y (1, 2) → 0,  gives h0(Y, IA∩Y,Y (1, 2)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Up to the isomorphism of Y and F1 we have L = h,  C1 ∈ |OF1(h + f)| and OY (1, 2) ∼= OF1(h + 3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since (A \\ C1) ∩ C1 = ∅, IA∩Y,Y (1, 3) ∼=  I(A\\C1)∩D,D(2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since L meets C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' , C4, (A \\ C1) ∩ D contains a set F ⊂ L such that  #F = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since each element of |f| contains a unique point of L, h0(Y, IA\\C)∩Y,Y (2f)) = 0,  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  We now conclude our paper with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5 which concerns surfaces of  bidegree (1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='5: Assume the existence of A ∈ T (6) and of an integral S ∈ |OF(1, 3)|  containing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='11 to get a contradiction it is sufficient to prove the existence  of a curve L of bidegree (1, 0) such that all the components of A intersects L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7 for any union A′ ⊂ A of 5 components of A there is a union A′′ ⊂ A′ of 3 connected  components intersecting some L of bidegree (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let L be a curve of bidegree (1, 0)  intersecting the maximal number, z, of components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We have that z ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hence, by  24  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' ALTAVILLA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BALLICO, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' BRAMBILLA  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='11 it is sufficient to prove that z ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Assume then that z ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We will now  exclude all the cases z = 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  For any connected component C of A, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='7 tells us that there is a curve L of  bidegree (1, 0) intersecting at least 3 connected components of A \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' In particular there  is an integral M ∈ |OF(1, 1)| containing at least 3 components of A (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Note that j(M) = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We may take M with the additional condition that it contains  the maximal number e of components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let E be the union of the components of A  contained in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus 3 ≤ e ≤ z ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since each twistor fiber is j-invariant, j(L) meets each connected component of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  B´ezout gives L ∪ j(L) ⊂ M and L ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' If e ≥ 4 B´ezout gives j(L) ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' However,  the one-dimensional cycle M ∩ S has bidegree (7, 5) and thus e ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Set Σ := S ∩ M  (as a scheme-theoretic intersection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since the one-dimensional scheme Σ is the complete  intersection of F with 2 very ample divisors, h0(OΣ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Set F := A \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (a) Assume e = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Hence E ∪ L ∪ j(L) ⊂ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since E ∪ L ∪ j(L) has bidegree  (5, 5) and h0(OΣ) = 1, Σ is the union of E ∪ L ∪ j(L) and a multiple structure on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Note that Σ ∈ |OF(1, 3)| and that Σ contains E ∪ j(L) with multiplicity 1 and L with  multiplicity 3 (as divisors of the smooth surface M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since Σ has multidegree (7, 5),  Σ = 3L ∪ j(L) ∪ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Note that Σ contains the degree 4 zero-dimensional scheme F ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since F ∩ E = ∅, F ∩ (j(L) ∪ L) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus at least one irreducible component, T, of  F meets L ∪ j(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since j(T) = T, T ∩ L ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Let C be a component  of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since C ∩ L ̸= ∅, C ∪ L is a connected and nodal curve of bidegree (2, 1) with  arithmetic genus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thus h0(OC∪L(0, 1)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Thus there is Y ∈ |IC∪L(0, 1)| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Since S is irreducible, Y is not an irreducible component of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus the residual exact  sequence of Y gives h0(Y, IA∩Y,Y (1, 3)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Up to the isomorphism of Y and F1 we  have L = h, C ∈ |OF1(h + f)| and OY (1, 3) ∼= OF1(h + 4f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since (A \\ C) ∩ C = ∅,  IA∩Y,Y (1, 3) ∼= I(A\\C)∩Y,Y (3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since z = 5, (A \\ C) ∩ Y contains a set H ⊂ L such that  #H = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since each element of |f| contains a unique point of L, h0(Y, IA\\C)∩Y,Y (3f)) = 0,  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (b) Assume e = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fix a connected component C of E and set B := A \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Set  {Y } := |IC(0, 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As in step (a) we have that L ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The following exact sequence  (23)  0 → IB(1, 2) → IA(1, 3) → IC∪(B∩Y ),Y (1, 3) → 0  is the residual exact sequence of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Since Y is not an irreducible component of S, we have  h0(Y, IC∪(B∩Y ),Y (1, 3)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' As in step (a) we have IC∪(B∩Y ),Y (1, 3) ∼= IB∩Y,F1(3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' We  now have two possibilities: either h0(IB(1, 2)) = 0 or h0(IB(1, 2)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (b1) Assume for the moment h0(IB(1, 2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus h0(Y, IC∪(B∩Y ),Y (1, 3)) ≥ 2 and  h1(Y, IC∪(B∩Y ),Y (1, 3)) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Up to the identification of Y and F1 we have IC,Y (1, 3) ∼=  OF1(3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Hence the 5 points B ∩ Y give at most one condition to the linear system  |OF1(3f)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus there is J ∈ |OF1(f)| such that B ∩ Y ⊂ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Note that J is a curve of  bidegree (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' The maximality of the integer e gives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  (b2) Assume that h0(IB(1, 2)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='4 any surface containing B is re-  ducible, say M1∪Y with M1 irreducible of bidegree (1, 1) containing at least 4 components  of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Thus e ≥ 4, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  □  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Altavilla, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Ballico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Twistor lines on algebraic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Global Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 55 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  3, 555–573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Altavilla, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Sarfatti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Slice-polynomial functions and twistor geometry of ruled surfaces in CP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 291 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 3-4, 1059–1092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Altavilla, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Ballico, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Brambilla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Surfaces in the flag threefold containing smooth conics and  twistor fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Mediterr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 19, 281 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1007/s00009-022-02202-3  TWISTOR FIBERS IN (1, d)-SURFACES OF THE FLAG THREEFOLD  25  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Altavilla, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Ballico, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Brambilla, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Salamon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Twistor geometry of the Flag manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 303, 24 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='1007/s00209-022-03161-x  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Armstrong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Povero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Salamon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Twistor lines on cubic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Semin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Politec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Torino 71 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 3-4, 317–338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Atiyah, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hitchin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Self-duality in four-dimensional Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' London Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' A 362 (1978), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 1711, 425–461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Chirka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Orthogonal complex structures in R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' (Russian) Uspekhi Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Nauk 73 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  1(439), 99–172;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' translation in Russian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Surveys 73 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 1, 91–159  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Fujiki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Pontecorvo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Twistors and bi-Hermitian surfaces of non-K¨ahler type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' SIGMA Symmetry  Integrability Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 10 (2014), Paper 042, 13 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Gentili, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Salamon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Stoppato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Twistor transforms of quaternionic functions and orthogonal  complex structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' (JEMS) 16 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 11, 2323–2353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hartshorne, Algebraic Geometry, Springer-Verlag, Berlin–Heidelberg–New York, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Hitchin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' K¨ahlerian twistor spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' (3) 43 (1981), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 1, 133–150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Pontecorvo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Complex structures on Riemannian four-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 309 (1997), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 1,  159–177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Salamon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Viaclovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Orthogonal complex structures on domains in R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 343 (2009),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content=' 4, 853–899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='  Dipartimento di Matematica, Universit`a degli Studi di Bari ‘Aldo Moro’, via Edoardo  Orabona, 4, 70125, Bari, Italia  Email address: amedeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='altavilla@uniba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='it  Dipartimento Di Matematica, Universit`a di Trento, Via Sommarive 14, 38123, Povo, Trento,  Italia  Email address: edoardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='ballico@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='it  Universit`a Politecnica delle Marche, via Brecce Bianche, I-60131 Ancona, Italia  Email address: brambilla@dipmat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='univpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
+page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE4T4oBgHgl3EQfEAug/content/2301.04874v1.pdf'}
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+arXiv:2301.03872v1  [cs.IT]  10 Jan 2023
+Towards Quantum Annealing for Multi-user
+NOMA-based Networks
+Eldar Gabdulsattarov, #Khaled Rabie, ◦Xingwang Li, and Galymzhan Nauryzbayev
+�
+School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan, Z05H0K4, Kazakhstan
+#Department of Engineering, Manchester Metropolitan University, Manchester, M15 6BH, UK
+◦School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454000, China
+Email: {eldar.gabdulsattarov, galymzhan.nauryzbayev}@nu.edu.kz, #k.rabie@mmu.ac.uk, ◦lixingwang@hpu.edu.cn
+Abstract—Quantum Annealing (QA) uses quantum fluctuations
+to search for a global minimum of an optimization-type problem
+faster than classical computers. To meet the demand for future
+internet traffic and mitigate the spectrum scarcity, this work
+presents the QA-aided maximum likelihood (ML) decoder for
+multi-user non-orthogonal multiple access (NOMA) networks as
+an alternative to the successive interference cancellation (SIC)
+method. The practical system parameters such as channel ran-
+domness and possible transmit power levels are taken into account
+for all individual signals of all involved users. The brute force (BF)
+and SIC signal detection methods are taken as benchmarks in the
+analysis. The QA-assisted ML decoder results in the same BER
+performance as the BF method outperforming the SIC technique,
+but the execution of QA takes more time than BF and SIC.
+The parallelization technique can be a potential aid to fasten
+the execution process. This will pave the way to fully realize the
+potential of QA decoders in NOMA systems.
+Index Terms—Quantum annealing, non-orthogonal multiple
+access (NOMA), maximum likelihood (ML).
+I. INTRODUCTION
+There has been a drastic rise in the traffic of advanced
+multimedia applications such as real-time conferencing, online
+video gaming and virtual reality in recent years. Moreover,
+according to the Cisco Annual Report, the total number of
+internet users will reach 5.3 billion by 2023 [1]. To meet this
+ever-growing demand on traffic, it is crucial to consider some
+optimization improvements in the quality of data transmission
+in terms of the bandwidth, error rate, and delay. One of the
+actions that can be taken into account is to implement Quantum
+Annealing (QA) technologies to solve NP-hard optimization
+problems in wireless communication.
+The authors in [2]–[5] suggested the centralized radio access
+network (C-RAN) as a promising and cost-effective design
+architecture for future wireless networks, where multiple base
+stations (BSs) are interconnected with a centralized data center.
+The C-RAN’s data centers support most of the BSs’ signal
+processing operations. It is envisioned that the QA comput-
+ers placed at the C-RAN centers can facilitate optimization
+problems, paving the way for full integration of the presented
+methodologies in real-world communication systems.
+One of the computational complex problems in wireless
+networks is to obtain an optimal scheduling in multi-user
+(MU) systems, which can help one effectively employ network
+resources. The authors in [6] introduced the QA approach to
+find an optimal solution for the time-division multiple access
+(TDMA) transmission scheduling problem in a wireless sensor
+network using a tree topology. The results revealed that QA
+outperforms the other scheduling solutions in terms of the
+runtime and proximity to the optimal solution.
+An essential part of cellular base-band processing that can
+enhance the quality of communication is the error correction
+code (ECC), which mitigates the errors in received bit-streams
+caused by the noise in a wireless environment. However, the
+error correction code protocols with exceptional performance
+require significant computational resources for the decoding
+process at the receiver. For this reason, the implementation of
+QA computers was proposed in [2] for efficient decoding of
+low-density parity-check codes in uplink systems. The method
+achieved a bit error rate (BER) of 10−8 at the signal-to-noise
+ratio (SNR) of 2.5-3.5 dB lower than the standard algorithm for
+decoding ECCs on a field-programmable gate array (FPGA).
+Another key aspect in the performance of wireless communi-
+cation systems is a signal detection. To achieve the full potential
+of multi-user systems, BSs have to efficiently decode superim-
+posed data streams received from different users. Maximum
+likelihood (ML) usually ends up with an optimum solution for
+decoding the received signals, but it is an NP-hard problem. The
+computational cost of ML rises exponentially with the number
+of users and data rate [3]. For large-scale networks, the ML
+decoding process might last expiration of acknowledgement
+timeout. The authors in [3]–[5] employed a QA approach to
+speed up this process. Particularly, [3] presented the QuAMax
+approach for ML decoding in MU MIMO systems using QA.
+The performance of the solution was tested on the quantum
+annealer with 2048 qubits considering different modulation
+schemes and multiple users. The method achieves the same
+BER as ZF does, but with significantly lower computation time
+in the case of binary phase-shift keying (BPSK) and quadrature
+phase-shift keying (QPSK) modulation types. Moreover, the
+QA approach outperforms the conventional decoders in terms of
+the computable size of ML problems. For instance, the work in
+[3] was extended by [4], where an experiment on the recently
+released QA hardware with 5640 qubits was performed. The
+results revealed that the new computer is able to provide better
+scaling for embedding the ML MIMO decoding problem. In
+addition, the study showed that today’s QA machines can
+achieve identical BER as the ML sphere decoder for 4 × 4
+MIMO using 16-QAM (quadrature amplitude modulation).
+
+The current communication technologies support multiple
+users using orthogonal multiple access (OMA) techniques such
+as TDMA, frequency- and code-division multiple access (i.e.,
+FDMA and CDMA) schemes. However, all these techniques
+have certain shortcomings such as precise time synchronization,
+limited frequency carriers and code length. These drawbacks
+get more perceptible for a large number of users [7]. Non-
+orthogonal multiple access (NOMA) is a technology that al-
+lows all users in a cell operate in the common time and
+frequency, that, in turn, improve the spectral efficiency (SE).
+The successive interference cancellation (SIC) is considered
+as a promising decoding technique for NOMA that decodes
+symbol-by-symbol until the desired one is detected. However,
+the high computational cost, latency and error propagation of
+SIC make it impractical to implement NOMA in the current
+wireless communication systems [7], [8]. The traditional ML
+decoder supports high-quality decoding but suffers from high
+computation time. To address this issue, a NOMA-ML decoder
+based on QA was proposed in [5] for a two-user system model,
+with a BPSK modulation and constant channel gains.
+In contrast to the previous studies, in this work, we consider
+the NOMA-based uplink network comprising multiple users
+under practical scenarios and different modulation types. The
+NOMA-ML decoding problem was firstly reformulated into
+a QUBO model for multiple modulation schemes including
+BPSK, QPSK, 16−QAM and 64−QAM. Moreover, the gen-
+eralized QUBO expressions of the proposed system model are
+derived for the multi-user network under the previously defined
+modulation types. We have investigated how the number of
+users in a cell affects the signal decoding performance. Fur-
+thermore, the QA-assisted ML decoder was compared with SIC
+and Brute Force (BF) techniques in terms of BER performance
+and simulation time. Finally, the influence of the parallelization
+method on the execution time of the QA machine is examined.
+II. SYSTEM MODEL
+The system model considered here represents an uplink
+NOMA-based network scenario (depicted in Fig. 1) compris-
+ing BS and N end-users, denoted by Uk, with k ∈ A =
+{1, 2, . . ., N}. It is assumed that all nodes are deployed with a
+single antenna. The channel between the user of interest and BS
+is given by ˜hk and assumed to be independent and identically
+distributed random variables (RVs) following Rayleigh fading.
+A. Non-orthogonal Multiple Access
+Due to the considered uplink mode of communication, all
+end-users simultaneously transmit their individual messages to
+BS while ensuring the NOMA principle to be true through the
+path loss associated with their corresponding distances1. Hence,
+the received signal at BS can be written as
+y =
+N
+�
+k=1
+skhk + n,
+(1)
+1Note that the similar assumption can be made using the principle of
+messages’ priorities.
+0
+U1
+50
+U2
+50N
+N−1
+U3
+50(N+1)
+N−1
+UN
+100
+m
+h1
+h2
+h3
+hN
+...
+...
+Fig. 1. An illustration of uplink NOMA with N users.
+where sk is the transmitted message of user k, with the transmit
+power defined as E{sks∗
+k} = Pk. For the sake of simplicity,
+we assume that all end-users transmit with the same power
+levels, i.e., P = P1 = P2 = . . . = PN. hk is modelled as
+hk = ˜hk/
+�
+dτ
+k, where dk and τ indicate the corresponding
+distance and path-loss exponent. n is an additive white Gaussian
+noise (AWGN) term, with zero mean and variance σ2
+n.
+B. Maximum Likelihood Detection
+The receiver implements the ML detection as a decoding
+method, which output, given by an optimal symbol vector ˆsML,
+is formulated as
+ˆsML = arg min
+ˆs1,ˆs2,...ˆsN
+|y − shT|2,
+(2)
+where y is the received signal, s and h are the 1 × N vectors
+representing the transmit symbols and channel gains of end-
+users, respectively.
+C. Quantum Annealing
+Quantum computing implements the concepts of quantum
+mechanics to solve high-computational problems faster than
+classical computers. There are several types of quantum com-
+puters. If the quantum circuit model controls the qubits using
+quantum logic gates to solve a wide range of problems, QA
+is an effective tool to solve optimization problems with given
+objective functions. Specifically, the quantum processing unit
+(QPU) uses quantum fluctuations to search for the global
+minimum solution. However, modern quantum computers are
+not that perfect yet and QA might end up with a local minimum
+solution due to the analogue noises in the device [9]. To
+mitigate the internal errors in QPU, the annealing parameters
+could be modified based on the specifics of the problem. To
+minimize these errors, QA can be simulated many times, and
+the optimal solution will be the answer with the lowest energy
+level and most frequent occurrences. The annealing parameter
+that controls the number of samples per simulation is the
+number of reads (R), it was decided to set it to 1000. Another
+annealing parameter that could impact the performance of the
+QA machine is the single-sample annealing time (Ta), which
+was set to 20 µs without any pausing time [10].
+D. Quadratic Unconstrained Binary Optimization
+To solve an optimization problem by a means of QA, the
+objective function should be reformulated in a form of the
+quadratic unconstrained binary optimization (QUBO) model or
+the Ising model. These formulations can be used interchange-
+ably, and it is easy to switch from one to another. At the end
+
+a
+7a
+5a
+3a
+a
+3a
+5a
+7a
+-b
+-3b
+-b
+-3b
+1
+-1
+j
+-j
+Im
+Re
+BPSK
+64-QAM
+16-QAM
+QPSK
+Fig. 2. Constellations of different modulation types.
+of annealing process, the best solution is given by the qubit
+sequence with a minimum energy state. The model is given by
+an upper-diagonal matrix Q with a size of M × M
+ˆq1, ˆq2, . . . , ˆqM = arg min
+q1,q2,...qM
+M
+�
+i≤j
+Qijqiqj,
+(3)
+where M denotes the number of qubits, and each variable qi
+takes binary numbers. By applying the property q2
+i = qiqi = qi,
+and substituting the symbols in (2) by the corresponding binary
+equivalent form, it becomes feasible to formulate the NOMA
+ML problem in terms of the QUBO form. The corresponding di-
+agonal Qii and non-diagonal Qij terms of the QUBO model are
+represented by linear and quadratic coefficients, accordingly.
+To implement the obtained QUBO model on QPU, the matrix
+coefficients have to be mapped into physical qubits of a QA
+chip. The linear coefficients define the qubit biases, while
+quadratic coefficients represent the coupling strengths [11].
+However, QPU’s physical qubits are not fully connected, in the
+Chimera graph each qubit has six neighbours, and, in Pegasus
+topology, each qubit is connected with 15 neighbour qubits
+[12]. Therefore, sometimes it is not possible to directly apply
+the QUBO model onto real QA machine.
+III. FORMULATION OF NOMA ML PROBLEM IN A QUBO
+MODEL
+The ML expression in (2) should be described by a QUBO
+model to solve it with QA. Taking into account (1) and the
+complex numbers due to the broadcast transmission, we split
+the received signal y and channel coefficients into real and
+imaginary parts as follows: y = yR + jyI and hk = hk,R +
+jhk,I. Each symbol, denoted by sk, can be transformed into
+the qubit form. The methodology for representing symbols in
+terms of the binary variables depends on the modulation type.
+The considered modulations are presented in Fig. 2.
+1) BPSK: The BPSK symbols of a form sk ∈ {±1}, can
+be reformulated into the QUBO form by sk = 2qk − 1. In this
+case, M in (3) takes a value of the number of users, N.
+2
+3
+4
+5
+6
+7
+8
+9
+10
+10
+-4
+10
+-3
+10
+-2
+BER
+No. of users per cell
+ User 1
+ User 2
+ User 3
+ User 4
+ User 5
+ User 6
+ User 7
+ User 8
+ User 9
+ User 10
+ Average BER
+Fig. 3. BER vs. the number of users per cell under BPSK.
+2) QPSK: As shown in Fig. 2, the QPSK symbols, sk ∈
+{ ±1±j
+√
+2 }, reside on a unit circle with 90◦ phase difference
+between two consecutive symbols. They can be represented by
+two qubits; therefore, the term M in (3) becomes the double
+of a number of users, 2N. Finally, the QPSK symbols can be
+expressed as sk = [(2q2k−1 − 1) + j (2q2k − 1)] /
+√
+2.
+3) 16-QAM: The 16-QAM modulation encodes four in-
+formation bits into a complex symbol and correspondingly
+includes 16 constellation points. The symbols sk ∈ {±b ±
+jb, ±b ± j3b, ±3b ± jb, ±3b ± j3b}, where b = 1/(3
+√
+2),
+require four qubits to be adapted for the QUBO model, i.e.,
+sk = [(4q4k−3 + 2q4k−2 − 3) + j (4q4k−1 + 2q4k − 3)] /3
+√
+2.
+M in (3) becomes the quadruple of a number of users, 4N.
+4) 64-QAM: On the other hand, the 64-QAM modulation
+returns a complex symbol by encoding 16 information bits.
+As shown in Fig. 2, the constellation points are defined as
+per Fig. 2 and require six qubits to be adopted for the
+QUBO model; therefore, the term M in (3) becomes the
+sixtuple of a number of users, 6N. These symbols can be
+re-expressed as sk = a (8q6k−5 + 4q6k−4 + 2q6k−3 − 7) +
+ja (8q6k−2 + 4q6k−1 + 2q6k − 7).
+IV. IMPLEMENTATION AND RESULTS DISCUSSION
+A. Implementation
+The system model considered in the analysis represents a N-
+end user uplink NOMA system that employs BPSK modulation.
+As illustrated in Fig. 1, the base station is located at a coordinate
+0. We assume that the first and the farthest users are located
+50 and 100 m away from the base station, respectively, while
+the other intermediate nodes are evenly distributed in-between.
+For practical purposes, we apply the standardized scenarios
+described in [13] and [14]. The performance of the exploited
+solvers (i.e., BF, SIC and QA) were examined by varying the
+transmit power from −40 dBm to 24 dBm. Moreover, the
+environment of data transmission is considered to be a free
+space with line-of-sight propagation which relates to the path-
+loss exponent τ = 2 [15].
+
+-40
+-30
+-20
+-10
+0
+10
+20
+10
+-2
+10
+-1
+10
+0
+BF
+Bit error rate
+Transmit power, dBm
+ U
+1
+ U
+2
+ U
+3
+ QA
+SIC
+Fig. 4. The BER performance vs. the transmit power for a three-
+user NOMA scenario, with different decoding techniques.
+15.07
+15.07
+15.07
+15.07
+20.00
+20.00
+20.00
+20.00
+84.94
+89.19
+84.39
+86.06
+20.54
+20.54
+20.54
+20.54
+4.07
+4.74
+5.51
+5.02
+1.51
+1.37
+1.49
+1.74
+4.74
+4.65
+5.50
+5.38
+9.00
+9.32
+8.81
+8.89
+QA:
+ 
+p
+ 
+ T
+D
+ T
+R
+ T
+A
+ T
+p
+14
+10
+-10
+-30
+Transmit power, dBm
+ SIC
+ BF
+Fig. 5. Comparison between the decoding techniques in terms
+of simulation time for five (5) samples (in ms).
+The QA method was implemented on the proposed system
+model using D-Wave’s Advantage QPU [12]. It is important
+to mention that the parallelization procedure was used to save
+the simulation time on the QA computer [3], i.e., 5 instances
+of the problem were run at the same time on QPU. Since
+the 3-user BPSK problem requires only 3 logical qubits, 5
+instances would occupy 15 logical qubits. In the analysis of
+simulation time, the following terms are used as follows. QPU
+Programming Time (Tp) is the time taken for programming
+the couplers and biases of the chip in accordance with the
+QUBO model. QPU Sampling Time (Ts) is the total time for
+simulation of R samples and consists of the annealing time
+(Ta), the readout time (Tr), and the delay time (Td) for every
+single sample [16]. For the sake of simplicity, we declare the
+following terms: the total QPU Annealing Time (TA = R ·Ta),
+the total QPU Readout Time (TR = R · Tr), and the QPU
+Delay Time (TD = R · Td). Tp, Ts, QPU Access Overhead
+Table I. Simulation time (in ms) of the decoding techniques.
+Technique
+for 5 samples
+for 1 sample
+SIC
+5.066
+3.958
+BF
+9.006
+4.535
+QA
+QPU Service Time
+148.116
+107.984
+Tp
+15.069
+15.068
+TA
+20
+20
+TR
+86.145
+46.68
+TD
+20.54
+20.54
+∆
+4.835
+3.86
+∆p
+1.527
+1.836
+Time (∆) and Post-processing Overhead Time (∆p) constitute
+QPU Service Time, which is considered as the total time taken
+for the QA simulation excluding internet delay.
+B. Discussion
+In Fig. 3, we examine the influence of the number of
+users per cell on the BER performance in the NOMA system.
+Particularly, the transmit power and the AWGN power were
+set to 10 dBm and −60 dBm, respectively. As expected, the
+average BER performance degrades with the increase in the
+number of users per cell. It is seen from the plot that, for the
+number of users greater than 6, the difference between the BER
+curves becomes quite similar to each other. Therefore, we apply
+the 3-user NOMA scenario in the further analysis.
+In Figs. 4 and 5, we compare the SIC, BF and QA decoding
+techniques in terms of the BER performance and simulation
+time, with the noise power set to σ2
+n = −30 dBm. QA
+was simulated for particular transmit power levels, i.e., P =
+{−30, −10, 10, 14} dBm. The number of QA simulations per
+power level was taken to ensure a 1% accuracy with respect
+to the BER performance of BF. Overall, about 380 simulations
+(with 5 problems at a time) were performed on QPU.
+Fig. 4 illustrates the BER performances of each decoding
+method. The BER curves can be characterized by different
+behaviour. It can be seen from the U1’s curve, the SIC and
+BF performances coincide up to −5 dBm, and then SIC starts
+outperforming BF over a short transmit power range (i.e.,
+from −5 dBm to 10 dBm). However, afterwards, SIC starts
+experiencing noticeable performance degradation. At the same
+time, U2 has identical BER for SIC and BF up to 10 dBm.
+After this point, there is a slight improvement of the SIC curve,
+but, in general, its performance is much worse than that of BF.
+What stands out for U3 is the equal performance of SIC and BF
+for transmit power less than 5 dBm, thereafter, the difference
+between them begins to grow considerably. The performance
+of BF improves remarkably after 5 dBm, whereas the SIC
+curve enhances minimally until 15 dBm with the following
+saturation after this point. In general, the QA, BF and SIC
+curves follow the same trend till 10 dBm, but after this level,
+the SIC performance degrades substantially, while the BER
+result of QA is approximately the same as BF. The reason
+for the SIC’s low performance could be insufficient differences
+between the end-users’ power levels.
+Fig. 5 presents the timing data for each technique. As
+can be seen from the histogram plot, SIC shows the fastest
+
+execution among the presented techniques. As expected, the
+second fastest technique is BF, since it needs to iterate over all
+possible combinations. The QA execution time interval consists
+of several timing sub-intervals. While Tp, TA and TD are the
+same for all cases, TR shows a slight variation, since the reading
+time depends on the position of exploited physical qubits on the
+chip and on the number of used qubits (more time is needed
+for a larger number of qubits) [16]. The dependence on the
+number of qubits could be noticed in Table I, i.e., TR for 1
+sample is twice as fast as TR for 5 samples. Overall, the total
+time taken for QA execution is about 30 times more than the
+SIC simulation. However, it is important to mention that QA
+needs to simulate the same problem R number of times (in
+our case it is set to 1000). Furthermore, the problem could be
+addressed by the parallelization method. From Table I, it is
+clearly noticeable that the execution of 5 instances in parallel
+is more than 3 times faster than executing the same number of
+samples in a sequential order. Therefore, one could potentially
+conclude that the overall QA simulation time can be decreased
+by simulating multiple problem instances at the same time.
+V. CONCLUSION
+To mitigate the problem of spectrum scarcity and present
+an alternative for SIC, this work aims at evaluating the per-
+formance of QA-aided ML detection for NOMA systems. The
+ML problem was firstly described in terms of the QUBO model
+to enable the integration of the problem with QPU. For 3-user
+NOMA under BPSK modulation, the BER performance of QA
+is approximately the same as the BF method, but QA takes
+longer time to execute due to the current hardware specifics.
+The parallelization method seems to be a potential solution
+that could decrease the execution time of QA. Additionally,
+the analogous noises in QA could be suppressed with coming
+of the next-generation QA hardware, and the time taken for
+error alleviation would be reduced. This will pave the way for
+computing time-dependent operations on QA including the QA-
+assisted ML decoding technique for future NOMA systems.
+VI. ACKNOWLEDGEMENT
+This work was supported by the Nazarbayev University (NU)
+Social Policy grant, the NU FDCRP Grant no. 240919FD3935
+and the NU CRP Grant no. 11022021CRP1513.
+APPENDIX A
+QUBO MODEL COEFFICIENTS
+In this section, we present the coefficients for the QUBO
+model for different modulation types2.
+1) BPSK: The corresponding QB
+ij values can be found as
+QB
+ii = P
+�
+−4hi,R
+� N
+�
+l=1
+hl,R−hi,R
+�
+−4hi,I
+� N
+�
+l=1
+hl,I−hi,I
+��
+−
+√
+P (4yRhi,R + 4yIhi,I) ,
+(A.1)
+QB
+ij = P (8hi,Rhj,R + 8hi,Ihj,I) .
+(A.2)
+2Note that the equations involving the positive integer variables i and j must
+satisfy the condition i < j.
+2) QPSK: The QQ
+ij values can be found using (A.3), (A.4)
+and the equations given below
+QQ
+(2i−1),(2i) = 0,
+∀i ≥ 1,
+(A.5)
+QQ
+(2i−1),(2j−1) = QQ
+(2i),(2j) = P
+2 (8hi,Rhj,R + 8hi,Ihj,I) ,
+(A.6)
+QQ
+(2i−1),(2j) = P
+2 (8hi,Ihj,R − 8hi,Rhj,I) ,
+(A.7)
+QQ
+(2i),(2j−1) = P
+2 (−8hi,Ihj,R + 8hi,Rhj,I) .
+(A.8)
+3) 16-QAM: The Q16Q
+ij
+values can be found using
+Q16Q
+(4i−3),(4i−1) = Q16Q
+(4i−3),(4i)
+= Q16Q
+(4i−2),(4i−1) = Q16Q
+(4i−2),(4i) = 0,
+(A.13)
+Q16Q
+(4i−3),(4i−2) = Q16Q
+(4i−1),(4i) = 8P
+9
+�
+|hi,R|2 + |hi,I|2�
+,
+(A.14)
+Q16Q
+(4i−3),(4j−2) = Q16Q
+(4i−2),(4j−3) = Q16Q
+(4i),(4j−1)
+= Q16Q
+(4i−1),(4j) = P
+9 (8hi,Rhj,R + 8hi,Ihj,I) ,
+(A.15)
+Q16Q
+(4i−2),(4j−2) = Q16Q
+(4i),(4j) = P
+9 (4hi,Rhj,R + 4hi,Ihj,I) ,
+(A.16)
+Q16Q
+(4i−1),(4j−1) = Q16Q
+(4i−3),(4j−3)
+= P
+9 (16hi,Rhj,R + 16hi,Ihj,I) ,
+(A.17)
+Q16Q
+(4i−2),(4j−1) = Q16Q
+(4i−3),(4j) = 8P
+9 (−hi,Rhj,I + hi,Ihj,R) ,
+(A.18)
+Q16Q
+(4i−1),(4j−2) = Q16Q
+(4i),(4j−3)
+= P
+9 (−8hi,Ihj,R + 8hi,Rhj,I) ,
+(A.19)
+Q16Q
+(4i−2),(4j) = P
+9 (−4hi,Rhj,I + 4hi,Ihj,R) ,
+(A.20)
+Q16Q
+(4i),(4j−2) = P
+9 (−4hi,Ihj,R + 4hi,Rhj,I) ,
+(A.21)
+Q16Q
+(4i−1),(4j−3) = P
+9 (−16hi,Ihj,R + 16hi,Rhj,I) ,
+(A.22)
+Q16Q
+(4i−3),(4j−1) = P
+9 (−16hi,Rhj,I + 16hi,Ihj,R) .
+(A.23)
+4) 64-QAM: The derivations of Q64Q
+ij
+values are omitted
+due to the submission page limit and will be included in the
+extended version, if accepted.
+REFERENCES
+[1] “Cisco
+Annual
+Internet
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+Cisco
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+(2018–2023)
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+
+QQ
+(2i−1),(2i−1) = P
+2
+�
+−4hi,R
+�� N
+�
+l=1
+hl,R − hi,R
+�
+−
+� N
+�
+l=1
+hl,I − hi,I
+��
+− 4hi,I
+�� N
+�
+l=1
+hl,R − hi,R
+�
++
+� N
+�
+l=1
+hl,I − hi,I
+���
+−
+√
+2P (2yRhi,R + 2yIhi,I)
+(A.3)
+QQ
+(2i),(2i) = P
+2
+�
+−4hi,R
+�� N
+�
+l=1
+hl,R − hi,R
+�
++
+� N
+�
+l=1
+hl,I − hi,I
+��
+− 4hi,I
+�
+−
+� N
+�
+l=1
+hl,R − hi,R
+�
++
+� N
+�
+l=1
+hl,I − hi,I
+���
+−
+√
+2P (−2yRhi,I + 2yIhi,R)
+(A.4)
+Q16Q
+(4i−3),(4i−3) = P
+9
+�
+−4hi,R
+�
+hi,R + 3
+�� N
+�
+k=1
+hk,R − hi,R
+�
+−
+� N
+�
+k=1
+hk,I − hi,I
+���
+−4hi,I
+�
+hi,I + 3
+�� N
+�
+k=1
+hk,R − hi,R
+�
++
+� N
+�
+k=1
+hk,I − hi,I
+����
+−
+√
+2P
+3
+(4yRhi,R + 4yIhi,I)
+(A.9)
+Q16Q
+(4i−2),(4i−2) = P
+9
+�
+−4hi,R
+�
+hi,R + 3
+2
+�� N
+�
+k=1
+hk,R − hi,R
+�
+−
+� N
+�
+k=1
+hk,I − hi,I
+���
+−4hi,I
+�
+hi,I + 3
+2
+�� N
+�
+k=1
+hk,R − hi,R
+�
++
+� N
+�
+k=1
+hk,I − hi,I
+����
+−
+√
+2P
+3
+(2yRhi,R + 2yIhi,I)
+(A.10)
+Q16Q
+(4i−1),(4i−1) = P
+9
+�
+−4hi,R
+�
+hi,R + 3
+�� N
+�
+k=1
+hk,R − hi,R
+�
++
+� N
+�
+k=1
+hk,I − hi,I
+���
+−4hi,I
+�
+hi,I + 3
+�
+−
+� N
+�
+k=1
+hk,R − hi,R
+�
++
+� N
+�
+k=1
+hk,I − hi,I
+����
+−
+√
+2P
+3
+(−4yRhi,I + 4yIhi,R)
+(A.11)
+Q16Q
+(4i),(4i) = P
+9
+�
+−4hi,R
+�
+hi,R + 3
+2
+�� N
+�
+k=1
+hk,R − hi,R
+�
++
+� N
+�
+k=1
+hk,I − hi,I
+���
+−4hi,I
+�
+hi,I + 3
+2
+�
+−
+� N
+�
+k=1
+hk,R − hi,R
+�
++
+� N
+�
+k=1
+hk,I − hi,I
+����
+−
+√
+2P
+3
+(−2yRhi,I + 2yIhi,R)
+(A.12)
+[3] M. Kim, D. Venturelli, and K. Jamieson, “Leveraging Quantum Annealing
+for Large MIMO Processing in Centralized Radio Access Networks,” in
+Proc. of the ACM Special Interest Group on Data Communication, pp.
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+[4] Z. I. Tabi et al., “Evaluation of Quantum Annealer Performance via the
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+[5] R. C. Kizilirmak, “Quantum Annealing Approach to NOMA Signal
+Detection,” 12th Int. Symp. Commun. Syst. Netw. Digital Signal Process.
+(CSNDSP), Porto, Portugal, pp. 1-5, Jul. 2020.
+[6] F. Ishizaki, “Computational Method Using Quantum Annealing for
+TDMA Scheduling Problem in Wireless Sensor Networks,” 13th Int. Conf.
+Signal Process. Commun. Syst. (ICSPCS), Australia, pp. 1-9, Dec. 2019.
+[7] R. C. Kizilirmak, “Non-Orthogonal Multiple Access (NOMA) for 5G
+Networks”, Towards 5G Wireless Networks - A Physical Layer Perspec-
+tive. London, United Kingdom: IntechOpen, 2016 [Online]. Available:
+https://www.intechopen.com/chapters/52822
+[8] B. Makki et al., “A Survey of NOMA: Current Status and Open Research
+Challenges,” IEEE Open J. Commun. Soc., vol. 1, pp. 179-189, 2020.
+[9] “Error
+Sources
+for
+Problem
+Representation”
+D-Wave
+System
+Documentation,
+2021
+[Online].
+Available:
+https://docs.dwavesys.com/docs/latest/c qpu ice.html
+[Accessed
+Apr.
+2022].
+[10] “Solver
+Parameters,”
+D-Wave
+System
+Doc-
+umentation,
+2021
+[Online].
+Available:
+https://docs.dwavesys.com/docs/latest/c solver parameters.html
+[Accessed Apr. 2022].
+[11] “Constraints
+Example:
+Minor-Embedding”
+D-Wave
+System
+Documentation,
+2021
+[Online].
+Available:
+https://docs.dwavesys.com/docs/latest/c gs 7.html[Accessed Apr. 2022].
+[12] “D-Wave
+QPU
+Architecture:
+Topologies”
+D-Wave
+System
+Documentation,
+2021
+[Online].
+Available:
+https://docs.dwavesys.com/docs/latest/c gs 7.html[Accessed Apr. 2022].
+[13] “LTE; Evolved Universal Terrestrial Radio Access (E-UTRA); Base Sta-
+tion (BS) radio transmission and reception (3GPP TS 36.104 version 9.4.0
+Release 9),” ETSI, France, Tech. Specification. RTS/TSGR-0436104v940,
+Jul. 2010.
+[14] “LTE; Evolved Universal Terrestrial Radio Access (E-UTRA); User
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+sion 14.3.0 Release 14),” ETSI, France, Tech. Specification. RTS/TSGR-
+0436101ve30, Apr. 2017.
+[15] T. Rappaport, “Mobile Radio Propagation: Large-Scale Path Loss,” Wire-
+less Communications Principles and Practice, 2nd ed., Prentice Hall,
+2001, pp. 69-138.
+[16] “Operation and Timing” in D-Wave System Documentation, 2021 [On-
+line]. Available: https://docs.dwavesys.com/docs/latest/c qpu timing.html
+[Accessed 5 Apr. 2022].
+
diff --git a/btE2T4oBgHgl3EQfaQc9/content/tmp_files/load_file.txt b/btE2T4oBgHgl3EQfaQc9/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..cd34790d1f5809accacb12c1db545a67ade6fea3
--- /dev/null
+++ b/btE2T4oBgHgl3EQfaQc9/content/tmp_files/load_file.txt
@@ -0,0 +1,432 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf,len=431
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='03872v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='IT]  10 Jan 2023  Towards Quantum Annealing for Multi-user  NOMA-based Networks  Eldar Gabdulsattarov, #Khaled Rabie, ◦Xingwang Li, and Galymzhan Nauryzbayev  �  School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan, Z05H0K4, Kazakhstan  #Department of Engineering, Manchester Metropolitan University, Manchester, M15 6BH, UK  School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454000, China  Email: {eldar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='gabdulsattarov, galymzhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='nauryzbayev}@nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='kz, #k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='rabie@mmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='uk, ◦lixingwang@hpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='cn  Abstract—Quantum Annealing (QA) uses quantum fluctuations  to search for a global minimum of an optimization-type problem  faster than classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' To meet the demand for future  internet traffic and mitigate the spectrum scarcity, this work  presents the QA-aided maximum likelihood (ML) decoder for  multi-user non-orthogonal multiple access (NOMA) networks as  an alternative to the successive interference cancellation (SIC)  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The practical system parameters such as channel ran-  domness and possible transmit power levels are taken into account  for all individual signals of all involved users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The brute force (BF)  and SIC signal detection methods are taken as benchmarks in the  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The QA-assisted ML decoder results in the same BER  performance as the BF method outperforming the SIC technique,  but the execution of QA takes more time than BF and SIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  The parallelization technique can be a potential aid to fasten  the execution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' This will pave the way to fully realize the  potential of QA decoders in NOMA systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  Index Terms—Quantum annealing, non-orthogonal multiple  access (NOMA), maximum likelihood (ML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' INTRODUCTION  There has been a drastic rise in the traffic of advanced  multimedia applications such as real-time conferencing, online  video gaming and virtual reality in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Moreover,  according to the Cisco Annual Report, the total number of  internet users will reach 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='3 billion by 2023 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' To meet this  ever-growing demand on traffic, it is crucial to consider some  optimization improvements in the quality of data transmission  in terms of the bandwidth, error rate, and delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' One of the  actions that can be taken into account is to implement Quantum  Annealing (QA) technologies to solve NP-hard optimization  problems in wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  The authors in [2]–[5] suggested the centralized radio access  network (C-RAN) as a promising and cost-effective design  architecture for future wireless networks, where multiple base  stations (BSs) are interconnected with a centralized data center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  The C-RAN’s data centers support most of the BSs’ signal  processing operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' It is envisioned that the QA comput-  ers placed at the C-RAN centers can facilitate optimization  problems, paving the way for full integration of the presented  methodologies in real-world communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  One of the computational complex problems in wireless  networks is to obtain an optimal scheduling in multi-user  (MU) systems, which can help one effectively employ network  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The authors in [6] introduced the QA approach to  find an optimal solution for the time-division multiple access  (TDMA) transmission scheduling problem in a wireless sensor  network using a tree topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The results revealed that QA  outperforms the other scheduling solutions in terms of the  runtime and proximity to the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  An essential part of cellular base-band processing that can  enhance the quality of communication is the error correction  code (ECC), which mitigates the errors in received bit-streams  caused by the noise in a wireless environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' However, the  error correction code protocols with exceptional performance  require significant computational resources for the decoding  process at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' For this reason, the implementation of  QA computers was proposed in [2] for efficient decoding of  low-density parity-check codes in uplink systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The method  achieved a bit error rate (BER) of 10−8 at the signal-to-noise  ratio (SNR) of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='5-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='5 dB lower than the standard algorithm for  decoding ECCs on a field-programmable gate array (FPGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  Another key aspect in the performance of wireless communi-  cation systems is a signal detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' To achieve the full potential  of multi-user systems, BSs have to efficiently decode superim-  posed data streams received from different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Maximum  likelihood (ML) usually ends up with an optimum solution for  decoding the received signals, but it is an NP-hard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The  computational cost of ML rises exponentially with the number  of users and data rate [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' For large-scale networks, the ML  decoding process might last expiration of acknowledgement  timeout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The authors in [3]–[5] employed a QA approach to  speed up this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Particularly, [3] presented the QuAMax  approach for ML decoding in MU MIMO systems using QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  The performance of the solution was tested on the quantum  annealer with 2048 qubits considering different modulation  schemes and multiple users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The method achieves the same  BER as ZF does, but with significantly lower computation time  in the case of binary phase-shift keying (BPSK) and quadrature  phase-shift keying (QPSK) modulation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Moreover, the  QA approach outperforms the conventional decoders in terms of  the computable size of ML problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' For instance, the work in  [3] was extended by [4], where an experiment on the recently  released QA hardware with 5640 qubits was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The  results revealed that the new computer is able to provide better  scaling for embedding the ML MIMO decoding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' In  addition, the study showed that today’s QA machines can  achieve identical BER as the ML sphere decoder for 4 × 4  MIMO using 16-QAM (quadrature amplitude modulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  The current communication technologies support multiple  users using orthogonal multiple access (OMA) techniques such  as TDMA, frequency- and code-division multiple access (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=',  FDMA and CDMA) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' However, all these techniques  have certain shortcomings such as precise time synchronization,  limited frequency carriers and code length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' These drawbacks  get more perceptible for a large number of users [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Non-  orthogonal multiple access (NOMA) is a technology that al-  lows all users in a cell operate in the common time and  frequency, that, in turn, improve the spectral efficiency (SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  The successive interference cancellation (SIC) is considered  as a promising decoding technique for NOMA that decodes  symbol-by-symbol until the desired one is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' However,  the high computational cost, latency and error propagation of  SIC make it impractical to implement NOMA in the current  wireless communication systems [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The traditional ML  decoder supports high-quality decoding but suffers from high  computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' To address this issue, a NOMA-ML decoder  based on QA was proposed in [5] for a two-user system model,  with a BPSK modulation and constant channel gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  In contrast to the previous studies, in this work, we consider  the NOMA-based uplink network comprising multiple users  under practical scenarios and different modulation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The  NOMA-ML decoding problem was firstly reformulated into  a QUBO model for multiple modulation schemes including  BPSK, QPSK, 16−QAM and 64−QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Moreover, the gen-  eralized QUBO expressions of the proposed system model are  derived for the multi-user network under the previously defined  modulation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' We have investigated how the number of  users in a cell affects the signal decoding performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Fur-  thermore, the QA-assisted ML decoder was compared with SIC  and Brute Force (BF) techniques in terms of BER performance  and simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Finally, the influence of the parallelization  method on the execution time of the QA machine is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' SYSTEM MODEL  The system model considered here represents an uplink  NOMA-based network scenario (depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 1) compris-  ing BS and N end-users, denoted by Uk, with k ∈ A =  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' It is assumed that all nodes are deployed with a  single antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The channel between the user of interest and BS  is given by ˜hk and assumed to be independent and identically  distributed random variables (RVs) following Rayleigh fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Non-orthogonal Multiple Access  Due to the considered uplink mode of communication, all  end-users simultaneously transmit their individual messages to  BS while ensuring the NOMA principle to be true through the  path loss associated with their corresponding distances1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Hence,  the received signal at BS can be written as  y =  N  �  k=1  skhk + n,  (1)  1Note that the similar assumption can be made using the principle of  messages’ priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  0  U1  50  U2  50N  N−1  U3  50(N+1)  N−1  UN  100  m  h1  h2  h3  hN  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' An illustration of uplink NOMA with N users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  where sk is the transmitted message of user k, with the transmit  power defined as E{sks∗  k} = Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' For the sake of simplicity,  we assume that all end-users transmit with the same power  levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=', P = P1 = P2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' = PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' hk is modelled as  hk = ˜hk/  �  dτ  k, where dk and τ indicate the corresponding  distance and path-loss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' n is an additive white Gaussian  noise (AWGN) term, with zero mean and variance σ2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Maximum Likelihood Detection  The receiver implements the ML detection as a decoding  method, which output, given by an optimal symbol vector ˆsML,  is formulated as  ˆsML = arg min  ˆs1,ˆs2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='ˆsN  |y − shT|2,  (2)  where y is the received signal, s and h are the 1 × N vectors  representing the transmit symbols and channel gains of end-  users, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Quantum Annealing  Quantum computing implements the concepts of quantum  mechanics to solve high-computational problems faster than  classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' There are several types of quantum com-  puters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' If the quantum circuit model controls the qubits using  quantum logic gates to solve a wide range of problems, QA  is an effective tool to solve optimization problems with given  objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Specifically, the quantum processing unit  (QPU) uses quantum fluctuations to search for the global  minimum solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' However, modern quantum computers are  not that perfect yet and QA might end up with a local minimum  solution due to the analogue noises in the device [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' To  mitigate the internal errors in QPU, the annealing parameters  could be modified based on the specifics of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' To  minimize these errors, QA can be simulated many times, and  the optimal solution will be the answer with the lowest energy  level and most frequent occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The annealing parameter  that controls the number of samples per simulation is the  number of reads (R), it was decided to set it to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Another  annealing parameter that could impact the performance of the  QA machine is the single-sample annealing time (Ta), which  was set to 20 µs without any pausing time [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Quadratic Unconstrained Binary Optimization  To solve an optimization problem by a means of QA, the  objective function should be reformulated in a form of the  quadratic unconstrained binary optimization (QUBO) model or  the Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' These formulations can be used interchange-  ably, and it is easy to switch from one to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' At the end  a  7a  5a  3a  a  3a  5a  7a  b  3b  b  3b  1  1  j  j  Im  Re  BPSK  64-QAM  16-QAM  QPSK  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Constellations of different modulation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  of annealing process, the best solution is given by the qubit  sequence with a minimum energy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The model is given by  an upper-diagonal matrix Q with a size of M × M  ˆq1, ˆq2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' , ˆqM = arg min  q1,q2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='qM  M  �  i≤j  Qijqiqj,  (3)  where M denotes the number of qubits, and each variable qi  takes binary numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' By applying the property q2  i = qiqi = qi,  and substituting the symbols in (2) by the corresponding binary  equivalent form, it becomes feasible to formulate the NOMA  ML problem in terms of the QUBO form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The corresponding di-  agonal Qii and non-diagonal Qij terms of the QUBO model are  represented by linear and quadratic coefficients, accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  To implement the obtained QUBO model on QPU, the matrix  coefficients have to be mapped into physical qubits of a QA  chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The linear coefficients define the qubit biases, while  quadratic coefficients represent the coupling strengths [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  However, QPU’s physical qubits are not fully connected, in the  Chimera graph each qubit has six neighbours, and, in Pegasus  topology, each qubit is connected with 15 neighbour qubits  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Therefore, sometimes it is not possible to directly apply  the QUBO model onto real QA machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' FORMULATION OF NOMA ML PROBLEM IN A QUBO  MODEL  The ML expression in (2) should be described by a QUBO  model to solve it with QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Taking into account (1) and the  complex numbers due to the broadcast transmission, we split  the received signal y and channel coefficients into real and  imaginary parts as follows: y = yR + jyI and hk = hk,R +  jhk,I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Each symbol, denoted by sk, can be transformed into  the qubit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The methodology for representing symbols in  terms of the binary variables depends on the modulation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  The considered modulations are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  1) BPSK: The BPSK symbols of a form sk ∈ {±1}, can  be reformulated into the QUBO form by sk = 2qk − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' In this  case, M in (3) takes a value of the number of users, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  2  3  4  5  6  7  8  9  10  10  4  10  3  10  2  BER  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' of users per cell  User 1  User 2  User 3  User 4  User 5  User 6  User 7  User 8  User 9  User 10  Average BER  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' BER vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' the number of users per cell under BPSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  2) QPSK: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 2, the QPSK symbols, sk ∈  { ±1±j  √  2 }, reside on a unit circle with 90◦ phase difference  between two consecutive symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' They can be represented by  two qubits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' therefore, the term M in (3) becomes the double  of a number of users, 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Finally, the QPSK symbols can be  expressed as sk = [(2q2k−1 − 1) + j (2q2k − 1)] /  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  3) 16-QAM: The 16-QAM modulation encodes four in-  formation bits into a complex symbol and correspondingly  includes 16 constellation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The symbols sk ∈ {±b ±  jb, ±b ± j3b, ±3b ± jb, ±3b ± j3b}, where b = 1/(3  √  2),  require four qubits to be adapted for the QUBO model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=',  sk = [(4q4k−3 + 2q4k−2 − 3) + j (4q4k−1 + 2q4k − 3)] /3  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  M in (3) becomes the quadruple of a number of users, 4N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  4) 64-QAM: On the other hand, the 64-QAM modulation  returns a complex symbol by encoding 16 information bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 2, the constellation points are defined as  per Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 2 and require six qubits to be adopted for the  QUBO model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' therefore, the term M in (3) becomes the  sixtuple of a number of users, 6N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' These symbols can be  re-expressed as sk = a (8q6k−5 + 4q6k−4 + 2q6k−3 − 7) +  ja (8q6k−2 + 4q6k−1 + 2q6k − 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' IMPLEMENTATION AND RESULTS DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Implementation  The system model considered in the analysis represents a N-  end user uplink NOMA system that employs BPSK modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 1, the base station is located at a coordinate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' We assume that the first and the farthest users are located  50 and 100 m away from the base station, respectively, while  the other intermediate nodes are evenly distributed in-between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  For practical purposes, we apply the standardized scenarios  described in [13] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The performance of the exploited  solvers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=', BF, SIC and QA) were examined by varying the  transmit power from −40 dBm to 24 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Moreover, the  environment of data transmission is considered to be a free  space with line-of-sight propagation which relates to the path-  loss exponent τ = 2 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  40  30  20  10  0  10  20  10  2  10  1  10  0  BF  Bit error rate  Transmit power, dBm  U  1  U  2  U  3  QA  SIC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The BER performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' the transmit power for a three-  user NOMA scenario, with different decoding techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='07  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='07  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='07  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='07  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='00  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='00  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='00  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='00  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='94  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='19  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='39  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='06  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='54  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='54  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='54  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='54  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='07  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='74  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='51  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='51  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='37  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='74  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='74  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='65  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='50  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='38  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='00  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='32  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='81  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='89  QA:  p  T  D  T  R  T  A  T  p  14  10  10  30  Transmit power, dBm  SIC  BF  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Comparison between the decoding techniques in terms  of simulation time for five (5) samples (in ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  The QA method was implemented on the proposed system  model using D-Wave’s Advantage QPU [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' It is important  to mention that the parallelization procedure was used to save  the simulation time on the QA computer [3], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=', 5 instances  of the problem were run at the same time on QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Since  the 3-user BPSK problem requires only 3 logical qubits, 5  instances would occupy 15 logical qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' In the analysis of  simulation time, the following terms are used as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' QPU  Programming Time (Tp) is the time taken for programming  the couplers and biases of the chip in accordance with the  QUBO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' QPU Sampling Time (Ts) is the total time for  simulation of R samples and consists of the annealing time  (Ta), the readout time (Tr), and the delay time (Td) for every  single sample [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' For the sake of simplicity, we declare the  following terms: the total QPU Annealing Time (TA = R ·Ta),  the total QPU Readout Time (TR = R · Tr), and the QPU  Delay Time (TD = R · Td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Tp, Ts, QPU Access Overhead  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Simulation time (in ms) of the decoding techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  Technique  for 5 samples  for 1 sample  SIC  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='066  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='958  BF  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='006  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='535  QA  QPU Service Time  148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='116  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='984  Tp  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='069  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='068  TA  20  20  TR  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='145  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='68  TD  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='54  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='54  ∆  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='835  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='86  ∆p  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='527  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='836  Time (∆) and Post-processing Overhead Time (∆p) constitute  QPU Service Time, which is considered as the total time taken  for the QA simulation excluding internet delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Discussion  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 3, we examine the influence of the number of  users per cell on the BER performance in the NOMA system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  Particularly, the transmit power and the AWGN power were  set to 10 dBm and −60 dBm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' As expected, the  average BER performance degrades with the increase in the  number of users per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' It is seen from the plot that, for the  number of users greater than 6, the difference between the BER  curves becomes quite similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Therefore, we apply  the 3-user NOMA scenario in the further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 4 and 5, we compare the SIC, BF and QA decoding  techniques in terms of the BER performance and simulation  time, with the noise power set to σ2  n = −30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' QA  was simulated for particular transmit power levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=', P =  {−30, −10, 10, 14} dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The number of QA simulations per  power level was taken to ensure a 1% accuracy with respect  to the BER performance of BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Overall, about 380 simulations  (with 5 problems at a time) were performed on QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 4 illustrates the BER performances of each decoding  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The BER curves can be characterized by different  behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' It can be seen from the U1’s curve, the SIC and  BF performances coincide up to −5 dBm, and then SIC starts  outperforming BF over a short transmit power range (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=',  from −5 dBm to 10 dBm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' However, afterwards, SIC starts  experiencing noticeable performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' At the same  time, U2 has identical BER for SIC and BF up to 10 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  After this point, there is a slight improvement of the SIC curve,  but, in general, its performance is much worse than that of BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  What stands out for U3 is the equal performance of SIC and BF  for transmit power less than 5 dBm, thereafter, the difference  between them begins to grow considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The performance  of BF improves remarkably after 5 dBm, whereas the SIC  curve enhances minimally until 15 dBm with the following  saturation after this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' In general, the QA, BF and SIC  curves follow the same trend till 10 dBm, but after this level,  the SIC performance degrades substantially, while the BER  result of QA is approximately the same as BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The reason  for the SIC’s low performance could be insufficient differences  between the end-users’ power levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 5 presents the timing data for each technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' As  can be seen from the histogram plot, SIC shows the fastest  execution among the presented techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' As expected, the  second fastest technique is BF, since it needs to iterate over all  possible combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The QA execution time interval consists  of several timing sub-intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' While Tp, TA and TD are the  same for all cases, TR shows a slight variation, since the reading  time depends on the position of exploited physical qubits on the  chip and on the number of used qubits (more time is needed  for a larger number of qubits) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The dependence on the  number of qubits could be noticed in Table I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=', TR for 1  sample is twice as fast as TR for 5 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Overall, the total  time taken for QA execution is about 30 times more than the  SIC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' However, it is important to mention that QA  needs to simulate the same problem R number of times (in  our case it is set to 1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Furthermore, the problem could be  addressed by the parallelization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' From Table I, it is  clearly noticeable that the execution of 5 instances in parallel  is more than 3 times faster than executing the same number of  samples in a sequential order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Therefore, one could potentially  conclude that the overall QA simulation time can be decreased  by simulating multiple problem instances at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' CONCLUSION  To mitigate the problem of spectrum scarcity and present  an alternative for SIC, this work aims at evaluating the per-  formance of QA-aided ML detection for NOMA systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' The  ML problem was firstly described in terms of the QUBO model  to enable the integration of the problem with QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' For 3-user  NOMA under BPSK modulation, the BER performance of QA  is approximately the same as the BF method, but QA takes  longer time to execute due to the current hardware specifics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  The parallelization method seems to be a potential solution  that could decrease the execution time of QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Additionally,  the analogous noises in QA could be suppressed with coming  of the next-generation QA hardware, and the time taken for  error alleviation would be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' This will pave the way for  computing time-dependent operations on QA including the QA-  assisted ML decoding technique for future NOMA systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' ACKNOWLEDGEMENT  This work was supported by the Nazarbayev University (NU)  Social Policy grant, the NU FDCRP Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 240919FD3935  and the NU CRP Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 11022021CRP1513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  APPENDIX A  QUBO MODEL COEFFICIENTS  In this section, we present the coefficients for the QUBO  model for different modulation types2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  1) BPSK: The corresponding QB  ij values can be found as  QB  ii = P  �  −4hi,R  � N  �  l=1  hl,R−hi,R  �  −4hi,I  � N  �  l=1  hl,I−hi,I  ��  −  √  P (4yRhi,R + 4yIhi,I) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='1)  QB  ij = P (8hi,Rhj,R + 8hi,Ihj,I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='2)  2Note that the equations involving the positive integer variables i and j must  satisfy the condition i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  2) QPSK: The QQ  ij values can be found using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='3), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='4)  and the equations given below  QQ  (2i−1),(2i) = 0,  ∀i ≥ 1,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='5)  QQ  (2i−1),(2j−1) = QQ  (2i),(2j) = P  2 (8hi,Rhj,R + 8hi,Ihj,I) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='6)  QQ  (2i−1),(2j) = P  2 (8hi,Ihj,R − 8hi,Rhj,I) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='7)  QQ  (2i),(2j−1) = P  2 (−8hi,Ihj,R + 8hi,Rhj,I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='8)  3) 16-QAM: The Q16Q  ij  values can be found using  Q16Q  (4i−3),(4i−1) = Q16Q  (4i−3),(4i)  = Q16Q  (4i−2),(4i−1) = Q16Q  (4i−2),(4i) = 0,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='13)  Q16Q  (4i−3),(4i−2) = Q16Q  (4i−1),(4i) = 8P  9  �  |hi,R|2 + |hi,I|2�  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='14)  Q16Q  (4i−3),(4j−2) = Q16Q  (4i−2),(4j−3) = Q16Q  (4i),(4j−1)  = Q16Q  (4i−1),(4j) = P  9 (8hi,Rhj,R + 8hi,Ihj,I) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='15)  Q16Q  (4i−2),(4j−2) = Q16Q  (4i),(4j) = P  9 (4hi,Rhj,R + 4hi,Ihj,I) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='16)  Q16Q  (4i−1),(4j−1) = Q16Q  (4i−3),(4j−3)  = P  9 (16hi,Rhj,R + 16hi,Ihj,I) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='17)  Q16Q  (4i−2),(4j−1) = Q16Q  (4i−3),(4j) = 8P  9 (−hi,Rhj,I + hi,Ihj,R) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='18)  Q16Q  (4i−1),(4j−2) = Q16Q  (4i),(4j−3)  = P  9 (−8hi,Ihj,R + 8hi,Rhj,I) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='19)  Q16Q  (4i−2),(4j) = P  9 (−4hi,Rhj,I + 4hi,Ihj,R) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='20)  Q16Q  (4i),(4j−2) = P  9 (−4hi,Ihj,R + 4hi,Rhj,I) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='21)  Q16Q  (4i−1),(4j−3) = P  9 (−16hi,Ihj,R + 16hi,Rhj,I) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='22)  Q16Q  (4i−3),(4j−1) = P  9 (−16hi,Rhj,I + 16hi,Ihj,R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='23)  4) 64-QAM: The derivations of Q64Q  ij  values are omitted  due to the submission page limit and will be included in the  extended version, if accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  REFERENCES  [1] “Cisco  Annual  Internet  Report Cisco  Annual  Internet  Report  (2018–2023)  White  Paper,”  Cisco,  March  2020,  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  Available:  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='cisco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='com/c/en/us/solutions/collateral/executive-  perspectives/annual-internet-report/white-paper-c11-741490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Kasi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Jamieson, “Towards quantum belief propagation for LDPC  decoding in wireless networks,” 26th Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Mobile Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=', London, UK, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 1-14, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  QQ  (2i−1),(2i−1) = P  2  �  −4hi,R  �� N  �  l=1  hl,R − hi,R  �  −  � N  �  l=1  hl,I − hi,I  ��  − 4hi,I  �� N  �  l=1  hl,R − hi,R  �  +  � N  �  l=1  hl,I − hi,I  ���  −  √  2P (2yRhi,R + 2yIhi,I)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='3)  QQ  (2i),(2i) = P  2  �  −4hi,R  �� N  �  l=1  hl,R − hi,R  �  +  � N  �  l=1  hl,I − hi,I  ��  − 4hi,I  �  −  � N  �  l=1  hl,R − hi,R  �  +  � N  �  l=1  hl,I − hi,I  ���  −  √  2P (−2yRhi,I + 2yIhi,R)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='4)  Q16Q  (4i−3),(4i−3) = P  9  �  −4hi,R  �  hi,R + 3  �� N  �  k=1  hk,R − hi,R  �  −  � N  �  k=1  hk,I − hi,I  ���  −4hi,I  �  hi,I + 3  �� N  �  k=1  hk,R − hi,R  �  +  � N  �  k=1  hk,I − hi,I  ����  −  √  2P  3  (4yRhi,R + 4yIhi,I)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='9)  Q16Q  (4i−2),(4i−2) = P  9  �  −4hi,R  �  hi,R + 3  2  �� N  �  k=1  hk,R − hi,R  �  −  � N  �  k=1  hk,I − hi,I  ���  −4hi,I  �  hi,I + 3  2  �� N  �  k=1  hk,R − hi,R  �  +  � N  �  k=1  hk,I − hi,I  ����  −  √  2P  3  (2yRhi,R + 2yIhi,I)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='10)  Q16Q  (4i−1),(4i−1) = P  9  �  −4hi,R  �  hi,R + 3  �� N  �  k=1  hk,R − hi,R  �  +  � N  �  k=1  hk,I − hi,I  ���  −4hi,I  �  hi,I + 3  �  −  � N  �  k=1  hk,R − hi,R  �  +  � N  �  k=1  hk,I − hi,I  ����  −  √  2P  3  (−4yRhi,I + 4yIhi,R)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='11)  Q16Q  (4i),(4i) = P  9  �  −4hi,R  �  hi,R + 3  2  �� N  �  k=1  hk,R − hi,R  �  +  � N  �  k=1  hk,I − hi,I  ���  −4hi,I  �  hi,I + 3  2  �  −  � N  �  k=1  hk,R − hi,R  �  +  � N  �  k=1  hk,I − hi,I  ����  −  √  2P  3  (−2yRhi,I + 2yIhi,R)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='12)  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content=' Venturelli, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content=', “A Survey of NOMA: Current Status and Open Research  Challenges,” IEEE Open J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content='  [10] “Solver  Parameters,”  D-Wave  System  Doc-  umentation,  2021  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content='  [11] “Constraints  Example:  Minor-Embedding”  D-Wave  System  Documentation,  2021  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content='  [12] “D-Wave  QPU  Architecture:  Topologies”  D-Wave  System  Documentation,  2021  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content=' Evolved Universal Terrestrial Radio Access (E-UTRA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='0  Release 9),” ETSI, France, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' RTS/TSGR-0436104v940,  Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  [14] “LTE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='0 Release 14),” ETSI, France, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' RTS/TSGR-  0436101ve30, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content='  [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
+page_content=' Rappaport, “Mobile Radio Propagation: Large-Scale Path Loss,” Wire-  less Communications Principles and Practice, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE2T4oBgHgl3EQfaQc9/content/2301.03872v1.pdf'}
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+arXiv:2301.01117v1  [math.AG]  3 Jan 2023
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A
+GIVEN CURVE
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+Abstract. In the present note we construct new families of free plane curves
+starting from a curve C and adding high order inflectional tangent lines of C,
+lines joining the singularities of the curve C, or lines in the tangent cone of some
+singularities of C.
+These lines L have in common that the intersection C ∩ L
+consists of a small number of points. We introduce the notion of a supersolvable
+plane curve and conjecture that such curves are always free, as in the known case of
+line arrangements. Some evidence for this conjecture is given as well, both in terms
+of a general result in the case of quasi homogeneous singularities and in terms of
+specific examples.
+1. Introduction
+Our goal in this paper is to construct new free or nearly curves by adding inflec-
+tional tangents or lines passing through the singularities of a given plane projective
+curve. Sometimes, when high order inflectional tangents are missing, lines in the
+tangent cones of the singularities may replace them successfully. The use of tangent
+cones is a must when we want to get supersolvable curves with the modular point
+belonging to a non-linear irreducible component, see Definition 1.9.
+To determine the existence of inflectional tangents, we start by recalling some
+facts about inflection points. Let C : F = 0 be a reduced plane curve in the complex
+projective plane P2 which is defined by a homogeneous polynomial F ∈ S = C[x, y, z]
+of degree d ≥ 2. The Hessian of F is given by the following well-known formula
+(1.1)
+H = det
+
+
+Fxx
+Fxy
+Fxz
+Fxy
+Fyy
+Fyz
+Fxz
+Fyz
+Fzz
+
+ .
+Let HC : H = 0 be the Hessian curve associated to C. It is known that the inter-
+section XC = C ∩ HC consists exactly of the set of inflection points IC of C union
+with the set of singular points YC of C. Recall that if p ∈ C is a smooth point of
+this curve, and TpC denotes the projective line tangent to C at p, then the inflection
+2010 Mathematics Subject Classification. Primary 14H50; Secondary 14B05, 13D02, 32S22.
+Key words and phrases. plane curve.
+A. Dimca was partially supported by the Romanian Ministry of Research and Innovation, CNCS -
+UEFISCDI, Grant PN-III-P4-ID-PCE-2020-0029, within PNCDI III.
+P. Pokora was partially supported by the National Science Center (Poland) Sonata Grant Nr
+2018/31/D/ST1/00177.
+1
+
+2
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+order of p is by definition
+(1.2)
+ιp(C) = (C, TpC)p − 2,
+where (C, TpC)p denotes the intersection multiplicity of the curves C and TpC at the
+common point p. Moreover, we say that p is an inflection point of C, i.e., p ∈ IC, if
+and only if ιp(C) > 0.
+The intersection multiplicity (C, HC)p of the curves C and HC at the point p is a
+key invariant in understanding the geometry of the plane curve C. When p ∈ IC is
+an inflection point, then the relation between the inflection order ιp(C) of p and the
+intersection multiplicity (C, HC)p is well-known, see for instance [14, Theorem 9.7
+and Corollary 9.10].
+Theorem 1.1. For any reduced plane curve C of degree d ≥ 2 and any smooth point
+p ∈ C, one has
+ιp(C) = (C, HC)p.
+It is clear that ιp(C) ≤ d − 2, except the case when p sits on a line L = TpC which
+is an irreducible component of C. In the later case, one has ιp(C) = ∞, and it is easy
+to check that the line L is also an irreducible component of the Hessian curve HC.
+From now on, while searching for the inflection points, we assume that no irreducible
+component of C is a line.
+When p ∈ YC is a singular point, then the description of the intersection multi-
+plicity (C, HC)p is more subtle. Let TCp(C) be the reduced projective tangent cone
+of the curve C at p. For any line L ∈ TCp(C) we define the corresponding tangential
+multiplicity
+mL(C) = (L, CL)p,
+where (CL, p) is the union of all branches of the singularity (C, p) whose tangent line
+at p coincides with L. With this notation, one has the following general result, which
+is a user-friendly reformulation of [13, Proposition 25].
+Theorem 1.2. Assume that C is a reduced plane curve and that p ∈ C is any
+singular point. Then one has
+(C, HC)p = 3µp(C) + mp(C) − 3 +
+�
+L∈TCp(C)
+mL(C),
+where µp(C) is the Milnor number and mp(C) is the multiplicity of the singularity
+(C, p).
+The case when (C, p) is irreducible is also stated in [16, Theorem 2.1.9]. Indeed, in
+this case, one has µp(C) = 2δp(C) with δp(C) being the δ-invariant of the singularity
+(C, p). We discuss in detail the statement of Theorem 1.2 and give some examples
+in the next section.
+Our new results are the following. Consider the case when (C, p) is an ordinary
+k-multiple point, that is there are k smooth branches C1, . . . , Ck at p, with distinct
+tangent lines L1, . . . , Lk. If we set mj = (Cj, Lj)p ≥ 2 for j ∈ {1, . . . , k}, then we
+call (C, p) an ordinary k-multiple point of type (m1, . . . , mk). Moreover, we say that
+(C, p) is an ordinary simple k-multiple point if mj = 2 for all j ∈ {1, . . . , k}.
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+3
+Theorem 1.3. For any reduced plane curve C of degree d ≥ 3 and any singular
+point p ∈ C with multiplicity mp(C) = k, one has
+(C, HC)p ≥ 3k(k − 1),
+and equality holds if and only if (C, p) is an ordinary simple k-multiple point.
+Corollary 1.4. For any reduced plane curve C of degree d ≥ 3 and having s singular
+points, one has
+i(C) ≤ 3d(d − 2) − 6s.
+More precisely, if nk denotes the number of singular points of C of multiplicity k,
+then
+i(C) ≤ 3d(d − 2) −
+�
+k
+3k(k − 1)nk.
+The equality occurs if and only if all the singularities of C are ordinary simple k-
+multiple points for various k.
+This result is a significant improvement of the inequality
+�
+p∈IC
+ιp ≤ 3d(d − 2) − s
+which is given in [14, Corollary 9.9]. Our next result is a general construction of free
+curves by adding inflectional tangents and lines passing through the singularities of
+the initial curve C, which is a curve of Thom-Sebastiani type. Let m ≥ 2 be a
+positive integer and let ℓj(x, y) with j ∈ {1, . . . , m} be m distinct linear forms in x
+and y. Consider the curve
+C : F =
+m
+�
+j=1
+ℓj(x, y)kj − zd = 0,
+in P2, where kj ≥ 1 are positive integers such that � kj = d, and the family of lines
+Lj : ℓj(x, y) = 0. The line Lj is the inflectional tangent at pj if kj = 1, and it is the
+reduced tangent cone at the singularity pj when kj > 1.
+Theorem 1.5. With the notation as above, the curve
+C′ = C ∪
+m
+�
+j=1
+Lj :
+F ′ = F ·
+m
+�
+j=1
+ℓj = 0
+of degree d + m is free with the exponents (m − 1, d). Moreover, if L : z = 0 is the
+line passing through all the points pj of C, then the curve
+C′′ = C′ ∪ L :
+F ′′ = zF ·
+m
+�
+j=1
+ℓj = 0
+of degree d + m + 1 is free with the exponents (m − 1, d + 1).
+
+4
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+Starting with a smooth Fermat type curve and adding all of its inflectional tangents
+and the 3 coordinate axes, one gets again a free curve, as the following result shows.
+Consider the Fermat curve C : xd + yd + zd = 0. Let ǫ be any root of the equation
+td+1 = 0 in C. Then the line Lǫ : y = ǫx intersects C only at the point pǫ = (1 : ǫ : 0).
+Hence pǫ is an inflection point of the maximal order and hence i(Lǫ) = d − 2, i.e.,
+one has the equality in (2.9). In this way, we get d inflection points, which are just
+the intersection of C with the line z = 0. Cyclically permuting x, y, z, we get all the
+3d inflection points of this type, and these are all the inflection points of C by (2.7).
+The following result is a special case of Theorem 1.5, when kj = 1 for all j.
+Corollary 1.6. The union C′ : F ′ = (xd + yd)F = 0 of the smooth Fermat curve
+C : F = xd + yd + zd = 0 of degree d with the d inflectional tangents Lǫ meeting at
+one point, is a free curve of degree 2d and the exponents are (d − 1, d). When d = 3,
+the curve C′ is maximizing of degree 6. The union C′′ : F ′′ = zF ′ = 0 of the curve
+C′ with the line L : z = 0 passing through all the flex points pǫ of C, is a free curve
+of degree 2d + 1 and the exponents are (d − 1, d). When d = 3, then the curve C′ is
+maximizing of degree 7.
+Note that if we continue to add just inflectional tangents, the resulting curve is no
+longer a free curve. For instance, the curve
+C′′ : F ′′ = (x3 + y3 + z3)(x3 + y3)(y3 + z3) = 0
+is nearly free with the exponents (4, 5), and the curve
+C′′′ : F ′′′ = (x3 + y3 + z3)(x3 + y3)(y3 + z3)(x3 + z3) = 0
+is not even nearly free. However, we have the following general result.
+Theorem 1.7. The smooth Fermat curve C : F = xd + yd + zd = 0 has exactly 3d
+inflectional tangents and their union forms the following line arrangement
+A : (xd + yd)(yd + zd)(xd + zd) = 0.
+The union of the curve C, the lines in A, and the 3 coordinate axes produce a new
+curve
+C′ : F ′ = xyz(xd + yd)(yd + zd)(xd + zd)(xd + yd + zd) = 0
+of degree 4d + 3, which is free with the exponents (2d + 1, 2d + 1). Moreover, the
+curve C′′ ⊂ C′ given by
+C′′ : F ′′ = xy(yd + zd)(xd + zd)(xd + yd + zd) = 0
+has degree 3d + 2 and it is free with the exponents (d + 1, 2d).
+When d = 2, then the curve C′′ is maximizing of degree 8.
+It is easy to prove that the curve
+C : F = xmym + ymzm + xmzm = 0
+has no infection points using Theorem 1.2, see Example 6.3 below. To get a free
+curve from C, we may add two of the three tangent cones, or just one tangent cone
+and two lines joining singular points. Indeed, one has the following result.
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+5
+Theorem 1.8. The curve
+C′ : F ′ = (xmym + ymzm + xmzm)(xm + ym)(ym + zm) = 0
+has degree 4m and it is free with exponents (2m − 1, 2m) for any m ≥ 2. The curve
+C′′ : F ′′ = yz(xmym + ymzm + xmzm)(ym + zm) = 0
+has degree 3m + 2 and it is free with exponents (m + 1, 2m) for any m ≥ 2.
+Definition 1.9. Given a reduced plane curve C, we say that p ∈ C is a modular
+point for C if the central projection
+πp : P2 \ {p} → P1
+induces a locally trivial fibration of the complement M(C) = P2 \ C. We say that a
+reduced plane curve C is supersolvable if it has at least one modular point.
+The map induced by πp is a locally trivial fibration if and only if for any line Lp
+passing through p and not an irreducible component of C, one has
+(C, Lp)p = multp(C) and (C, Lp)q = 1 for any q ∈ C ∩ Lp, q ̸= p.
+This fibration has as base and as fiber a projective line P1 with a number of points
+deleted, and hence the complement M(C) is a K(π, 1) space.
+When C is a line
+arrangement, this definition of a modular point coincides with the usual one, and a
+line arrangement is supersolvable by definition if it has a modular point. In particular,
+the existence of a modular point for a line arrangement C implies that C is free, see
+for all these well known facts [2, 17]. We venture to make the following.
+Conjecture 1.10. A supersolvable plane curve C is free.
+One setting where this conjecture holds is the following.
+Theorem 1.11. Let C0 be a reduced plane curve, let p ∈ M(C0) be a point and let
+A be the set of lines L passing through p such that there is a point q ∈ L ∩ C0 with
+(C0, L)q > 1. Assume that all the singularities of the curve C obtained by adding all
+the lines in A to C0 are quasi homogeneous. Then C is supersolvable and free. In
+particular, this holds when all the singularities s of C0 have multiplicity 2, and p is
+not on any tangent cone TCs(C0) for (C0, s) a singularity with µ(C0, 0) ≥ 3.
+When C0 is a plane curve having only nodes A1 and cusps A2 as singularities, and
+in addition p is a generic point, this result is known, see [3, Theorem 1.12]. Moreover,
+it is easy to see that the free curves C′ and C′′ constructed above in Theorem 1.5
+are special cases of the construction in Theorem 1.11. The free curve C′′ constructed
+above in Theorem 1.8 is of a different nature, since in this case p ∈ C0 and the curve
+C has not only quasi homogeneous singularities. It is interesting that it gives new
+examples where Conjecture 1.10 holds, in view of the following result.
+Proposition 1.12. The free curve C′′ constructed in Theorem 1.8 is supersolvable.
+In particular, the associated complement M(C′′) is a K(π, 1) space.
+
+6
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+We explain in Example 4.11 that the other free curves constructed above in The-
+orems 1.7 and 1.8 are not supersolvable.
+The organization of the paper goes as follows. In Section 2, we discuss the proof
+of Theorem 1.2 explaining all the necessary details. In Section 3 we recall basic facts
+on the free, nearly free and maximizing curves.
+In Section 4 we deliver our proofs of Theorems 1.3, 1.5, 1.7, 1.8 and 1.11, and of
+Proposition 1.12. Then, in Section 5, we describe all smooth plane quartic curves
+admitting the maximal possible number of flexes of maximal order, i.e., flexes of order
+2. There are two such curves, and only one of them, the Fermat curve, yields free
+curves as in Corollary 1.6 and in Theorem 1.7. In the final section we discuss some
+singular plane curves and the free curves obtained from them, which are sometimes
+supersolvable as well, given rise to supersolvable free curves not covered by our
+general Theorem 1.11.
+2. Discussion on Theorem 1.2 and some examples
+The paper [13] uses rather heavy notations, and perhaps due to this fact has a
+smaller impact than it deserves. Let us introduce some notation. For a reduced
+plane curve C : F = 0 and any point q = (α : β : γ) ∈ P2, we define the polar ∆q(C)
+of C with respect to q by the equation
+(2.1)
+∆q(C) : αFx + βFy + γFz = 0.
+For a property P, the authors of [13] use the notation 1P to denote 1 if P is true
+and 0 otherwise, see the discussion before Theorem 2 in [13]. The first equality in
+[13, Proposition 25] gives the expression of the intersection multiplicity (C, ∆q(C))p
+for any singular point p ∈ C and any point q ∈ P2. Using our discussion above,
+we see that for q ̸= p, q not on any line L in the tangent cone TCp(C), this mul-
+tiplicity (C, ∆q(C))p is given by a double sum S, i.e., the second sum involving the
+characteristic functions 1P vanishes. With this observation, the second equality in
+[13, Proposition 25] can be stated as
+(2.2)
+(C, HC)p = 3(C, ∆q(C))p + Ip,
+where Ip = �
+i∈I(i(i)
+p − 2). Here I is a set of indices parametrizing the pro-branches
+of (C, p) and i(i)
+p
+is the tangential intersection number of the pro-branch Bi, see [13,
+Definition 22]. Recall that any branch B of a plane curve singularity (C, 0) at the
+origin of C2, such that x = 0 is not a tangent line, can be defined by a Weierstrass
+polynomial
+ΓB(x, y) =
+�
+j=1,mB
+(y − φB,j(x))
+where mB is the multiplicity of the branch B and there is an analytic function
+φB(x) ∈ C{x} such that
+φB,j(x) = φB(exp(2πij/mB)x
+1
+mB ).
+With this notation, the branch B has mB associated pro-branches
+Bj : y − φB,j(x) = 0
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+7
+and the corresponding tangential intersection number is given by
+i
+Bj
+0
+= ord(φB,j(x) − φ′
+B,j(0)x) ∈ Q.
+It follows that i
+Bj
+0
+= iBk
+0
+for any 0 ≤ j ≤ k ≤ mB and hence
+mBi
+Bj
+0
+= (L, CL)0,
+for any j, with L the tangent line to B and CL = B. This discussion implies that
+one has
+(2.3)
+Ip =
+�
+i∈I
+(i(i)
+p − 2) =
+�
+L∈TCp(C)
+mL(C) − 2mp(C),
+since clearly |I| = mp(C), each branch having exactly a number of pro-branches given
+by the multiplicity of that branch. Next, we return to the intersection multiplicity
+(C, ∆q(C))p. We can assume that p = (0 : 0 : 1) and set f(x, y) = F(x, y, 1), then
+one has
+fx(x, y) = Fx(x, y, 1), fy(x, y) = Fy(x, y, 1), and xfx + yfy + Fz(x, y, 1) = f.
+If we define the generic local polar variety of the singularity
+(C, 0) : f(x, y) = 0
+by the equation
+∆0(C) : α′fx + β′fy = 0,
+with (α′ : β′) ∈ P1 being a generic point, it is easy to see that
+(2.4)
+(C, ∆0(C))0 = (C, ∆q(C))p.
+In fact, the line L′ : z = 0 is clearly not in the tangent cone TCp(C) since p /∈ L′,
+and hence we may take γ = 0 and (α : β) ∈ P1 generic in the formula (2.1). In order
+to compute this local intersection number
+κ0(C) = (C, ∆0(C))0,
+which is also called the κ-invariant of the singularity (C, 0), we can use for instance
+[12, Proposition 3.38] and get
+(2.5)
+κ0(C) = µ(C, 0) + m0(C, 0) − 1.
+If we use this formula for the singularity (C, p), we get from (2.2) and (2.3) the
+following equality
+(C, HC)p = 3(µp(C) + mp(C) − 1) +
+�
+L∈TCp(C)
+mL(C) − 2mp(C) =
+= 3µp(C) + mp(C) − 3 +
+�
+L∈TCp(C)
+mL(C).
+This proves our reformulation of [13, Proposition 25] in Theorem 1.2.
+
+8
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+In order to construct free plane curves starting with a curve C of degree d, by
+adding lines, in particular inflectional tangents, we have to look for lines L such that
+the sum
+(2.6)
+i(L) =
+�
+p∈L∩IC, TpC=L
+ιp(C)
+is as large as possible with respect to the degree d. Given a curve C, first we use
+Theorem 1.2 to count the total number of inflection points of C, namely
+(2.7)
+i(C) =
+�
+p∈IC
+ιp(C) = 3d(d − 2) −
+�
+p∈YC
+(C, HC)p.
+One clearly has for any line L
+(2.8)
+i(L) ≤ i(C),
+and the equality holds if and only if for any point p ∈ IC one has TpC = L. Moreover,
+(2.9)
+i(L) ≤
+�
+p∈L∩C
+((C, L)p − 2) = d − 2|L ∩ C|,
+and the equality holds if and only if for any point p ∈ L ∩ C one has TpC = L.
+Example 2.1. Let us consider the case when (C, p) is a node A1, that is there are
+two smooth branches (C1, p) and (C2, p) meeting transversally at p. Let T1 = TpC1
+and T2 = TpC2 be the associated tangent lines and define the type of the node (C, p)
+to be the pair of integers
+(m1, m2) = ((C1, T1)p, (C2, T2)p).
+It is clear that mj ≥ 2 for j = 1, 2. When m1 = m2 = 2, then (C, p) is said to be a
+simple node, and one knows that (C, HC)p = 6, see [11, pp. 68–69]. In the general
+situation, Theorem 1.2 gives the equality
+(C, HC)p = 3 + 2 − 3 + m1 + m2 = m1 + m2 + 2.
+More generally, consider the case when (C, p) is an ordinary m-multiple point, that
+is there are m smooth branches C1, . . . , Cm with distinct tangent lines L1, . . . , Lm.
+If we set mj = (Cj, Lj)p for j = 1, . . . , m, then Theorem 1.2 gives the equality
+(C, HC)p = 3(m − 1)2 + m − 3 +
+�
+j=1,m
+mj = m(3m − 5) +
+�
+j=1,m
+mj.
+Example 2.2. Let us consider the case when (C, p) is a singularity A2m−1 with
+m ≥ 2. Then there are two tangent smooth branches with a common tangent line
+L. If we set mL = (C, L)p, then Theorem 1.2 gives the equality
+(C, HC)p = 3(2m − 1) + 2 − 3 + mL = 2(3m − 2) + mL.
+Example 2.3. Let us consider the case when (C, p) is a singularity A2m with m ≥ 1.
+Then there is a unique branch, with a tangent line L. If we set mL = (C, L)p, then
+Theorem 1.2 gives the equality
+(C, HC)p = 3(2m) + 2 − 3 + mL = 6m − 1 + mL.
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+9
+When (C, p) is a cusp A2, only the value mL = 3 is possible, and hence (C, HC)p = 8
+in this case.
+3. Free, nearly free and maximizing curves
+In this section we recall some basic facts on free, nearly free and, maximizing
+curves in P2 following [6, 7].
+Let
+Der(S) = {∂ := a · ∂x + b · ∂y + c · ∂z, a, b, c ∈ S}
+be the free S-module of C-linear derivations of the polynomial ring S. For a reduced
+curve C : F = 0, we introduce
+D(F) = {∂ ∈ Der(S) : ∂ F ∈ ⟨F⟩},
+the graded S-module of derivations preserving the ideal ⟨F⟩. We have the following
+decomposition
+D(F) = D0(F) ⊕ S · δE,
+where δE = x∂x + y∂y + z∂z is the Euler derivation and
+D0(F) = {∂ ∈ Der(S) : ∂ F = 0}
+is the set of all C-linear derivations of S killing the polynomial F.
+Definition 3.1. We say that a reduced curve C : F = 0 is free if D(F), or equiv-
+alently D0(F), is a free graded S-module. The exponents (d1, d2) of a free curve C
+are the degrees of a basis for the free graded S-module D0(F) which rank 2.
+Remark 3.2. The exponents (d1, d2) of a free curve C : F = 0 of degree d are known
+to satisfy d1 +d2 = d−1. Conversely, if there are two elements r1, r2 ∈ D0(F), which
+are S-linearly independent and satisfy
+d1 + d2 = d − 1
+then the curve C is free with exponents (d1, d2), see [18, 19].
+Definition 3.3. The minimal degree of derivations killing F, or of Jacobian syzygies
+involving the partial derivatives of F, is defined as
+mdr(F) = min{r ∈ N : D0(F)r ̸= 0}.
+To check whether a given plane curve is free, one may use the following result by
+du Plessis and Wall [9].
+Theorem 3.4. Let C : F = 0 be a reduced plane curve of degree d and let r =
+mdr(F). Then the following two cases hold.
+a) If r < d/2, then τ(C) ≤ τ(d, r)max = (d − 1)2 − r(d − r − 1) and the equality
+holds if and only if the curve C is free.
+b) If d/2 ≤ r ≤ d − 1, then τ(C) ≤ τ(d, r)′
+max, where, in this case, we set
+τ(d, r)′
+max = τ(d, r)max −
+�2r − d + 2
+2
+�
+.
+
+10
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+Definition 3.5. A reduced curve C : F = 0 of degree d is nearly free if either
+mdr(F) < d/2 and τ(C) = τ(d, r)max − 1, or mdr(F) = d/2 and τ(C) = τ(d, r)max.
+In addition, the exponents of a nearly free curve C : F = 0 of degree d are given by
+the pair (mdr F, d − mdr F).
+Definition 3.6. A curve C : F = 0 of degree d having only ADE-singularities is
+maximizing if either d = 2m and τ(C) = 3m(m − 1) + 1, or d = 2m + 1 and
+τ(C) = 3m2 + 1.
+The relation between maximizing curves and free curves is the following, see [6].
+Theorem 3.7. A curve C : F = 0 of degree d having only ADE-singularities is
+maximizing if and only if either d = 2m and C is a free curve with the exponents
+(m − 1, m), or d = 2m + 1 and C is a free curve with the exponents (m − 1, m + 1).
+In the sequel we need the following version of [3, Theorem 1.10].
+Theorem 3.8. Let C : F = 0 be a reduced plane curve of degree d and let p be any
+point of C. Let A be the union of the irreducible components of C which are lines
+passing through p, and let C′ : G = 0 be the union of the other irreducible components
+of C. We assume that p ∈ C′. Let m = |A| and e = deg G. Then r = mdr(F) can
+be in one of the following cases.
+a) r = e;
+b) r = m − 1 and C is free with exponents (m − 1, e);
+c) m ≤ r ≤ e − 1.
+The only difference of this result with respect to [3, Theorem 1.10] is that here
+p ∈ C′.
+Proof. The key part of the proof of [3, Theorem 1.10] is contained in [3, Lemma 4.3],
+where a Jacobian syzygy
+ρ : aFx + bFy + cFz = 0
+is constructed using a differential 2-form ω, and it is shown that this syzygy is
+primitive, that is there is no common factor for a, b, c ∈ S. It is in this latter part
+that the condition p /∈ C′ was used. If we assume that p = (0 : 0 : 1), it is easy to
+see that the Jacobian relation ρ constructed there is still a Jacobian syzygy in our
+situation. Moreover, it is a primitive syzygy if and only if G and Gz have no common
+factor. Let M ∈ S be an irreducible polynomial which is a common factor for G and
+Gz. Note that M cannot involve only x and y, since this would correspond to a line
+in C′ passing through p, which is impossible by the definition of C′. It follows that
+Mz ̸= 0. If G = MN, then Gz = MzN + MNz, which implies that either M divides
+N or M divides Mz. But M cannot divide N, since C′ is reduced. And M cannot
+divide Mz, since the degree of Mz as a polynomial in z is strictly smaller than the
+corresponding degree of M. This contradiction proves our claim.
+□
+4. The proofs of our main results
+4.1. Proof of Theorem 1.3. We can assume that p = (0 : 0 : 1) and set f(x, y) =
+F(x, y, 1), g(x, y) = jkf(x, y) the initial form of f. Let n be the number of distinct
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+11
+factors of g. If n = k, then using Example 2.1 we have
+(C, HC)p = k(3k − 5) +
+�
+j=1,k
+mj ≥ 3k(k − 1),
+since mj ≥ 2. Moreover, it is obvious that the equality holds if and only if (C, p) is
+an ordinary simple k-multiple point.
+Assume now that n < k and let
+g(x, y) = ℓa1
+1 · ... · ℓan
+n
+be the decomposition of g as a product of linear factors. Recall that for two isolated
+plane curve singularities (X, 0) and (Y, 0) with no common component one has
+(4.1)
+µ(X ∪ Y, 0) = µ(X, 0) + µ(Y, 0) + 2(X, Y )0 − 1,
+see [20, Theorem 6.5.1]. Let Cj : fj = 0 be the union of the branches of (C, p)
+which are tangent to the line Lj : ℓj = 0 with j ∈ {1, . . . , n}.
+Then (C, p) =
+(C1, p) ∪ ... ∪ (Cn, p) and we estimate µp(C) = µ(C, p) using the above formula. To
+start with, note that since jajfj = ℓ
+aj
+j it follows that
+µ(Cj, 0) ≥ aj(aj − 1).
+Assume that for some m < n one has
+µ((C1, p) ∪ ... ∪ (Cm, p)) ≥ (a1 + . . . + am)2 − (a1 + . . . + am) − m + 1.
+Then it follows that
+µ((C1, p) ∪ ... ∪ (Cm, p) ∪ (Cm+1, p)) ≥ ((a1 + . . . + am)2 − (a1 + . . . + am) − m + 1)+
++am+1(am+1 − 1) + 2am+1(a1 + . . . + am) − 1 =
+= (a1 + . . . + am+1)2 − (a1 + . . . + am+1) − m.
+Since a1 + . . . + an = k, this yields the inequality
+µ(C, p) ≥ k2 − k − n + 1.
+On the other hand, one has
+mLj = (Cj, Lj)p ≥ aj + 1
+and hence
+�
+j
+mLj ≥ k + n.
+Using Theorem 1.2 we get
+(C, HC)p ≥ 3(k2 − k − n + 1) + k − 3 + k + n = 3k(k − 1) + 2(k − n) > 3k(k − 1),
+since we have assumed n < k. This completes the proof of Theorem 1.3.
+Corollary 1.4 is an obvious consequence of Theorem 1.3.
+
+12
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+4.2. Proof of Theorem 1.5. The curve C was studied in [8, Example 4.5] and it
+was shown that the minimal degree of a Jacobian relation for C is given by
+mdr(F) = m − 1.
+Since the minimal degree of a Jacobian relation can only increase when one adds
+lines to a given curve, see [5, Proposition 3.1], it follows that
+mdr(F ′) ≥ mdr(F) = m − 1.
+The curve C has m inflection points and singularities on the line z = 0, locally
+given by equations ukj − vd = 0, located at the points pj = (aj : bj : 0), with
+ℓj(aj, bj) = 0 for j ∈ {1, . . . , m}. When we add the line Lj, we get at the point pj
+a weighted homogeneous singularity of degree dj = 1 with respect to the weights
+w1 = wt(uj) =
+1
+kj+1 and w2 = wt(z) =
+kj
+d(kj+1), where uj = ℓj is a local coordinate
+at pj on the line Lj. It follows the following equality involving Tjurina and Milnor
+numbers:
+τ(C′, pj) = µ(C′, pj) = (1 − w1)(1 − w2)
+w1w2
+= (d − 1)kj + d.
+Hence the total Tjurina number of C′ is
+τ(C′) = d(d − 1) + md + (m − 1)2,
+since clearly
+τ(C′, p) = µ(C′, p) = (m − 1)2.
+Now a curve of degree d′ = d + m with r′ = mdr(F ′) satisfies the inequality
+τ(C′) ≤ τ(d′, r′)max
+where the function
+τ(d′, r′)max = (d′ − 1)2 − r′(d′ − 1 − r′)
+is a decreasing function of r′ for 2r′ < d′ which follows from Theorem 3.4, and
+the equality τ(C′) = τ(d′, r′)max implies that 2r′ < d′ and that C′ is free with the
+exponents (r′, d′ − r′ − 1). In our case, we get
+τ(d′, m − 1)max = (d + m − 1)2 − d(m − 1) = τ(C′),
+and this proves our claim. The proof of the second claim goes analogously.
+Remark 4.3. One can check that the curve C′, resp. C′′, can be regarded as a
+special case of the curve C constructed in Theorem 1.11, starting from the curve
+C0 : F = 0 and p = (0 : 0 : 1) for C′, resp. C0 : zF = 0 and p = (0 : 0 : 1) for C′′.
+This is an alternative way to proving Theorem 1.5.
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+13
+4.4. Proof of Theorem 1.7. We start with the following.
+Lemma 4.5. Consider the line arrangement
+B : g = xyz(xd + yd)(yd + zd)(xd + zd) = 0.
+Then mdr(g) = 2d + 1.
+Proof. We consider first the subarrangement of B given by
+B0 : g0 = xyz(xd + yd)(yd + zd) = 0.
+Note that in this arrangement B0 there are two points of multiplicity d+2, connected
+by the line y = 0. All the lines pass through one of these two points, and the other
+intersection points are all double points. It follows that B0 is a line arrangement of
+type ˆL(d + 2, d + 2) as in [4, Definition 4.9], and
+mdr(g0) = d + 1,
+see [4, Example 4.11]. To get the arrangement B from B0, we have to add the d lines
+L1, ..., Ld given by xd + zd = 0. At the stage k, where 1 ≤ k ≤ d, we have to add the
+line Lk to the arrangement
+Bk = B0 ∪ L1 ∪ ... ∪ Lk−1.
+Note that the intersection of Lk and Bk consists of exactly 2d + 2 points. If Bk is
+given by the reduced equation fk = 0, for 1 ≤ k ≤ d, it follows from [5, Corollary
+6.4] that one has
+mdr(fk) = mdr(fk−1) + 1
+for all 1 ≤ k ≤ d. Hence
+mdr(g) = mdr(fd) = d + 1 + d = 2d + 1.
+□
+Using [5, Theorem 5.1 (b)] we see that mdr(F ′) ≥ 2d + 1. We compute now the
+total Tjurina number of the curve C′. This curve has 3d2 nodes A1 and 3 points with
+local equation uv(ud + vd) coming from the double points of the line arrangement B
+not situated on xyz = 0 and the 3 points of multiplicity d + 2. The line x = 0 in B
+contains d double points of this line arrangement, which are precisely the inflection
+points of order d − 2 of the Fermat curve C situated on this line. The corresponding
+inflectional tangents are the lines given by yd + zd = 0. Therefore, when we add C,
+each of these d points becomes a singularity of type D2d+2. Similar remarks apply
+to the lines y = 0 and z = 0. It follows that
+τ(C′) = 3d2 + 3(d + 1)2 + 3d(2d + 2) = 12d2 + 12d + 3.
+On the other hand, we have
+τ(4d + 3, 2d + 1)max = (4d + 2)2 − (2d + 1)2 = 12d2 + 12d + 3.
+The equality
+τ(C′) = τ(4d + 3, 2d + 1)max
+
+14
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+implies as above that r′ = mdr(F ′) = 2d + 1 and that C′ is a free curve with the
+exponents (2d + 1, 2d + 1). The claims for the curve C′′ are proved in a similar way.
+The line arrangement
+B′ : g′ = xy(yd + zd)(xd + zd) = 0
+satisfies mdr(g′) = d+1, see [4, Proposition 4.10]. The lines x = 0 and y = 0 contain
+each d points of type D2d+2 as above. Besides these points, the line arrangement B′
+has two points of multiplicity d + 1 and d2 + 1 double points. It follows that
+τ(C′′) = 2d(2d + 2) + 2d2 + (d2 + 1) = 7d2 + 4d + 1 = τ(3d + 2, d + 1)max.
+Remark 4.6. It is interesting to note that the line arrangement formed by the
+corresponding 3d inflectional tangent lines is the arrangement
+(xd + yd)(yd + zd)(xd + zd) = 0,
+which is far from being free. On the other hand, the line dual to the inflection point
+pǫ = (1 : ǫ : 0) is L′
+ǫ : x + ǫy = 0, and the union of all these d dual lines obtained
+when ǫ varies, is given by xd − yd = 0 when d is odd. Therefore, for d odd, the line
+arrangement formed by the corresponding 3d dual lines is precisely the free monomial
+(or Fermat) line arrangement
+(xd − yd)(yd − zd)(xd − zd) = 0.
+The case d = 3 is of course well-known.
+4.7. Proof of Theorem 1.8. First we consider the curve C′. The reader can check
+the following Jacobian relations r1, r2 ∈ D0(F ′)
+r1 : zm−1(xm + ym)F ′
+x − xm−1(ym + zm)F ′
+z = 0
+and
+r2 : xym−1(2xm + 3ym)F ′
+x − (2F + y2m)F ′
+y + zym−1(2zm + 3ym)F ′
+z = 0,
+where F = xmym + ymzm + xmzm. Since
+deg r1 + deg r2 = (2m − 1) + 2m = deg F ′ − 1,
+our claim is proved by Remark 3.2.
+We consider now the curve C′′. First we show that mdr F ′′ = m + 1. To do this,
+we first determine a minimal degree Jacobian syzygy r1 for F. One has for instance
+r1 : a1Fx + b1Fy + c1Fz = 0,
+where a1 = x(ym − zm), b1 = −y(ym + zm) and c1 = z(ym + zm). Now we apply [5,
+Theorem 3.3] and see that if we add a line L0 to C given by an equation ℓ = sy+ty = 0
+such that ℓ divides
+sb1 + tc1 = (−sy + tz)(ym + zm)
+the resulting curve C0 = C ∪ L0 : F0 = ℓF = 0 has again
+mdr F0 = mdr F = m + 1.
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+15
+Moreover, the coefficients of a minimal degree syzygy for F0 can be obtained from
+the discussion just before [5, Theorem 3.3]. It follows that one can add one by one
+all the lines in the arrangement
+yz(ym + zm) = 0
+and get at the end mdr F ′′ = mdr F = m + 1 as we have claimed. To show now that
+C′′ is free with the given exponents it is enough to apply Theorem 3.8.
+Remark 4.8. If one likes to use Theorem 3.4 as above to prove that the curve
+C′ is free, one needs to compute the total Tjurina number τ(C′).
+This in turn
+is complicated, since the singularities of C′ at the points p1 = (1 : 0 : 0) and
+p3 = (0 : 0 : 1) are no longer quasi homogeneous, and hence τ(C′′, pj) < µ(C′′, pj),
+for j = 1 and j = 3. In fact, our Theorem 1.8 combined with Theorem 3.4 implies
+that
+τ(C′′, p1) = τ(C′′, p3) = 5m2 − 2m,
+maybe a result not easy to prove otherwise.
+4.9. The proof of Theorem 1.11. By its very construction, it is clear that p is a
+modular point for C. Let e = deg C0 and m = |A| = multp(C). Hence d = deg(C) =
+e + m. Then the fibration
+πp : M(C) → B
+induced by the central projection with center p has as fiber F the projective line P1
+minus e + 1 points, and as base B the projective line P1 minus m points. It follows
+that the Euler number E(M(C)) of the complement M(C) is given by
+E(M(C)) = E(F)E(B) = (1 − e)(2 − m).
+On the other hand, we know that
+E(M(C)) = E(P2) − E(C) = 3 − (µ(C) − d(d − 3)),
+where µ(C) is the total Milnor number of C. The above two equations give
+µ(C) = (e + m)2 − em − 2m − e + 1.
+Since all the singularities of C are supposed to be quasi homogeneous, we get
+τ(C) = µ(C) = (e + m)2 − em − 2m − e + 1.
+To show that C : F = 0 is free we apply [3, Theorem 1.10]. It follows that r = mdr F
+satisfies one of the following properties.
+a) r = e. Then
+τ(e + m, e)max = (e + m − 1)2 − e(m − 1) = τ(C),
+which implies that e < m and C is free with exponents (e, m − 1) using
+Theorem 3.4.
+b) r = m − 1. Then
+τ(e + m, m − 1)max = (e + m − 1)2 − (m − 1)e = τ(C),
+which implies that m ≤ e and C is free with exponents (m − 1, e) again by
+Theorem 3.4.
+
+16
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+c) m ≤ r < e. This case is impossible, since it implies
+τ(C) ≤ τ(e + m, r)max < τ(e + m, m − 1)max = τ(C).
+The first inequality follows from Theorem 3.4, and the second one by the fact
+that the function t �→ τ(e + m, t)max is decreasing for 2t < e + m.
+To prove the last claim in Theorem 1.11, first notice that the possible non quasi
+homogeneous singularities of C may occur only at the intersection s = L∩C0, where
+s is a singular point of C0 and L is a line in A. If mults C0 = 2 and L is not in
+the corresponding tangent cone TCs(C0) as we have assumed, then mults C = 3 and
+the 3-jet j3g of a local equation (C, s) : g = 0 is a binary cubic form with at least
+2 distinct factors. It follows from the classification of singularities, see for instance
+[1], that such a singularity has type Dk, for some k ≥ 4, and in particular it is quasi
+homogeneous. If µ(C0, s) = 1 and L is in the corresponding tangent cone TCs(C0),
+then (C0, s) is a node A1, and the same argument as above works, namely (C, s) is
+a Dk singularity. Finally, when µ(C0, s) = 2 and L is in the corresponding tangent
+cone TCs(C0), then (C0, s) is a cusp A2, and the new singularity (C, s) is easily seen
+to be of type E7, hence again quasi homogeneous.
+4.10. The proof of Proposition 1.12. The point p = (1 : 0 : 0) is a modular point
+in this case, since any line Lp through p, not an irreducible component for C′′, is
+given by z = ty with t ̸= 0 and tm + 1 ̸= 0. The intersection Lp ∩ C′′ is described by
+the equation
+ty2(xmym + tmy2m + tmxmym)(ym + tmym) = t(tm + 1)y2m+2((1 + tm)xm + tmym) = 0.
+The solution y = 0 corresponds to the point p, which has multiplicity 2m + 2 on
+C′′, and there are m = deg F ′′ − (2m + 2) other intersection points coming from the
+solutions of (1 + tm)xm + tmym = 0.
+Example 4.11. Here we show first that the free curves C′ and C′′ coming from The-
+orem 1.7 are not supersolvable. For the curve C′, it is clear that the only candidates
+for modular points are the point p = (0 : 0 : 1) and the 2 other points obtained
+from p by permutation of coordinates. Indeed, a modular point has to contain all
+the tangents to the Fermat curve issued from it. Now the point p is not a modular
+point for C′, since the line Lp : y − x = 0 is not an irreducible component of C′ and
+it satisfies
+2d+2 = |Lp ∩C′| < |L′ ∩C′| = deg C′ −multp(C′)+1 = 4d+3−(d+2)+1 = 3d+2.
+The same line Lp : y − x = 0 shows that the point p is not a modular point for C′′
+either. The point p′ = (1 : 0 : 0) is also not a modular point, as the choice of the
+line Lp′ : z = 0 shows. The point p′′ = (0 : 1 : 0) has the same property, as our curve
+is invariant under the coordinate change x �→ y and y �→ x. To show that the free
+curve C′ coming from Theorem 1.8 is not supersolvable, we use the same approach as
+above, the lines Lp to use in this case are given by x = 0, y = 0 or z = 0 respectively.
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+17
+5. Examples: the case of smooth quartic curves
+In this section we discuss examples of smooth quartic curves having the maximal
+possible number of flexes of high order.
+Let us recall that by Theorem 1.2 the
+maximal possible number of flex points of order 2 for smooth quartics is 12. It is
+natural to wonder whether there exists a complete classification of smooth quartics
+which have exactly 12 flexes of order 2. In order to do so, we discuss interesting
+properties of the following pencil of quartics which was studied by Ciani in the 19th
+century.
+Let us define
+(5.1)
+Cλ : x4 + y4 + z4 + λ · (y2z2 + z2x2 + x2y2) = 0.
+It is easy to observe that each curve in the pencil is invariant under the natural
+action of an octahedral group of collineations.
+There are some values of λ which lead to special members of the pencil, namely
+• λ = 0 gives us the Fermat quartic curve, or Dyck’s curve, which has a group
+of 96 collineations;
+• if λ is a root of λ2 + 3λ + 18 = 0, then we get the Klein quartic curve having
+a group of 168 collineations.
+In the case of the Fermat quartic, by a discussion presented above, we know that
+it has exactly 12 flexes of order 2, so the maximal possible number in the class of
+smooth quartics. In the case of the Klein quartic curve, we know that this curve has
+only flex points of order 1, so exactly 24 flexes. Now we pass to another interesting
+element in the pencil of quartics by taking λ = 3. The resulting quartic C3 is smooth
+and it has the group of collineations of order 24. It was verified directly by Edge in
+[10] that the curve C3 admits exactly 12 flexes of order 2 and he provided both the
+coordinates of these points and the equations of the associated tangent lines. Now
+we recover Edge’s calculations. Looking precisely on the Hessian H of C3, which is
+H = 2x6 + x4(3z2 + 3y2) + x2(8y2z2 + 3z4 + 3y4) + 2z6 + 2y6 + 3z4y2 + 3z2y4,
+one can show that flexes of order 2 are just the intersection points of the curve C3
+with the 6 lines given by the linear factors of
+(5.2)
+F = (x2 + y2)(y2 + z2)(z2 + x2).
+The flex points have the following coordinates:
+P1 : (i : 1 : −1),
+P2 : (−i : 1 : −1),
+P3 : (−1 : i : 1),
+P4 : (−1 : −i : 1),
+P5 : (1 : −1 : i),
+P6 : (1 : −1 : −i),
+P7 : (−i : 1 : 1),
+P8 : (i : 1 : 1),
+P9 : (1 : −i : 1),
+P10 : (1 : i : 1),
+P11 : (1 : 1 : −i),
+P12 : (1 : 1 : i).
+Observe that these 12 flexes of order 2 are uniformly distributed, four on each of the
+lines defined by F.
+Up to now we described exactly two smooth quartics having the maximal possible
+number of flexes of order 2. However, as it turns out by a result due to Kuribayashi
+
+18
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+and Komiya [15], these are the only smooth plane quartic curves having 12 flexes of
+order 2, and this is rather surprising.
+Remark 5.1. We have seen in Theorem 1.7 that if we add to the Fermat quartic
+its 12 inflectional tangents of order 2 and the triangle ∆ : xyz = 0 determined by
+the inflection points, then we get a free curve of degree 19. If we try to apply the
+same construction to the quartic curve C3, the resulting curves are far from being
+free. One explanation for this fact may be the following. The union AF of the 12
+inflectional tangents of the Fermat quartic is a line arrangement having 3 points of
+multiplicity 4. On the other hand, the union A3 of the 12 inflectional tangents of
+the quartic C3 is a line arrangement having only double points, and hence the total
+Tjurina number τ(A3) is much smaller than τ(AF). If we add the triangle ∆ to
+AF, we get a line arrangement having 3 points of multiplicity 6. On the other hand,
+if we add to A3 the 6 lines determined by (5.2), we get a line arrangement having
+only points of multiplicity 2 and 3, and hence having small total Tjurina number
+compared with respect to its degree.
+6. Examples: the case of singular curves
+Example 6.1. Any nodal cubic is projectively equivalent to the cubic
+C : F = xyz + x3 + y3 = 0.
+The corresponding Hessian is H = −2(3(x3 + y3) − xyz). Hence the intersection
+C ∩ HC consists of the following 4 points:
+p1 = (0 : 0 : 1) and pj = (1 : j : 0),
+where j3 + 1 = 0. The point p1 is a simple node, and the points pj give rise to 3
+inflection points of order 1. This is reflected in the equality
+(C, HC)p1 +
+�
+j
+(C, HC)pj = 6 + 1 + 1 + 1 = 9,
+recall Example 2.1. The 3 inflectional tangents Lk for k = 1, 2, 3 are given by the
+equations Lk : 3x+3j3
+ky+jkz = 0, where jk are the 3 roots of the equation j3+1 = 0.
+It follows that these 3 inflectional tangents Lk are not concurrent, so their addition
+to C will not give free curves as in Remark 5.1 above. On the other hand, if we add
+to C the tangent cone at the singular point, we obtain the curve
+C′ : F ′ = xy(xyz + x3 + y3) = 0,
+which is free with exponents (2, 2) as a direct computation with SINGULAR shows.
+Moreover, this curve C′ is supersolvable, since clearly p1 is a modular point for C′.
+Any cuspidal cubic is projectively equivalent to the cubic
+C : F = x2z + y3 = 0.
+The corresponding Hessian is H = −24x2y. Hence the intersection C ∩ HC consists
+of the following 2 points
+p1 = (0 : 0 : 1) and p2 = (1 : 0 : 0).
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+19
+The point p1 is a cusp A2, and the point p2 is an inflection point of order 1. This is
+reflected in the equality
+(C, HC)p1 + (C, HC)p2 = 8 + 1 = 9,
+recall Example 2.3. This is a special case of Theorem 1.5, and gives rise to two free
+curves by adding one or two lines, as explained there.
+Example 6.2. In this example we consider some plane quartic curves.
+Consider the quartic C : F = (x3 + y3)z + x4 + y4 = 0, which has a D4-singularity
+at p1 = (0 : 0 : 1). The corresponding Hessian is
+H = −54(xyz(x3 + y3) + 2x2y2(x2 + y2)).
+The point p1 is an ordinary simple singularity of multiplicity k = 3, and hence
+(C, HC)p1 = 18 by Theorem 1.3. There are in addition 6 inflection points of order
+1, with are the points (1 : 0 : −1), (0 : 1 : −1) and the 4 points (u : v : w), where
+(u : v) is coming from the 4 solutions of the equation
+u4 + v4 − 2uv(u2 + v2) = 0
+in P1 and w = −(u4 +v4)/(u3 +v3). If we add the tangent cone of the singular point,
+namely the lines x3 + y3 = 0, we get a free curve
+C′ : F ′ = (x3 + y3)F = 0,
+of degree 7 and exponents (3, 3). Moreover, this curve C′ is supersolvable, since
+clearly p1 is a modular point for C′.
+Next, consider the quartic
+C : F = x2y2 + y2z2 + x2z2 = 0,
+which has 3 nodes. It is easy to see that all of them have type (3, 3), and hence C
+has no inflection points by Example 2.1. The corresponding Hessian is
+H = −24(x4y2 + x2y4 + y4z2 + y2z4 + x4z2 + x2z4 − 6x2y2z2).
+If we add to C one tangent line at each of the 3 nodes, namely the lines
+(x + iy)(y + iz)(z + ix) = 0,
+we get a free curve of degree 7 with exponents (3, 3). All the singularities of this
+curve are simple, but this curve is not maximizing, recall our discussion in Section 3
+on these curves. Finally the quartic
+C : F = x2y2 + y2z2 + x2z2 − 2xyz(x + y + z) = 0,
+which has 3 cusps A2.
+Hence C has no inflection points by Example 2.3.
+The
+corresponding Hessian is
+H = 144(x3y3 + y3z3 + x3z3 − x3(y2z + yz2) − y3(x2z + xz2) − z3(x2y + xy2).
+Let L1, L2 and L3 be the 3 lines which are the reduced tangent cones corresponding
+to the 3 cusps, which are given up to an order by the equations x − y = 0, y − z = 0
+and z − x = 0. Then the curves
+C1 = C ∪ L1, C2 = C1 ∪ L2 and C3 = C2 ∪ L3
+
+20
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+are all free, with exponents respectively
+(2, 2), (2, 3) and (2, 4).
+We get in this way a free curve C1 of degree 5 and maximizing curves C2 and C3, of
+degree 6 and 7, respectively, as already pointed out in [6]. It is interesting to note
+that the curve C3 is supersolvable, and the point p = (1 : 1 : 1) is a modular point for
+it. Indeed, the lines joining p to the singularities of C are already in C3. It remains
+to show that any line Lp through p, different from L1, L2, L3 meets C in exactly 4
+points, that is Lp is not a tangent line to C. If q = (u : v : w) ∈ C is a smooth point
+such that the tangent line TqC passes through p, then we have
+Fx(q) + Fy(q) + Fz(q) = 0.
+A direct computation shows that
+Fx(q) + Fy(q) + Fz(q) = −12uvw
+and hence at least one of the coordinates of q vanishes. But then q ∈ C implies that
+2 coordinates vanish, and therefore q is a singularity of C, a contradiction. Note that
+the curve C3 is an example of curve satisfying both the first assumption in Theorem
+1.11, since all of its singularities are quasi homogeneous, and the second assumption,
+even if p belongs to the tangent cones TCs(C) of the three cusps, as they have Milnor
+numbers equal to 2.
+Example 6.3. In this example we consider the curve
+C : F = xmym + ymzm + xmzm = 0
+of degree d = 2m ≥ 4. This curve has 3 ordinary m multiple points
+p1 = (1 : 0 : 0), p2 = (0 : 1 : 0) and p3 = (0 : 0 : 1)
+which have type (m + 1, . . . , m + 1)
+�
+��
+�
+m times
+. These singularities are easily seen to be quasi-
+homogeneous and hence
+µ(C, pj) = τ(C, pj) = (m − 1)2,
+for j ∈ {1, 2, 3}. It follows by Theorem 1.2 that
+(C, HC)pj = 3(m − 1)2 + m − 3 + m(m + 1) = 4m(m − 1).
+The equality (2.7) implies that C has no inflection points. The reader can check that
+this curve C is not even nearly free, e.g. for m = 3. On the other hand, we show
+now that the curve
+C′ : F ′ = xyzF = 0
+is free with exponents (m + 1, m + 1). In order to show this we note first that the
+only singularities of C′ are again the points pj for j ∈ {1, 2, 3}, which are ordinary
+quasihomogeneous singularities of multiplicity (m + 2). For the last claim one can
+use [1, Exercise (7.33)]. It follows that
+τ(C′) = 3(m + 1)2.
+
+CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE
+21
+The equality
+x(ym − zm)Fx − y(ym + zm)Fy + z(ym + zm)Fz = 0
+shows that mdr(F) = m + 1. This implies that one has mdr F ′ ≥ mdr F = m + 1
+and
+τ(2m + 3, m + 1)max = (2m + 2)2 − (m + 1)2 = τ(C′).
+This equality implies our claim by Theorem 3.4.
+References
+[1] A. Dimca, Topics on real and complex singularities. An introduction. Advanced Lectures in
+Mathematics. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. (1987). 4.9, 6.3
+[2] A. Dimca, Hyperplane Arrangements: An Introduction, Universitext, Springer-Verlag, 2017. 1
+[3] A. Dimca, Curve arrangements, pencils, and Jacobian syzygies, Michigan Math. J. 66 (2017),
+347–365. 1, 3, 3, 4.9
+[4] A. Dimca, D. Ibadula, A. M˘acinic, Numerical invariants and moduli spaces for line arrange-
+ments. Osaka J. Math. 57: 847 – 870 (2020). 4.4
+[5] A. Dimca, G. Ilardi, G. Sticlaru, Addition-deletion results for the minimal degree of a Jacobian
+syzygy of a union of two curves. J. Algebra 615(1): 77 – 102 (2023). 4.2, 4.4, 4.7
+[6] A. Dimca and P. Pokora, Maximizing curves viewed as free curves. arXiv:2208.13399. 3, 3,
+6.2
+[7] A. Dimca and G. Sticlaru, Free and Nearly Free Curves vs. Rational Cuspidal Plane Curves.
+Publ. Res. Inst. Math. Sci. 54: 163 – 179 (2018). 3
+[8] A. Dimca and G. Sticlaru, Plane curves with three syzygies, minimal Tjurina curves, and
+nearly cuspidal curves. Geom. Dedicata: 207: 29 – 49 (2020). 4.2
+[9] A. A. Du Plessis and C. T. C. Wall, Application of the theory of the discriminant to highly
+singular plane curves. Math. Proc. Camb. Philos. Soc. 126(2): 259 – 266 (1999). 3
+[10] W.L. Edge, A plane quartic curve with twelve undulations. Edinburgh Math. Notes 35: 10 –
+13 (1945). 5
+[11] G. Fischer, Ebene algebraische Kurven. Vieweg, 1994. 2.1
+[12] G.-M. Greuel, C. Lossen and E. Shustin, Introduction to Singularities and Deformations.
+Springer (2007). 2
+[13] A. Josse and F. P`ene, On the degree of caustics of reflection. Comm. Algebra 42: 2442 – 2475
+(2014). 1, 2, 2, 2, 2
+[14] E. Kunz, Introduction to Plane Algebraic Curves, Birkh¨auser, 2005. 1, 1
+[15] A. Kuribayashi and K. Komiya, On Weierstrass points and automorphisms of curves of genus
+three. Algebraic geometry, Proc. Summer Meet., Copenh. 1978, Lect. Notes Math. 732: 253 –
+299 (1979). 5
+[16] T. K. Moe, Cuspidal curves on Hirzebruch surfaces. PhD thesis, University of Oslo, 2013, pp.
+xx+154. 1
+[17] P. Orlik, H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin Heidelberg New
+York, 1992. 1
+[18] A. Simis, S.O. Toh˘aneanu, Homology of homogeneous divisors, Israel J. Math. 200 (2014),
+449-487. 3.2
+[19] S. O. Toh˘aneanu, On freeness of divisors on P2, Communications in Algebra, 41 (2013), 2916–
+2932. 3.2
+[20] C. T. C. Wall, Singular Points of Plane Curves. Cambridge University Press, 2004.
+
+22
+ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU
+Universit´e Cˆote d’Azur, CNRS, LJAD, France and Simion Stoilow Institute of
+Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania
+Email address: dimca@unice.fr
+Dipartimento Matematica Ed Applicazioni “R. Caccioppoli” Universit`a Degli Studi
+Di Napoli “Federico II” Via Cintia - Complesso Universitario Di Monte S. Angelo
+80126 - Napoli - Italia
+Email address: giovanna.ilardi@unina.it
+Department of Mathematics, Pedagogical University of Krakow, Podchora¸˙zych
+2, PL-30-084 Krak´ow, Poland.
+Email address: piotr.pokora@up.krakow.pl
+Faculty of Mathematics and Informatics, Ovidius University Bd.
+Mamaia 124,
+900527 Constanta, Romania
+Email address: gabriel.sticlaru@gmail.com
+
diff --git a/cdAzT4oBgHgl3EQfLfvk/content/tmp_files/load_file.txt b/cdAzT4oBgHgl3EQfLfvk/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,793 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf,len=792
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='01117v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='AG]  3 Jan 2023  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A  GIVEN CURVE  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In the present note we construct new families of free plane curves  starting from a curve C and adding high order inflectional tangent lines of C,  lines joining the singularities of the curve C, or lines in the tangent cone of some  singularities of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  These lines L have in common that the intersection C ∩ L  consists of a small number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We introduce the notion of a supersolvable  plane curve and conjecture that such curves are always free, as in the known case of  line arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Some evidence for this conjecture is given as well, both in terms  of a general result in the case of quasi homogeneous singularities and in terms of  specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Introduction  Our goal in this paper is to construct new free or nearly curves by adding inflec-  tional tangents or lines passing through the singularities of a given plane projective  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Sometimes, when high order inflectional tangents are missing, lines in the  tangent cones of the singularities may replace them successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The use of tangent  cones is a must when we want to get supersolvable curves with the modular point  belonging to a non-linear irreducible component, see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  To determine the existence of inflectional tangents, we start by recalling some  facts about inflection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let C : F = 0 be a reduced plane curve in the complex  projective plane P2 which is defined by a homogeneous polynomial F ∈ S = C[x, y, z]  of degree d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The Hessian of F is given by the following well-known formula  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1)  H = det  \uf8eb  \uf8ed  Fxx  Fxy  Fxz  Fxy  Fyy  Fyz  Fxz  Fyz  Fzz  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Let HC : H = 0 be the Hessian curve associated to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It is known that the inter-  section XC = C ∩ HC consists exactly of the set of inflection points IC of C union  with the set of singular points YC of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Recall that if p ∈ C is a smooth point of  this curve, and TpC denotes the projective line tangent to C at p, then the inflection  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Primary 14H50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Secondary 14B05, 13D02, 32S22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' plane curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dimca was partially supported by the Romanian Ministry of Research and Innovation, CNCS -  UEFISCDI, Grant PN-III-P4-ID-PCE-2020-0029, within PNCDI III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Pokora was partially supported by the National Science Center (Poland) Sonata Grant Nr  2018/31/D/ST1/00177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  1  2  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  order of p is by definition  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2)  ιp(C) = (C, TpC)p − 2,  where (C, TpC)p denotes the intersection multiplicity of the curves C and TpC at the  common point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moreover, we say that p is an inflection point of C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=', p ∈ IC, if  and only if ιp(C) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The intersection multiplicity (C, HC)p of the curves C and HC at the point p is a  key invariant in understanding the geometry of the plane curve C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' When p ∈ IC is  an inflection point, then the relation between the inflection order ιp(C) of p and the  intersection multiplicity (C, HC)p is well-known, see for instance [14, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7  and Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' For any reduced plane curve C of degree d ≥ 2 and any smooth point  p ∈ C, one has  ιp(C) = (C, HC)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  It is clear that ιp(C) ≤ d − 2, except the case when p sits on a line L = TpC which  is an irreducible component of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In the later case, one has ιp(C) = ∞, and it is easy  to check that the line L is also an irreducible component of the Hessian curve HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  From now on, while searching for the inflection points, we assume that no irreducible  component of C is a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  When p ∈ YC is a singular point, then the description of the intersection multi-  plicity (C, HC)p is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let TCp(C) be the reduced projective tangent cone  of the curve C at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' For any line L ∈ TCp(C) we define the corresponding tangential  multiplicity  mL(C) = (L, CL)p,  where (CL, p) is the union of all branches of the singularity (C, p) whose tangent line  at p coincides with L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' With this notation, one has the following general result, which  is a user-friendly reformulation of [13, Proposition 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Assume that C is a reduced plane curve and that p ∈ C is any  singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then one has  (C, HC)p = 3µp(C) + mp(C) − 3 +  �  L∈TCp(C)  mL(C),  where µp(C) is the Milnor number and mp(C) is the multiplicity of the singularity  (C, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The case when (C, p) is irreducible is also stated in [16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Indeed, in  this case, one has µp(C) = 2δp(C) with δp(C) being the δ-invariant of the singularity  (C, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We discuss in detail the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 and give some examples  in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Our new results are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Consider the case when (C, p) is an ordinary  k-multiple point, that is there are k smooth branches C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , Ck at p, with distinct  tangent lines L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If we set mj = (Cj, Lj)p ≥ 2 for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , k}, then we  call (C, p) an ordinary k-multiple point of type (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moreover, we say that  (C, p) is an ordinary simple k-multiple point if mj = 2 for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  3  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' For any reduced plane curve C of degree d ≥ 3 and any singular  point p ∈ C with multiplicity mp(C) = k, one has  (C, HC)p ≥ 3k(k − 1),  and equality holds if and only if (C, p) is an ordinary simple k-multiple point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' For any reduced plane curve C of degree d ≥ 3 and having s singular  points, one has  i(C) ≤ 3d(d − 2) − 6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  More precisely, if nk denotes the number of singular points of C of multiplicity k,  then  i(C) ≤ 3d(d − 2) −  �  k  3k(k − 1)nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The equality occurs if and only if all the singularities of C are ordinary simple k-  multiple points for various k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  This result is a significant improvement of the inequality  �  p∈IC  ιp ≤ 3d(d − 2) − s  which is given in [14, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Our next result is a general construction of free  curves by adding inflectional tangents and lines passing through the singularities of  the initial curve C, which is a curve of Thom-Sebastiani type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let m ≥ 2 be a  positive integer and let ℓj(x, y) with j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , m} be m distinct linear forms in x  and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Consider the curve  C : F =  m  �  j=1  ℓj(x, y)kj − zd = 0,  in P2, where kj ≥ 1 are positive integers such that � kj = d, and the family of lines  Lj : ℓj(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The line Lj is the inflectional tangent at pj if kj = 1, and it is the  reduced tangent cone at the singularity pj when kj > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' With the notation as above, the curve  C′ = C ∪  m  �  j=1  Lj :  F ′ = F ·  m  �  j=1  ℓj = 0  of degree d + m is free with the exponents (m − 1, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moreover, if L : z = 0 is the  line passing through all the points pj of C, then the curve  C′′ = C′ ∪ L :  F ′′ = zF ·  m  �  j=1  ℓj = 0  of degree d + m + 1 is free with the exponents (m − 1, d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  4  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  Starting with a smooth Fermat type curve and adding all of its inflectional tangents  and the 3 coordinate axes, one gets again a free curve, as the following result shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Consider the Fermat curve C : xd + yd + zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let ǫ be any root of the equation  td+1 = 0 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then the line Lǫ : y = ǫx intersects C only at the point pǫ = (1 : ǫ : 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Hence pǫ is an inflection point of the maximal order and hence i(Lǫ) = d − 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=',  one has the equality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In this way, we get d inflection points, which are just  the intersection of C with the line z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Cyclically permuting x, y, z, we get all the  3d inflection points of this type, and these are all the inflection points of C by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The following result is a special case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5, when kj = 1 for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The union C′ : F ′ = (xd + yd)F = 0 of the smooth Fermat curve  C : F = xd + yd + zd = 0 of degree d with the d inflectional tangents Lǫ meeting at  one point, is a free curve of degree 2d and the exponents are (d − 1, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' When d = 3,  the curve C′ is maximizing of degree 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The union C′′ : F ′′ = zF ′ = 0 of the curve  C′ with the line L : z = 0 passing through all the flex points pǫ of C, is a free curve  of degree 2d + 1 and the exponents are (d − 1, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' When d = 3, then the curve C′ is  maximizing of degree 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Note that if we continue to add just inflectional tangents, the resulting curve is no  longer a free curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' For instance, the curve  C′′ : F ′′ = (x3 + y3 + z3)(x3 + y3)(y3 + z3) = 0  is nearly free with the exponents (4, 5), and the curve  C′′′ : F ′′′ = (x3 + y3 + z3)(x3 + y3)(y3 + z3)(x3 + z3) = 0  is not even nearly free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' However, we have the following general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The smooth Fermat curve C : F = xd + yd + zd = 0 has exactly 3d  inflectional tangents and their union forms the following line arrangement  A : (xd + yd)(yd + zd)(xd + zd) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The union of the curve C, the lines in A, and the 3 coordinate axes produce a new  curve  C′ : F ′ = xyz(xd + yd)(yd + zd)(xd + zd)(xd + yd + zd) = 0  of degree 4d + 3, which is free with the exponents (2d + 1, 2d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moreover, the  curve C′′ ⊂ C′ given by  C′′ : F ′′ = xy(yd + zd)(xd + zd)(xd + yd + zd) = 0  has degree 3d + 2 and it is free with the exponents (d + 1, 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  When d = 2, then the curve C′′ is maximizing of degree 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  It is easy to prove that the curve  C : F = xmym + ymzm + xmzm = 0  has no infection points using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2, see Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' To get a free  curve from C, we may add two of the three tangent cones, or just one tangent cone  and two lines joining singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Indeed, one has the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  5  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The curve  C′ : F ′ = (xmym + ymzm + xmzm)(xm + ym)(ym + zm) = 0  has degree 4m and it is free with exponents (2m − 1, 2m) for any m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The curve  C′′ : F ′′ = yz(xmym + ymzm + xmzm)(ym + zm) = 0  has degree 3m + 2 and it is free with exponents (m + 1, 2m) for any m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Given a reduced plane curve C, we say that p ∈ C is a modular  point for C if the central projection  πp : P2 \\ {p} → P1  induces a locally trivial fibration of the complement M(C) = P2 \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We say that a  reduced plane curve C is supersolvable if it has at least one modular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The map induced by πp is a locally trivial fibration if and only if for any line Lp  passing through p and not an irreducible component of C, one has  (C, Lp)p = multp(C) and (C, Lp)q = 1 for any q ∈ C ∩ Lp, q ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  This fibration has as base and as fiber a projective line P1 with a number of points  deleted, and hence the complement M(C) is a K(π, 1) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  When C is a line  arrangement, this definition of a modular point coincides with the usual one, and a  line arrangement is supersolvable by definition if it has a modular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In particular,  the existence of a modular point for a line arrangement C implies that C is free, see  for all these well known facts [2, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We venture to make the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' A supersolvable plane curve C is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  One setting where this conjecture holds is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let C0 be a reduced plane curve, let p ∈ M(C0) be a point and let  A be the set of lines L passing through p such that there is a point q ∈ L ∩ C0 with  (C0, L)q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Assume that all the singularities of the curve C obtained by adding all  the lines in A to C0 are quasi homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then C is supersolvable and free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In  particular, this holds when all the singularities s of C0 have multiplicity 2, and p is  not on any tangent cone TCs(C0) for (C0, s) a singularity with µ(C0, 0) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  When C0 is a plane curve having only nodes A1 and cusps A2 as singularities, and  in addition p is a generic point, this result is known, see [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moreover,  it is easy to see that the free curves C′ and C′′ constructed above in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5  are special cases of the construction in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The free curve C′′ constructed  above in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8 is of a different nature, since in this case p ∈ C0 and the curve  C has not only quasi homogeneous singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It is interesting that it gives new  examples where Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='10 holds, in view of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The free curve C′′ constructed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8 is supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  In particular, the associated complement M(C′′) is a K(π, 1) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  6  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  We explain in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11 that the other free curves constructed above in The-  orems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8 are not supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The organization of the paper goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In Section 2, we discuss the proof  of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 explaining all the necessary details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In Section 3 we recall basic facts  on the free, nearly free and maximizing curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  In Section 4 we deliver our proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11, and of  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then, in Section 5, we describe all smooth plane quartic curves  admitting the maximal possible number of flexes of maximal order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=', flexes of order  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' There are two such curves, and only one of them, the Fermat curve, yields free  curves as in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='6 and in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In the final section we discuss some  singular plane curves and the free curves obtained from them, which are sometimes  supersolvable as well, given rise to supersolvable free curves not covered by our  general Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Discussion on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 and some examples  The paper [13] uses rather heavy notations, and perhaps due to this fact has a  smaller impact than it deserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let us introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' For a reduced  plane curve C : F = 0 and any point q = (α : β : γ) ∈ P2, we define the polar ∆q(C)  of C with respect to q by the equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1)  ∆q(C) : αFx + βFy + γFz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  For a property P, the authors of [13] use the notation 1P to denote 1 if P is true  and 0 otherwise, see the discussion before Theorem 2 in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The first equality in  [13, Proposition 25] gives the expression of the intersection multiplicity (C, ∆q(C))p  for any singular point p ∈ C and any point q ∈ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Using our discussion above,  we see that for q ̸= p, q not on any line L in the tangent cone TCp(C), this mul-  tiplicity (C, ∆q(C))p is given by a double sum S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=', the second sum involving the  characteristic functions 1P vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' With this observation, the second equality in  [13, Proposition 25] can be stated as  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2)  (C, HC)p = 3(C, ∆q(C))p + Ip,  where Ip = �  i∈I(i(i)  p − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Here I is a set of indices parametrizing the pro-branches  of (C, p) and i(i)  p  is the tangential intersection number of the pro-branch Bi, see [13,  Definition 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Recall that any branch B of a plane curve singularity (C, 0) at the  origin of C2, such that x = 0 is not a tangent line, can be defined by a Weierstrass  polynomial  ΓB(x, y) =  �  j=1,mB  (y − φB,j(x))  where mB is the multiplicity of the branch B and there is an analytic function  φB(x) ∈ C{x} such that  φB,j(x) = φB(exp(2πij/mB)x  1  mB ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  With this notation, the branch B has mB associated pro-branches  Bj : y − φB,j(x) = 0  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  7  and the corresponding tangential intersection number is given by  i  Bj  0  = ord(φB,j(x) − φ′  B,j(0)x) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  It follows that i  Bj  0  = iBk  0  for any 0 ≤ j ≤ k ≤ mB and hence  mBi  Bj  0  = (L, CL)0,  for any j, with L the tangent line to B and CL = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This discussion implies that  one has  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3)  Ip =  �  i∈I  (i(i)  p − 2) =  �  L∈TCp(C)  mL(C) − 2mp(C),  since clearly |I| = mp(C), each branch having exactly a number of pro-branches given  by the multiplicity of that branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Next, we return to the intersection multiplicity  (C, ∆q(C))p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We can assume that p = (0 : 0 : 1) and set f(x, y) = F(x, y, 1), then  one has  fx(x, y) = Fx(x, y, 1), fy(x, y) = Fy(x, y, 1), and xfx + yfy + Fz(x, y, 1) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  If we define the generic local polar variety of the singularity  (C, 0) : f(x, y) = 0  by the equation  ∆0(C) : α′fx + β′fy = 0,  with (α′ : β′) ∈ P1 being a generic point, it is easy to see that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4)  (C, ∆0(C))0 = (C, ∆q(C))p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  In fact, the line L′ : z = 0 is clearly not in the tangent cone TCp(C) since p /∈ L′,  and hence we may take γ = 0 and (α : β) ∈ P1 generic in the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In order  to compute this local intersection number  κ0(C) = (C, ∆0(C))0,  which is also called the κ-invariant of the singularity (C, 0), we can use for instance  [12, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='38] and get  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5)  κ0(C) = µ(C, 0) + m0(C, 0) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  If we use this formula for the singularity (C, p), we get from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3) the  following equality  (C, HC)p = 3(µp(C) + mp(C) − 1) +  �  L∈TCp(C)  mL(C) − 2mp(C) =  = 3µp(C) + mp(C) − 3 +  �  L∈TCp(C)  mL(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  This proves our reformulation of [13, Proposition 25] in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  8  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  In order to construct free plane curves starting with a curve C of degree d, by  adding lines, in particular inflectional tangents, we have to look for lines L such that  the sum  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='6)  i(L) =  �  p∈L∩IC, TpC=L  ιp(C)  is as large as possible with respect to the degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Given a curve C, first we use  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 to count the total number of inflection points of C, namely  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7)  i(C) =  �  p∈IC  ιp(C) = 3d(d − 2) −  �  p∈YC  (C, HC)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  One clearly has for any line L  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8)  i(L) ≤ i(C),  and the equality holds if and only if for any point p ∈ IC one has TpC = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moreover,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9)  i(L) ≤  �  p∈L∩C  ((C, L)p − 2) = d − 2|L ∩ C|,  and the equality holds if and only if for any point p ∈ L ∩ C one has TpC = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let us consider the case when (C, p) is a node A1, that is there are  two smooth branches (C1, p) and (C2, p) meeting transversally at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let T1 = TpC1  and T2 = TpC2 be the associated tangent lines and define the type of the node (C, p)  to be the pair of integers  (m1, m2) = ((C1, T1)p, (C2, T2)p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  It is clear that mj ≥ 2 for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' When m1 = m2 = 2, then (C, p) is said to be a  simple node, and one knows that (C, HC)p = 6, see [11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 68–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In the general  situation, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 gives the equality  (C, HC)p = 3 + 2 − 3 + m1 + m2 = m1 + m2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  More generally, consider the case when (C, p) is an ordinary m-multiple point, that  is there are m smooth branches C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , Cm with distinct tangent lines L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  If we set mj = (Cj, Lj)p for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , m, then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 gives the equality  (C, HC)p = 3(m − 1)2 + m − 3 +  �  j=1,m  mj = m(3m − 5) +  �  j=1,m  mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let us consider the case when (C, p) is a singularity A2m−1 with  m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then there are two tangent smooth branches with a common tangent line  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If we set mL = (C, L)p, then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 gives the equality  (C, HC)p = 3(2m − 1) + 2 − 3 + mL = 2(3m − 2) + mL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let us consider the case when (C, p) is a singularity A2m with m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Then there is a unique branch, with a tangent line L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If we set mL = (C, L)p, then  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 gives the equality  (C, HC)p = 3(2m) + 2 − 3 + mL = 6m − 1 + mL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  9  When (C, p) is a cusp A2, only the value mL = 3 is possible, and hence (C, HC)p = 8  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Free, nearly free and maximizing curves  In this section we recall some basic facts on free, nearly free and, maximizing  curves in P2 following [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Let  Der(S) = {∂ := a · ∂x + b · ∂y + c · ∂z, a, b, c ∈ S}  be the free S-module of C-linear derivations of the polynomial ring S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' For a reduced  curve C : F = 0, we introduce  D(F) = {∂ ∈ Der(S) : ∂ F ∈ ⟨F⟩},  the graded S-module of derivations preserving the ideal ⟨F⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We have the following  decomposition  D(F) = D0(F) ⊕ S · δE,  where δE = x∂x + y∂y + z∂z is the Euler derivation and  D0(F) = {∂ ∈ Der(S) : ∂ F = 0}  is the set of all C-linear derivations of S killing the polynomial F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We say that a reduced curve C : F = 0 is free if D(F), or equiv-  alently D0(F), is a free graded S-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The exponents (d1, d2) of a free curve C  are the degrees of a basis for the free graded S-module D0(F) which rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The exponents (d1, d2) of a free curve C : F = 0 of degree d are known  to satisfy d1 +d2 = d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Conversely, if there are two elements r1, r2 ∈ D0(F), which  are S-linearly independent and satisfy  d1 + d2 = d − 1  then the curve C is free with exponents (d1, d2), see [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The minimal degree of derivations killing F, or of Jacobian syzygies  involving the partial derivatives of F, is defined as  mdr(F) = min{r ∈ N : D0(F)r ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  To check whether a given plane curve is free, one may use the following result by  du Plessis and Wall [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let C : F = 0 be a reduced plane curve of degree d and let r =  mdr(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then the following two cases hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  a) If r < d/2, then τ(C) ≤ τ(d, r)max = (d − 1)2 − r(d − r − 1) and the equality  holds if and only if the curve C is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  b) If d/2 ≤ r ≤ d − 1, then τ(C) ≤ τ(d, r)′  max, where, in this case, we set  τ(d, r)′  max = τ(d, r)max −  �2r − d + 2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  10  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' A reduced curve C : F = 0 of degree d is nearly free if either  mdr(F) < d/2 and τ(C) = τ(d, r)max − 1, or mdr(F) = d/2 and τ(C) = τ(d, r)max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  In addition, the exponents of a nearly free curve C : F = 0 of degree d are given by  the pair (mdr F, d − mdr F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' A curve C : F = 0 of degree d having only ADE-singularities is  maximizing if either d = 2m and τ(C) = 3m(m − 1) + 1, or d = 2m + 1 and  τ(C) = 3m2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The relation between maximizing curves and free curves is the following, see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' A curve C : F = 0 of degree d having only ADE-singularities is  maximizing if and only if either d = 2m and C is a free curve with the exponents  (m − 1, m), or d = 2m + 1 and C is a free curve with the exponents (m − 1, m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  In the sequel we need the following version of [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let C : F = 0 be a reduced plane curve of degree d and let p be any  point of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let A be the union of the irreducible components of C which are lines  passing through p, and let C′ : G = 0 be the union of the other irreducible components  of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We assume that p ∈ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let m = |A| and e = deg G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then r = mdr(F) can  be in one of the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  a) r = e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  b) r = m − 1 and C is free with exponents (m − 1, e);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  c) m ≤ r ≤ e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The only difference of this result with respect to [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='10] is that here  p ∈ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The key part of the proof of [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='10] is contained in [3, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3],  where a Jacobian syzygy  ρ : aFx + bFy + cFz = 0  is constructed using a differential 2-form ω, and it is shown that this syzygy is  primitive, that is there is no common factor for a, b, c ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It is in this latter part  that the condition p /∈ C′ was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If we assume that p = (0 : 0 : 1), it is easy to  see that the Jacobian relation ρ constructed there is still a Jacobian syzygy in our  situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moreover, it is a primitive syzygy if and only if G and Gz have no common  factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let M ∈ S be an irreducible polynomial which is a common factor for G and  Gz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Note that M cannot involve only x and y, since this would correspond to a line  in C′ passing through p, which is impossible by the definition of C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows that  Mz ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If G = MN, then Gz = MzN + MNz, which implies that either M divides  N or M divides Mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' But M cannot divide N, since C′ is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' And M cannot  divide Mz, since the degree of Mz as a polynomial in z is strictly smaller than the  corresponding degree of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This contradiction proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The proofs of our main results  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We can assume that p = (0 : 0 : 1) and set f(x, y) =  F(x, y, 1), g(x, y) = jkf(x, y) the initial form of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let n be the number of distinct  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  11  factors of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If n = k, then using Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1 we have  (C, HC)p = k(3k − 5) +  �  j=1,k  mj ≥ 3k(k − 1),  since mj ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moreover, it is obvious that the equality holds if and only if (C, p) is  an ordinary simple k-multiple point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Assume now that n < k and let  g(x, y) = ℓa1  1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' · ℓan  n  be the decomposition of g as a product of linear factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Recall that for two isolated  plane curve singularities (X, 0) and (Y, 0) with no common component one has  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1)  µ(X ∪ Y, 0) = µ(X, 0) + µ(Y, 0) + 2(X, Y )0 − 1,  see [20, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let Cj : fj = 0 be the union of the branches of (C, p)  which are tangent to the line Lj : ℓj = 0 with j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Then (C, p) =  (C1, p) ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' ∪ (Cn, p) and we estimate µp(C) = µ(C, p) using the above formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' To  start with, note that since jajfj = ℓ  aj  j it follows that  µ(Cj, 0) ≥ aj(aj − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Assume that for some m < n one has  µ((C1, p) ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' ∪ (Cm, p)) ≥ (a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' + am)2 − (a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' + am) − m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Then it follows that  µ((C1, p) ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' ∪ (Cm, p) ∪ (Cm+1, p)) ≥ ((a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' + am)2 − (a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' + am) − m + 1)+  +am+1(am+1 − 1) + 2am+1(a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' + am) − 1 =  = (a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' + am+1)2 − (a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' + am+1) − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Since a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' + an = k, this yields the inequality  µ(C, p) ≥ k2 − k − n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  On the other hand, one has  mLj = (Cj, Lj)p ≥ aj + 1  and hence  �  j  mLj ≥ k + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 we get  (C, HC)p ≥ 3(k2 − k − n + 1) + k − 3 + k + n = 3k(k − 1) + 2(k − n) > 3k(k − 1),  since we have assumed n < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4 is an obvious consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  12  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The curve C was studied in [8, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5] and it  was shown that the minimal degree of a Jacobian relation for C is given by  mdr(F) = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Since the minimal degree of a Jacobian relation can only increase when one adds  lines to a given curve, see [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1], it follows that  mdr(F ′) ≥ mdr(F) = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The curve C has m inflection points and singularities on the line z = 0, locally  given by equations ukj − vd = 0, located at the points pj = (aj : bj : 0), with  ℓj(aj, bj) = 0 for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' When we add the line Lj, we get at the point pj  a weighted homogeneous singularity of degree dj = 1 with respect to the weights  w1 = wt(uj) =  1  kj+1 and w2 = wt(z) =  kj  d(kj+1), where uj = ℓj is a local coordinate  at pj on the line Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows the following equality involving Tjurina and Milnor  numbers:  τ(C′, pj) = µ(C′, pj) = (1 − w1)(1 − w2)  w1w2  = (d − 1)kj + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Hence the total Tjurina number of C′ is  τ(C′) = d(d − 1) + md + (m − 1)2,  since clearly  τ(C′, p) = µ(C′, p) = (m − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Now a curve of degree d′ = d + m with r′ = mdr(F ′) satisfies the inequality  τ(C′) ≤ τ(d′, r′)max  where the function  τ(d′, r′)max = (d′ − 1)2 − r′(d′ − 1 − r′)  is a decreasing function of r′ for 2r′ < d′ which follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4, and  the equality τ(C′) = τ(d′, r′)max implies that 2r′ < d′ and that C′ is free with the  exponents (r′, d′ − r′ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In our case, we get  τ(d′, m − 1)max = (d + m − 1)2 − d(m − 1) = τ(C′),  and this proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The proof of the second claim goes analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' One can check that the curve C′, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' C′′, can be regarded as a  special case of the curve C constructed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11, starting from the curve  C0 : F = 0 and p = (0 : 0 : 1) for C′, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' C0 : zF = 0 and p = (0 : 0 : 1) for C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  This is an alternative way to proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We start with the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Consider the line arrangement  B : g = xyz(xd + yd)(yd + zd)(xd + zd) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Then mdr(g) = 2d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We consider first the subarrangement of B given by  B0 : g0 = xyz(xd + yd)(yd + zd) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Note that in this arrangement B0 there are two points of multiplicity d+2, connected  by the line y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' All the lines pass through one of these two points, and the other  intersection points are all double points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows that B0 is a line arrangement of  type ˆL(d + 2, d + 2) as in [4, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9], and  mdr(g0) = d + 1,  see [4, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' To get the arrangement B from B0, we have to add the d lines  L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=', Ld given by xd + zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' At the stage k, where 1 ≤ k ≤ d, we have to add the  line Lk to the arrangement  Bk = B0 ∪ L1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' ∪ Lk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Note that the intersection of Lk and Bk consists of exactly 2d + 2 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If Bk is  given by the reduced equation fk = 0, for 1 ≤ k ≤ d, it follows from [5, Corollary  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4] that one has  mdr(fk) = mdr(fk−1) + 1  for all 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Hence  mdr(g) = mdr(fd) = d + 1 + d = 2d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  □  Using [5, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1 (b)] we see that mdr(F ′) ≥ 2d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We compute now the  total Tjurina number of the curve C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This curve has 3d2 nodes A1 and 3 points with  local equation uv(ud + vd) coming from the double points of the line arrangement B  not situated on xyz = 0 and the 3 points of multiplicity d + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The line x = 0 in B  contains d double points of this line arrangement, which are precisely the inflection  points of order d − 2 of the Fermat curve C situated on this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The corresponding  inflectional tangents are the lines given by yd + zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Therefore, when we add C,  each of these d points becomes a singularity of type D2d+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Similar remarks apply  to the lines y = 0 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows that  τ(C′) = 3d2 + 3(d + 1)2 + 3d(2d + 2) = 12d2 + 12d + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  On the other hand, we have  τ(4d + 3, 2d + 1)max = (4d + 2)2 − (2d + 1)2 = 12d2 + 12d + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The equality  τ(C′) = τ(4d + 3, 2d + 1)max  14  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  implies as above that r′ = mdr(F ′) = 2d + 1 and that C′ is a free curve with the  exponents (2d + 1, 2d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The claims for the curve C′′ are proved in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The line arrangement  B′ : g′ = xy(yd + zd)(xd + zd) = 0  satisfies mdr(g′) = d+1, see [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The lines x = 0 and y = 0 contain  each d points of type D2d+2 as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Besides these points, the line arrangement B′  has two points of multiplicity d + 1 and d2 + 1 double points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows that  τ(C′′) = 2d(2d + 2) + 2d2 + (d2 + 1) = 7d2 + 4d + 1 = τ(3d + 2, d + 1)max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It is interesting to note that the line arrangement formed by the  corresponding 3d inflectional tangent lines is the arrangement  (xd + yd)(yd + zd)(xd + zd) = 0,  which is far from being free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' On the other hand, the line dual to the inflection point  pǫ = (1 : ǫ : 0) is L′  ǫ : x + ǫy = 0, and the union of all these d dual lines obtained  when ǫ varies, is given by xd − yd = 0 when d is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Therefore, for d odd, the line  arrangement formed by the corresponding 3d dual lines is precisely the free monomial  (or Fermat) line arrangement  (xd − yd)(yd − zd)(xd − zd) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The case d = 3 is of course well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' First we consider the curve C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The reader can check  the following Jacobian relations r1, r2 ∈ D0(F ′)  r1 : zm−1(xm + ym)F ′  x − xm−1(ym + zm)F ′  z = 0  and  r2 : xym−1(2xm + 3ym)F ′  x − (2F + y2m)F ′  y + zym−1(2zm + 3ym)F ′  z = 0,  where F = xmym + ymzm + xmzm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Since  deg r1 + deg r2 = (2m − 1) + 2m = deg F ′ − 1,  our claim is proved by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  We consider now the curve C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' First we show that mdr F ′′ = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' To do this,  we first determine a minimal degree Jacobian syzygy r1 for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' One has for instance  r1 : a1Fx + b1Fy + c1Fz = 0,  where a1 = x(ym − zm), b1 = −y(ym + zm) and c1 = z(ym + zm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Now we apply [5,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3] and see that if we add a line L0 to C given by an equation ℓ = sy+ty = 0  such that ℓ divides  sb1 + tc1 = (−sy + tz)(ym + zm)  the resulting curve C0 = C ∪ L0 : F0 = ℓF = 0 has again  mdr F0 = mdr F = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  15  Moreover, the coefficients of a minimal degree syzygy for F0 can be obtained from  the discussion just before [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows that one can add one by one  all the lines in the arrangement  yz(ym + zm) = 0  and get at the end mdr F ′′ = mdr F = m + 1 as we have claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' To show now that  C′′ is free with the given exponents it is enough to apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If one likes to use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4 as above to prove that the curve  C′ is free, one needs to compute the total Tjurina number τ(C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  This in turn  is complicated, since the singularities of C′ at the points p1 = (1 : 0 : 0) and  p3 = (0 : 0 : 1) are no longer quasi homogeneous, and hence τ(C′′, pj) < µ(C′′, pj),  for j = 1 and j = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In fact, our Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8 combined with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4 implies  that  τ(C′′, p1) = τ(C′′, p3) = 5m2 − 2m,  maybe a result not easy to prove otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' By its very construction, it is clear that p is a  modular point for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Let e = deg C0 and m = |A| = multp(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Hence d = deg(C) =  e + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then the fibration  πp : M(C) → B  induced by the central projection with center p has as fiber F the projective line P1  minus e + 1 points, and as base B the projective line P1 minus m points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows  that the Euler number E(M(C)) of the complement M(C) is given by  E(M(C)) = E(F)E(B) = (1 − e)(2 − m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  On the other hand, we know that  E(M(C)) = E(P2) − E(C) = 3 − (µ(C) − d(d − 3)),  where µ(C) is the total Milnor number of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The above two equations give  µ(C) = (e + m)2 − em − 2m − e + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Since all the singularities of C are supposed to be quasi homogeneous, we get  τ(C) = µ(C) = (e + m)2 − em − 2m − e + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  To show that C : F = 0 is free we apply [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows that r = mdr F  satisfies one of the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  a) r = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then  τ(e + m, e)max = (e + m − 1)2 − e(m − 1) = τ(C),  which implies that e < m and C is free with exponents (e, m − 1) using  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  b) r = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then  τ(e + m, m − 1)max = (e + m − 1)2 − (m − 1)e = τ(C),  which implies that m ≤ e and C is free with exponents (m − 1, e) again by  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  16  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  c) m ≤ r < e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This case is impossible, since it implies  τ(C) ≤ τ(e + m, r)max < τ(e + m, m − 1)max = τ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The first inequality follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4, and the second one by the fact  that the function t �→ τ(e + m, t)max is decreasing for 2t < e + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  To prove the last claim in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11, first notice that the possible non quasi  homogeneous singularities of C may occur only at the intersection s = L∩C0, where  s is a singular point of C0 and L is a line in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If mults C0 = 2 and L is not in  the corresponding tangent cone TCs(C0) as we have assumed, then mults C = 3 and  the 3-jet j3g of a local equation (C, s) : g = 0 is a binary cubic form with at least  2 distinct factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows from the classification of singularities, see for instance  [1], that such a singularity has type Dk, for some k ≥ 4, and in particular it is quasi  homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If µ(C0, s) = 1 and L is in the corresponding tangent cone TCs(C0),  then (C0, s) is a node A1, and the same argument as above works, namely (C, s) is  a Dk singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Finally, when µ(C0, s) = 2 and L is in the corresponding tangent  cone TCs(C0), then (C0, s) is a cusp A2, and the new singularity (C, s) is easily seen  to be of type E7, hence again quasi homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The point p = (1 : 0 : 0) is a modular point  in this case, since any line Lp through p, not an irreducible component for C′′, is  given by z = ty with t ̸= 0 and tm + 1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The intersection Lp ∩ C′′ is described by  the equation  ty2(xmym + tmy2m + tmxmym)(ym + tmym) = t(tm + 1)y2m+2((1 + tm)xm + tmym) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The solution y = 0 corresponds to the point p, which has multiplicity 2m + 2 on  C′′, and there are m = deg F ′′ − (2m + 2) other intersection points coming from the  solutions of (1 + tm)xm + tmym = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Here we show first that the free curves C′ and C′′ coming from The-  orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7 are not supersolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' For the curve C′, it is clear that the only candidates  for modular points are the point p = (0 : 0 : 1) and the 2 other points obtained  from p by permutation of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Indeed, a modular point has to contain all  the tangents to the Fermat curve issued from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Now the point p is not a modular  point for C′, since the line Lp : y − x = 0 is not an irreducible component of C′ and  it satisfies  2d+2 = |Lp ∩C′| < |L′ ∩C′| = deg C′ −multp(C′)+1 = 4d+3−(d+2)+1 = 3d+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The same line Lp : y − x = 0 shows that the point p is not a modular point for C′′  either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The point p′ = (1 : 0 : 0) is also not a modular point, as the choice of the  line Lp′ : z = 0 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The point p′′ = (0 : 1 : 0) has the same property, as our curve  is invariant under the coordinate change x �→ y and y �→ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' To show that the free  curve C′ coming from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='8 is not supersolvable, we use the same approach as  above, the lines Lp to use in this case are given by x = 0, y = 0 or z = 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Examples: the case of smooth quartic curves  In this section we discuss examples of smooth quartic curves having the maximal  possible number of flexes of high order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Let us recall that by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 the  maximal possible number of flex points of order 2 for smooth quartics is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It is  natural to wonder whether there exists a complete classification of smooth quartics  which have exactly 12 flexes of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In order to do so, we discuss interesting  properties of the following pencil of quartics which was studied by Ciani in the 19th  century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Let us define  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1)  Cλ : x4 + y4 + z4 + λ · (y2z2 + z2x2 + x2y2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  It is easy to observe that each curve in the pencil is invariant under the natural  action of an octahedral group of collineations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  There are some values of λ which lead to special members of the pencil, namely  λ = 0 gives us the Fermat quartic curve, or Dyck’s curve, which has a group  of 96 collineations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  if λ is a root of λ2 + 3λ + 18 = 0, then we get the Klein quartic curve having  a group of 168 collineations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  In the case of the Fermat quartic, by a discussion presented above, we know that  it has exactly 12 flexes of order 2, so the maximal possible number in the class of  smooth quartics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In the case of the Klein quartic curve, we know that this curve has  only flex points of order 1, so exactly 24 flexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Now we pass to another interesting  element in the pencil of quartics by taking λ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The resulting quartic C3 is smooth  and it has the group of collineations of order 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It was verified directly by Edge in  [10] that the curve C3 admits exactly 12 flexes of order 2 and he provided both the  coordinates of these points and the equations of the associated tangent lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Now  we recover Edge’s calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Looking precisely on the Hessian H of C3, which is  H = 2x6 + x4(3z2 + 3y2) + x2(8y2z2 + 3z4 + 3y4) + 2z6 + 2y6 + 3z4y2 + 3z2y4,  one can show that flexes of order 2 are just the intersection points of the curve C3  with the 6 lines given by the linear factors of  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2)  F = (x2 + y2)(y2 + z2)(z2 + x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The flex points have the following coordinates:  P1 : (i : 1 : −1),  P2 : (−i : 1 : −1),  P3 : (−1 : i : 1),  P4 : (−1 : −i : 1),  P5 : (1 : −1 : i),  P6 : (1 : −1 : −i),  P7 : (−i : 1 : 1),  P8 : (i : 1 : 1),  P9 : (1 : −i : 1),  P10 : (1 : i : 1),  P11 : (1 : 1 : −i),  P12 : (1 : 1 : i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Observe that these 12 flexes of order 2 are uniformly distributed, four on each of the  lines defined by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Up to now we described exactly two smooth quartics having the maximal possible  number of flexes of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' However, as it turns out by a result due to Kuribayashi  18  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  and Komiya [15], these are the only smooth plane quartic curves having 12 flexes of  order 2, and this is rather surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' We have seen in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7 that if we add to the Fermat quartic  its 12 inflectional tangents of order 2 and the triangle ∆ : xyz = 0 determined by  the inflection points, then we get a free curve of degree 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If we try to apply the  same construction to the quartic curve C3, the resulting curves are far from being  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' One explanation for this fact may be the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The union AF of the 12  inflectional tangents of the Fermat quartic is a line arrangement having 3 points of  multiplicity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' On the other hand, the union A3 of the 12 inflectional tangents of  the quartic C3 is a line arrangement having only double points, and hence the total  Tjurina number τ(A3) is much smaller than τ(AF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If we add the triangle ∆ to  AF, we get a line arrangement having 3 points of multiplicity 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' On the other hand,  if we add to A3 the 6 lines determined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2), we get a line arrangement having  only points of multiplicity 2 and 3, and hence having small total Tjurina number  compared with respect to its degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Examples: the case of singular curves  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Any nodal cubic is projectively equivalent to the cubic  C : F = xyz + x3 + y3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The corresponding Hessian is H = −2(3(x3 + y3) − xyz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Hence the intersection  C ∩ HC consists of the following 4 points:  p1 = (0 : 0 : 1) and pj = (1 : j : 0),  where j3 + 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The point p1 is a simple node, and the points pj give rise to 3  inflection points of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This is reflected in the equality  (C, HC)p1 +  �  j  (C, HC)pj = 6 + 1 + 1 + 1 = 9,  recall Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The 3 inflectional tangents Lk for k = 1, 2, 3 are given by the  equations Lk : 3x+3j3  ky+jkz = 0, where jk are the 3 roots of the equation j3+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  It follows that these 3 inflectional tangents Lk are not concurrent, so their addition  to C will not give free curves as in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' On the other hand, if we add  to C the tangent cone at the singular point, we obtain the curve  C′ : F ′ = xy(xyz + x3 + y3) = 0,  which is free with exponents (2, 2) as a direct computation with SINGULAR shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Moreover, this curve C′ is supersolvable, since clearly p1 is a modular point for C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Any cuspidal cubic is projectively equivalent to the cubic  C : F = x2z + y3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The corresponding Hessian is H = −24x2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Hence the intersection C ∩ HC consists  of the following 2 points  p1 = (0 : 0 : 1) and p2 = (1 : 0 : 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  19  The point p1 is a cusp A2, and the point p2 is an inflection point of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This is  reflected in the equality  (C, HC)p1 + (C, HC)p2 = 8 + 1 = 9,  recall Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This is a special case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='5, and gives rise to two free  curves by adding one or two lines, as explained there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In this example we consider some plane quartic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Consider the quartic C : F = (x3 + y3)z + x4 + y4 = 0, which has a D4-singularity  at p1 = (0 : 0 : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The corresponding Hessian is  H = −54(xyz(x3 + y3) + 2x2y2(x2 + y2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The point p1 is an ordinary simple singularity of multiplicity k = 3, and hence  (C, HC)p1 = 18 by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' There are in addition 6 inflection points of order  1, with are the points (1 : 0 : −1), (0 : 1 : −1) and the 4 points (u : v : w), where  (u : v) is coming from the 4 solutions of the equation  u4 + v4 − 2uv(u2 + v2) = 0  in P1 and w = −(u4 +v4)/(u3 +v3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If we add the tangent cone of the singular point,  namely the lines x3 + y3 = 0, we get a free curve  C′ : F ′ = (x3 + y3)F = 0,  of degree 7 and exponents (3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moreover, this curve C′ is supersolvable, since  clearly p1 is a modular point for C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Next, consider the quartic  C : F = x2y2 + y2z2 + x2z2 = 0,  which has 3 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It is easy to see that all of them have type (3, 3), and hence C  has no inflection points by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The corresponding Hessian is  H = −24(x4y2 + x2y4 + y4z2 + y2z4 + x4z2 + x2z4 − 6x2y2z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  If we add to C one tangent line at each of the 3 nodes, namely the lines  (x + iy)(y + iz)(z + ix) = 0,  we get a free curve of degree 7 with exponents (3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' All the singularities of this  curve are simple, but this curve is not maximizing, recall our discussion in Section 3  on these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Finally the quartic  C : F = x2y2 + y2z2 + x2z2 − 2xyz(x + y + z) = 0,  which has 3 cusps A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Hence C has no inflection points by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The  corresponding Hessian is  H = 144(x3y3 + y3z3 + x3z3 − x3(y2z + yz2) − y3(x2z + xz2) − z3(x2y + xy2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Let L1, L2 and L3 be the 3 lines which are the reduced tangent cones corresponding  to the 3 cusps, which are given up to an order by the equations x − y = 0, y − z = 0  and z − x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Then the curves  C1 = C ∪ L1, C2 = C1 ∪ L2 and C3 = C2 ∪ L3  20  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  are all free, with exponents respectively  (2, 2), (2, 3) and (2, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  We get in this way a free curve C1 of degree 5 and maximizing curves C2 and C3, of  degree 6 and 7, respectively, as already pointed out in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It is interesting to note  that the curve C3 is supersolvable, and the point p = (1 : 1 : 1) is a modular point for  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Indeed, the lines joining p to the singularities of C are already in C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It remains  to show that any line Lp through p, different from L1, L2, L3 meets C in exactly 4  points, that is Lp is not a tangent line to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' If q = (u : v : w) ∈ C is a smooth point  such that the tangent line TqC passes through p, then we have  Fx(q) + Fy(q) + Fz(q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  A direct computation shows that  Fx(q) + Fy(q) + Fz(q) = −12uvw  and hence at least one of the coordinates of q vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' But then q ∈ C implies that  2 coordinates vanish, and therefore q is a singularity of C, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Note that  the curve C3 is an example of curve satisfying both the first assumption in Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='11, since all of its singularities are quasi homogeneous, and the second assumption,  even if p belongs to the tangent cones TCs(C) of the three cusps, as they have Milnor  numbers equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In this example we consider the curve  C : F = xmym + ymzm + xmzm = 0  of degree d = 2m ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This curve has 3 ordinary m multiple points  p1 = (1 : 0 : 0), p2 = (0 : 1 : 0) and p3 = (0 : 0 : 1)  which have type (m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' , m + 1)  �  ��  �  m times  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' These singularities are easily seen to be quasi-  homogeneous and hence  µ(C, pj) = τ(C, pj) = (m − 1)2,  for j ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2 that  (C, HC)pj = 3(m − 1)2 + m − 3 + m(m + 1) = 4m(m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  The equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7) implies that C has no inflection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' The reader can check that  this curve C is not even nearly free, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' for m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' On the other hand, we show  now that the curve  C′ : F ′ = xyzF = 0  is free with exponents (m + 1, m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' In order to show this we note first that the  only singularities of C′ are again the points pj for j ∈ {1, 2, 3}, which are ordinary  quasihomogeneous singularities of multiplicity (m + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' For the last claim one can  use [1, Exercise (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='33)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' It follows that  τ(C′) = 3(m + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  CONSTRUCTION OF FREE CURVES BY ADDING LINES TO A GIVEN CURVE  21  The equality  x(ym − zm)Fx − y(ym + zm)Fy + z(ym + zm)Fz = 0  shows that mdr(F) = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' This implies that one has mdr F ′ ≥ mdr F = m + 1  and  τ(2m + 3, m + 1)max = (2m + 2)2 − (m + 1)2 = τ(C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  This equality implies our claim by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dimca, Topics on real and complex singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' An introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Advanced Lectures in  Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Braunschweig/Wiesbaden: Friedr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Vieweg & Sohn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='3  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dimca, Hyperplane Arrangements: An Introduction, Universitext, Springer-Verlag, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 1  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dimca, Curve arrangements, pencils, and Jacobian syzygies, Michigan Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 66 (2017),  347–365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 1, 3, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='9  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dimca, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Ibadula, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' M˘acinic, Numerical invariants and moduli spaces for line arrange-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Osaka J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 57: 847 – 870 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dimca, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Ilardi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Sticlaru, Addition-deletion results for the minimal degree of a Jacobian  syzygy of a union of two curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Algebra 615(1): 77 – 102 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='7  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dimca and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Pokora, Maximizing curves viewed as free curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='13399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 3, 3,  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dimca and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Sticlaru, Free and Nearly Free Curves vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Rational Cuspidal Plane Curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 54: 163 – 179 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 3  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dimca and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Sticlaru, Plane curves with three syzygies, minimal Tjurina curves, and  nearly cuspidal curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Dedicata: 207: 29 – 49 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Du Plessis and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Wall, Application of the theory of the discriminant to highly  singular plane curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Camb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 126(2): 259 – 266 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 3  [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Edge, A plane quartic curve with twelve undulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Edinburgh Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Notes 35: 10 –  13 (1945).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 5  [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Fischer, Ebene algebraische Kurven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Vieweg, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='1  [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Greuel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Lossen and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Shustin, Introduction to Singularities and Deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Springer (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 2  [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Josse and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' P`ene, On the degree of caustics of reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Algebra 42: 2442 – 2475  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 1, 2, 2, 2, 2  [14] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Kunz, Introduction to Plane Algebraic Curves, Birkh¨auser, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
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+page_content=' Kuribayashi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Komiya, On Weierstrass points and automorphisms of curves of genus  three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Algebraic geometry, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Summer Meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=', Copenh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
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+page_content=' Notes Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 732: 253 –  299 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Moe, Cuspidal curves on Hirzebruch surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' PhD thesis, University of Oslo, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  xx+154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
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+page_content=' Orlik, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin Heidelberg New  York, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 1  [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Simis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Toh˘aneanu, Homology of homogeneous divisors, Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 200 (2014),  449-487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2  [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Toh˘aneanu, On freeness of divisors on P2, Communications in Algebra, 41 (2013), 2916–  2932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='2  [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Wall, Singular Points of Plane Curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Cambridge University Press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  22  ALEXANDRU DIMCA, GIOVANNA ILARDI, PIOTR POKORA, AND GABRIEL STICLARU  Universit´e Cˆote d’Azur, CNRS, LJAD, France and Simion Stoilow Institute of  Mathematics, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Box 1-764, RO-014700 Bucharest, Romania  Email address: dimca@unice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='fr  Dipartimento Matematica Ed Applicazioni “R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Caccioppoli” Universit`a Degli Studi  Di Napoli “Federico II” Via Cintia - Complesso Universitario Di Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content=' Angelo  80126 - Napoli - Italia  Email address: giovanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='ilardi@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='it  Department of Mathematics, Pedagogical University of Krakow, Podchora¸˙zych  2, PL-30-084 Krak´ow, Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Email address: piotr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='pokora@up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='krakow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='pl  Faculty of Mathematics and Informatics, Ovidius University Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='  Mamaia 124,  900527 Constanta, Romania  Email address: gabriel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='sticlaru@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf'}
diff --git a/ddAyT4oBgHgl3EQfwvlY/content/tmp_files/2301.00653v1.pdf.txt b/ddAyT4oBgHgl3EQfwvlY/content/tmp_files/2301.00653v1.pdf.txt
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index 0000000000000000000000000000000000000000..93776331b00d1d2e2ba5e6335c043342c15318e3
--- /dev/null
+++ b/ddAyT4oBgHgl3EQfwvlY/content/tmp_files/2301.00653v1.pdf.txt
@@ -0,0 +1,1748 @@
+ 
+1 
+The original Gibbs paradox is the consequence  
+of the erroneous identification of non-identical functions 
+Volodymyr Ihnatovych 
+Igor Sikorsky Kyiv Polytechnic Institute 
+e-mail: v.ihnatovych@kpi.ua 
+ 
+Abstract 
+This article presents the results of research into the causes of the Gibbs paradox in the 
+formulation discussed by J. W. Gibbs himself. In this formulation, we are talking about an 
+inexplicable (paradoxical) jump in the entropy of mixing of two ideal gases during the transition 
+from mixing different to mixing identical gases. 
+It is shown that the entropy of mixing of different ideal gases and the entropy of mixing 
+of identical ideal gases are different (non-identical) functions of the same gas parameters. That, 
+called a paradoxical jump in the entropy of mixing, is not a jump in the value of some function, 
+but is the difference in the values of various functions, on condition that the variables and 
+parameters on which these functions depend remain constant. Those who were looking for an 
+explanation of the original Gibbs paradox did not notice this and tried to solve an unsolvable 
+falsely posed problem: to find a parameter that change during the transition from different to 
+identical gases caused the difference in the values of non-identical functions. 
+ 
+I. Introduction 
+The Gibbs Paradox is one of the most mysterious physical paradoxes. It has been known 
+for over a hundred years. It was explained or discussed by J. W. Gibbs, M. Planck, J. D. van der 
+Waals, H. A. Lorentz, A. Sommerfeld, E. Schrödinger, P. W. Bridgman and other well-known 
+physicists (see, e.g. [1–6]). Various explanations to this paradox have long been outlined in 
+textbooks (see, e.g. [2,3,7–9]), but papers devoted to it appear again and again (see, e.g., [5,10–
+19]). 
+There are several formulations (versions, kinds) of the Gibbs paradox. The formulation, 
+which we, following the example of J. van Lith ([16]), call the original Gibbs paradox, arises 
+when considering in the framework of classical thermodynamics the question of the value of the 
+entropy of mixing of two ideal gases separated an initially by impenetrable partition. 
+
+ 
+2 
+By reasoning, there is the following conclusion that the entropy of mixing of various ideal 
+gases with the same temperatures and pressures is not equal to zero and does not depend on the 
+properties of the gases being mixed. Also, by reasoning, there is an obtained conclusion that the 
+entropy of mixing of identical ideal gases with the same temperatures and pressures is equal to 
+zero. Thus, if we assume that there is a mixing of gases that are increasingly similar in properties 
+and are with the same temperatures and pressures, then the entropy of mixing remains constant 
+as long as there remains any, even vanishingly small, difference between the mixed gases. When 
+a transition to identical ideal gases occurs, the entropy mixing turns to zero by a jump, and the 
+magnitude of the jump does not depend on what and how much different these gases were. This 
+behavior of the entropy of mixing is paradoxical: a quantity that does not depend on the degree 
+of difference in properties of the gases suddenly vanishes when the difference in the properties 
+disappears. This formulation is also called the thermodynamic version of the Gibbs paradox 
+[13,17], the Gibbs paradox of the second kind [14], the discontinuity puzzle [18], the first Gibbs 
+paradox [19]. 
+The jump in the entropy of mixing looks especially paradoxical because the entropy of 
+mixing, like the entropy of gases, is a function of the properties of gases, i.e., a quantity that 
+depends on other quantities. The value of a function cannot change if the quantities on which it 
+depends have not changed. Consequently, the change in the entropy of mixing in the transition 
+from mixing different to mixing identical gases must be the result of a change in the value of 
+some variable or parameter. But in the case of the specified jump in the entropy of mixing, it is 
+not clear which quantity changes to cause such a change in this function. Therefore, those who 
+were looking for an explanation for this behavior of the entropy of mixing tried to find this 
+quantity. 
+For more than 100 years, many explanations of the Gibbs paradox in this formulation have 
+been proposed (see, e. g., [2–4, 7, 11, 13, 17]), but there is no one that would not cause 
+reasonable objections, and which could be called the final or generally accepted. 
+A common feature of all works devoted to this formulation of the Gibbs paradox is that 
+their authors did not take into account the fact that the conclusion about a paradoxical jump in 
+the entropy of mixing is not based on processing of empirical data, but arises as a result of 
+reasoning based on some premises. They did not analyze this reasoning and did not notice that 
+the conclusion about a paradoxical jump in the entropy of mixing is made by comparing the 
+values of non-identical functions, which are erroneously considered identical. To demonstrate 
+this, we derive a number of equations and consider the question: why do the values of the 
+entropy of mixing of different and identical ideal gases differ? 
+
+ 
+3 
+II. Derivation of equations for the entropy of mixing of ideal gases, initially 
+separated by a partition, and obtaining a conclusion about a paradoxical 
+jump 
+When deriving equations for the entropy of mixing of different and identical ideal gases, 
+we will use as initial equations the well-known equations for the entropy of an ideal gas, systems 
+of ideal gases and change in entropy with change in the state of a thermodynamic system: 
+
+
+
+
+
+
+
+
+
+i
+i
+i
+i
+Vi
+i
+i
+i
+i
+i
+s
+n
+V
+R
+T
+c
+n
+T
+V
+n
+S
+0
+ln
+ln
+)
+,
+,
+(
+, 
+(1) 
+)
+,
+,
+(
+)
+,
+,
+(
+)
+,
+,
+,
+,
+,
+(
+2
+2
+2
+2
+1
+1
+1
+1
+2
+2
+2
+1
+1
+1
+T
+V
+n
+S
+T
+V
+n
+S
+T
+V
+n
+T
+V
+n
+S b
+
+
+, 
+(2) 
+)
+,
+,
+(
+)
+,
+,
+(
+)
+,
+,
+,
+(
+2
+2
+1
+1
+2
+1
+T
+V
+n
+S
+T
+V
+n
+S
+T
+V
+n
+n
+S m
+
+
+, 
+(3) 
+I
+II
+S
+S
+S
+
+
+
+, 
+(4) 
+i
+i
+i
+i
+RT
+n
+V
+p
+
+, 
+(5) 
+where 
+)
+,
+,
+(
+i
+i
+i
+i
+T
+V
+n
+S
+ is the entropy of 
+in  moles of the ith ideal gas, its volume is 
+iV  and temperature 
+is 
+iT ; 
+Vi
+c  is the molar heat capacity of the ith ideal gas at a constant volume, which depends on 
+the nature of the gas; R is universal gas constant; 
+is0  is a constant, which depends on the nature 
+of the gas; 
+)
+,
+,
+,
+,
+,
+(
+2
+2
+2
+1
+1
+1
+T
+V
+n
+T
+V
+n
+Sb
+ is the entropy of a system consisting of two ideal gases 
+separated by an impermeable partition, the quantities of which are 
+1n  and 
+2
+n  moles, the volumes 
+are 
+1
+V  and 
+2
+V , and the temperatures are 1T  и 
+2T ; 
+)
+,
+,
+,
+(
+2
+1
+T
+V
+n
+n
+Sm
+ is the entropy of a mixture of 
+1n  
+and 
+2
+n  moles of different ideal gases, its volume is V and temperature is T; ΔS is change in the 
+entropy of the system during its transition from initial state I to final state II; 
+IS  is the entropy of 
+the system in the initial state, 
+II
+S  is the entropy of the system in the final state; 
+ip  is the pressure 
+of the ith ideal gas. 
+The equation (3) expresses the Gibbs’ theorem on the entropy of a mixture of ideal gases 
+(see e. g. [7,8]). 
+The equations (1)–(5) or equivalent equations are used by other authors who considered 
+the Gibbs paradox (see, e.g. [1–3,7–10,12,16]). 
+Function designations in Eqs. (1)–(3) contain variables on which these functions depend. 
+In what follows, the designations of various functions will also indicate the variables on which 
+they depend. The values related to systems of different gases will be denoted by superscripts d, 
+and the values related to systems of identical ideal gases will be denoted by superscripts i. 
+
+ 
+4 
+In order not to be distracted by details that have nothing to do with the paradoxical jump 
+in the entropy of mixing, we consider the mixing of ideal gases with the same temperatures and 
+pressures. Since the gases are ideal, the temperature of the system after mixing will be equal to 
+the temperature in the initial state. Note: Gibbs considered the case of mixing ideal gases with 
+the same temperatures, pressures and volumes [1]. Many authors consider the case of mixing 
+one and one mole of ideal gases with the same temperatures, pressures and volumes. We will 
+consider a more general case — mixing 
+1n  and 
+2
+n  moles of ideal gases that have the same initial 
+temperatures and pressures. 
+Suppose that 
+1n  and 
+2
+n  moles of different ideal gases 1 and 2, that temperatures are equal 
+to T, and the volumes are 
+1
+V  and 
+2
+V , are separated by an impermeable partition. The next 
+equation for the entropy of this system follows from Eqs. (1) and (2): 
+02
+2
+01
+1
+2
+2
+2
+1
+1
+1
+2
+2
+1
+1
+2
+2
+1
+1
+ln
+ln
+ln
+)
+(
+)
+,
+,
+,
+,
+(
+s
+n
+s
+n
+n
+V
+n
+n
+V
+n
+R
+T
+c
+n
+c
+n
+T
+V
+n
+V
+n
+S
+V
+V
+d
+b
+
+
+
+
+
+
+
+
+
+
+
+
+. (6) 
+After removing the partition, a mixture of 
+1n  and 
+2
+n  moles of ideal gases 1 and 2 of 
+volume 
+2
+1
+V
+V 
+ is formed. The next equation for the entropy of this mixture follows from Eqs. 
+(1) and (3): 
+02
+2
+01
+1
+2
+2
+1
+2
+1
+2
+1
+1
+2
+2
+1
+1
+2
+1
+2
+1
+ln
+ln
+ln
+)
+(
+)
+,
+,
+,
+(
+s
+n
+s
+n
+n
+V
+V
+n
+n
+V
+V
+n
+R
+T
+c
+n
+c
+n
+T
+V
+V
+n
+n
+S
+V
+V
+m
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+. (7) 
+The next equation for the entropy of mixing of different ideal gases 1 and 2 with equal 
+initial temperatures follows from Eqs. (4), (6), (7), taking into account that 
+)
+,
+,
+,
+,
+(
+2
+2
+1
+1
+T
+V
+n
+V
+n
+S
+S
+d
+b
+d
+I 
+ and 
+)
+,
+,
+,
+(
+2
+1
+2
+1
+T
+V
+V
+n
+n
+S
+S
+m
+d
+II
+
+
+: 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+2
+2
+2
+1
+1
+1
+2
+2
+1
+2
+1
+2
+1
+1
+2
+2
+1
+1
+ln
+ln
+ln
+ln
+)
+,
+,
+,
+(
+n
+V
+n
+n
+V
+n
+R
+n
+V
+V
+n
+n
+V
+V
+n
+R
+V
+n
+V
+n
+S d
+. 
+(8) 
+If ideal gases have the same temperature and pressure, then it follows from Eq. (5): 
+2
+1
+2
+1
+2
+2
+1
+1
+n
+n
+V
+V
+n
+V
+n
+V
+
+
+
+
+. 
+(9) 
+The equation (8) can be converted to: 
+)]
+ln
+ln
+(
+)
+ln(
+)
+[(
+ln
+ln
+ln
+)
+(
+)
+,
+,
+,
+(
+2
+2
+1
+1
+2
+1
+2
+1
+2
+2
+2
+1
+1
+1
+2
+1
+2
+1
+2
+1
+2
+2
+1
+1
+n
+n
+n
+n
+n
+n
+n
+n
+R
+n
+V
+n
+n
+V
+n
+R
+n
+n
+V
+V
+n
+n
+R
+V
+n
+V
+n
+S d
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+(10) 
+From (9) and (10) follows the equation for the entropy of mixing of different ideal gases 
+with the same temperatures and pressures: 
+
+ 
+5 
+)]
+ln
+ln
+(
+)
+ln(
+)
+[(
+)
+,
+(
+2
+2
+1
+1
+2
+1
+2
+1
+2
+1
+n
+n
+n
+n
+n
+n
+n
+n
+R
+n
+n
+S d
+
+
+
+
+
+
+. 
+(11) 
+If also 
+n
+n
+n
+
+
+2
+1
+, then from (11) it follows: 
+2
+ln
+2
+)
+,
+(
+2
+1
+Rn
+n
+n
+S d
+
+
+. 
+(12) 
+If also 
+1
+2
+1
+
+ n
+n
+, then from (11) it follows: 
+2
+ln
+2
+)
+,
+(
+2
+1
+R
+n
+n
+S d
+
+
+. 
+(13) 
+Now let us consider the case of mixing of identical ideal gases with the same temperatures. 
+Suppose, 
+1n  and 
+2
+n  moles of identical gases 3 and 3, that temperatures are equal to T, and 
+the volumes are 
+1
+V  and 
+2
+V , are separated by an impenetrable partition. The next equation for the 
+entropy of this system follows from Eqs. (1) and (2): 
+03
+2
+1
+2
+2
+2
+1
+1
+1
+3
+2
+1
+2
+2
+1
+1
+)
+(
+ln
+ln
+ln
+)
+(
+)
+,
+,
+,
+(
+s
+n
+n
+n
+V
+n
+n
+V
+n
+R
+T
+c
+n
+n
+V
+n
+V
+n
+S
+V
+i
+b
+
+
+
+
+
+
+
+
+
+
+
+
+. 
+(14) 
+After removing the partition, 
+2
+1
+n
+n 
+ moles of pure ideal gas 3 of volume 
+2
+1
+V
+V 
+ are 
+formed, that entropy according to Eq. (1) is: 
+03
+2
+1
+2
+1
+2
+1
+2
+1
+3
+2
+1
+2
+1
+2
+1
+3
+)
+(
+ln
+)
+(
+ln
+)
+(
+)
+,
+,
+(
+s
+n
+n
+n
+n
+V
+V
+n
+n
+R
+T
+c
+n
+n
+T
+V
+V
+n
+n
+S
+V
+
+
+
+
+
+
+
+
+
+
+. 
+(15) 
+The next equation for the entropy of mixing of identical ideal gases with equal initial 
+temperatures follows from Eqs. (4), (14), (15), taking into account that 
+)
+,
+,
+,
+,
+(
+2
+2
+1
+1
+T
+V
+n
+V
+n
+S
+S
+i
+b
+i
+I 
+ 
+and 
+)
+,
+,
+(
+2
+1
+2
+1
+3
+T
+V
+V
+n
+n
+S
+S i
+II
+
+
+
+: 
+
+
+
+
+
+
+
+
+
+
+
+
+
+2
+2
+2
+1
+1
+1
+2
+1
+2
+1
+2
+1
+2
+2
+1
+1
+ln
+ln
+ln
+)
+(
+)
+,
+,
+,
+(
+n
+V
+n
+n
+V
+n
+R
+n
+n
+V
+V
+n
+n
+R
+V
+n
+V
+n
+S i
+. 
+(16) 
+From (16) and (11) follows the equation for the entropy of mixing of identical ideal gases 
+with the same temperatures and pressures: 
+0
+)
+,
+(
+2
+1
+
+
+n
+n
+S i
+. 
+(17) 
+Based on Eqs. (11) and (17), we obtain a conclusion about the paradoxical behavior of the 
+entropy of mixing in the transition from mixing different to mixing identical gases. The entropy 
+of mixing 
+1n  and 
+2
+n moles of different ideal gases with the same temperatures and pressures 
+)
+,
+(
+2
+1 n
+n
+S d
+
+, according to Eq. (11), is not equal to zero, it depends only on the quantities of gases 
+1n  and 
+2
+n  and does not depend on gases properties. The entropy of mixing 
+1n and 
+2
+n  moles of 
+identical ideal gases with the same temperatures and pressures 
+)
+,
+(
+2
+1 n
+n
+S i
+
+, according to Eq. 
+(17), is equal to zero. Let us assume that the quantities of mixed gases do not change, and the 
+
+ 
+6 
+properties of gases 1 and 2 approach the properties of gas 3 and, in the limit, gases 1 and 2 
+become identical to gas 3. The entropy of mixing, according to Eq. (11), remains constant as 
+long as the gases are different. When gases 1 and 2 become identical to gas 3, their entropy of 
+mixing, according to Eq. (17), is equal to zero. Thus, during the transition from mixing different 
+ideal gases to mixing identical ideal gases, the entropy of mixing turns to zero by a jump. 
+In the same way, by comparing the values of 
+)
+,
+(
+2
+1 n
+n
+S i
+
+ and 
+)
+,
+(
+2
+1 n
+n
+S d
+
+ for cases of 
+mixing different and identical ideal gases with the same temperatures and pressures, other 
+authors also obtain a conclusion on the paradoxical jump in the entropy of mixing (see, e.g. [1–
+3,7–10,12,16]). The conclusion about the paradoxical jump in the entropy of mixing seems 
+absolutely reliable. This conclusion follows from a comparison of Eq. (17) with Eqs. (11)–(13). 
+Eqs. (11)–(13), (17) are obtained on the basis of generally accepted initial equations by 
+mathematical derivations given above. The correctness of the derivation of each of these 
+equations can be easily verified. All the initial equations and all the assumptions used in 
+obtaining each equation are given above. When obtaining each intermediate equation, it is 
+indicated from which initial and intermediate equations it follows. 
+ 
+III. Identification of the error, the consequence of which is the conclusion 
+about the paradoxical jump in the entropy of mixing 
+Consider the question: what causes the difference in the values of the entropy of mixing 
+of the same quantities of different and identical ideal gases, i.e., the difference in the values of 
+the functions 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S d
+
+ and 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S i
+
+? 
+To find the answer to this question, we obtain a equation for the difference in values 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S d
+
+ and 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S i
+
+. It follows from (10) and (16): 
+)]
+ln
+ln
+(
+)
+ln(
+)
+[(
+)
+,
+,
+,
+(
+)
+,
+,
+,
+(
+2
+2
+1
+1
+2
+1
+2
+1
+2
+2
+1
+1
+2
+2
+1
+1
+n
+n
+n
+n
+n
+n
+n
+n
+R
+V
+n
+V
+n
+S
+V
+n
+V
+n
+S
+i
+d
+
+
+
+
+
+
+
+
+ 
+(18) 
+It follows from (18): 
+)]
+ln
+ln
+(
+)
+ln(
+)
+[(
+)
+,
+,
+,
+(
+)
+,
+,
+,
+(
+2
+2
+1
+1
+2
+1
+2
+1
+2
+2
+1
+1
+i
+2
+2
+1
+1
+n
+n
+n
+n
+n
+n
+n
+n
+R
+V
+n
+V
+n
+S
+V
+n
+V
+n
+S d
+
+
+
+
+
+
+
+
+ (19) 
+It follows from Eq. (19) that the function called the entropy of mixing of different ideal 
+gases 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S d
+
+ is equal to the sum of two functions: the function called the entropy of 
+mixing 
+of 
+identical 
+ideal 
+gases 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S i
+
+ 
+and 
+the 
+function 
+)]
+ln
+ln
+(
+)
+ln(
+)
+([
+2
+2
+1
+1
+2
+1
+2
+1
+n
+n
+n
+n
+n
+n
+n
+n
+R
+
+
+
+
+. Therefore, the function called the entropy of 
+mixing of different ideal gases 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S d
+
+ is not identical to the function called the 
+
+ 
+7 
+entropy of mixing of identical ideal gases 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S i
+
+. As a result, these functions have 
+different values even if the variables and parameters on which these functions depend, namely 
+1n , 
+2
+n , 
+1
+V , 
+2
+V , have the same values. The value 
+)]
+ln
+ln
+(
+)
+ln(
+)
+([
+2
+2
+1
+1
+2
+1
+2
+1
+n
+n
+n
+n
+n
+n
+n
+n
+R
+
+
+
+
+ 
+called the magnitude of the jump in the entropy of mixing ideal gases during the transition from 
+mixing different to mixing identical gases, is not a change in the value of some function, but is 
+the difference in the values of various functions. The difference of this value from zero is due 
+precisely to the difference in the functions 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S d
+
+ and 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S i
+
+. 
+However, the authors who, following Gibbs, draw the conclusion about a paradoxical 
+jump in the entropy of mixing to zero in the transition from mixing different to mixing identical 
+gases, erroneously believe that the quantities 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S d
+
+, that is expressed by Eq. (10), 
+and 
+)
+,
+,
+,
+(
+2
+2
+1
+1
+V
+n
+V
+n
+S i
+
+, that is expressed by Eq. (16), are different values of the same function 
+— the entropy of mixing of ideal gases. But a function cannot have different values if the values 
+of the parameters and variables on that it depends do not change. Therefore, those who are 
+looking for the reason for this jump face the problem of finding the parameter of gases, due to 
+which, 
+when 
+changing 
+from 
+different 
+to 
+identical 
+gases, 
+the 
+function 
+)]
+ln
+ln
+(
+)
+ln(
+)
+([
+2
+2
+1
+1
+2
+1
+2
+1
+n
+n
+n
+n
+n
+n
+n
+n
+R
+
+
+
+
+ turns to zero, on condition that 
+1
+n and 
+2
+n  do not 
+change. This problem arises due to the erroneous identification of non-identical functions and 
+is unsolvable. Therefore, no one has solved it for more than a hundred years. 
+ 
+IV. Conclusions 
+The appearance of original Gibbs paradox in the formulation of the conclusion on a 
+paradoxical jump in the entropy of mixing of ideal gases during the transition from mixing of 
+different gases to mixing of identical gases is due to the erroneous identification of non-identical 
+functions: the entropy of mixing of different ideal gases and the entropy of mixing of identical 
+ideal gases, the first of which is the sum of the second and the function 
+)]
+ln
+ln
+(
+)
+ln(
+)
+([
+2
+2
+1
+1
+2
+1
+2
+1
+n
+n
+n
+n
+n
+n
+n
+n
+R
+
+
+
+
+. This error entails a completely unsolvable 
+problem: to find an parameter of ideal gas different from 
+1
+n  and 
+2
+n , changing when passing 
+from 
+different 
+gases 
+to 
+identical 
+ones 
+and 
+causing 
+the 
+function 
+)]
+ln
+ln
+(
+)
+ln(
+)
+([
+2
+2
+1
+1
+2
+1
+2
+1
+n
+n
+n
+n
+n
+n
+n
+n
+R
+
+
+
+
+ turns to zero. 
+If this error is not made and it is recognized that the function called the entropy of mixing 
+of different ideal gases is not identical to the function called the entropy of mixing of identical 
+
+ 
+8 
+ideal gases, then the difference in their values will become impossible to interpret as a change 
+(jump) in the value of some function, and the original formulation Gibbs paradox will not arise. 
+ 
+Acknowledgements 
+The author is grateful to professor Oleksandr Andriiko, senior lecturer Vasiliy 
+Pikhorovich and the late associate professor Viktor Haidey for helpful discussions and 
+comments. 
+ 
+References 
+[1] Gibbs J W 1928 The Collected Works Vol 1 Thermodynamics (New York: Longmans) 
+[2] Planck M 1903 Treatise on Thermodynamics (New York: Longmans) 
+[3] Sommerfeld A 1956 Thermodynamics and Statistical Mechanics (New York: Academic 
+Press) 
+[4] Khaytun S D 2010 The History of the Gibbs Paradox 3th edition (Moscow: KomKniga) (in 
+Russian) 
+[5] Gibbs paradox and its resolutions / Compiled by S-K Lin [Internet]; 2009 Nov 7 [cited 2022 
+Dec 12]; [about 1 screen]. Available from: http://www.mdpi.org/lin/entropy/gibbs-
+paradox.htm 
+[6] Darrigol O 2018 The Gibbs Paradox: Early History and Solutions Entropy 20, 443 
+[7] Bazarov I P 1991 Thermodynamics 4th edition, rev and enl (Moscow: Vysshaya Shkola) (in 
+Russian) 
+[8] Zemansky M W and Dittman R H 1997Heat and Thermodynamics (7th edition) (New York: 
+McGraw-Hill) 
+[9] Kondepudi D and Prigogine I 1998 Modern thermodynamics: From heat engines to 
+dissipative structures (New York: Wiley) 
+[10] Gelfer Ya M, Luboshitz V L, and Podgoretskii M I 1975 Gibbs Paradox and Identity of 
+Particles in Quantum Mechanics (Nauka, Moscow) (in Russian) 
+[11] Denbigh K G and Redhead M L G 1989 Gibbs’ paradox and non-uniform convergence 
+Synthese 81 283–312 
+[12] Urusov V S Gibbs paradox and symmetrization of a multicomponent system 2007 Dokl 
+Phys Chem 417 337–41 
+[13] Ainsworth P M 2012 The Gibbs Paradox and the Definition of Entropy in Statistical 
+Mechanics Philos Sci 79 542–60 
+
+ 
+9 
+[14] Peters H 2014 Demonstration and resolution of the Gibbs paradox of the first kind Eur J 
+Phys 35, 015023 
+[15] Murashita Y and Ueda M 2017Gibbs Paradox Revisited from the Fluctuation Theorem with 
+Absolute Irreversibility Phys Rev Lett 118 060601 
+[16] van Lith J 2018 The Gibbs Paradox Lessons from Thermodynamics Entropy 20 328 
+[17] Dieks D 2018 The Gibbs Paradox and Particle Individuality Entropy 20 466 
+[18] Saunders S 2018 The Gibbs Paradox Entropy 20 552 
+[19] Swendsen R H 2018Probability, Entropy, and Gibbs’ Paradox(es) Entropy 20 450 
+
diff --git a/ddAyT4oBgHgl3EQfwvlY/content/tmp_files/load_file.txt b/ddAyT4oBgHgl3EQfwvlY/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f9b449de8af187bec3d137cfce74ce2c543a4b5b
--- /dev/null
+++ b/ddAyT4oBgHgl3EQfwvlY/content/tmp_files/load_file.txt
@@ -0,0 +1,198 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf,len=197
+page_content='1  The original Gibbs paradox is the consequence  of the erroneous identification of non-identical functions  Volodymyr Ihnatovych  Igor Sikorsky Kyiv Polytechnic Institute  e-mail: v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='ihnatovych@kpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='ua  Abstract  This article presents the results of research into the causes of the Gibbs paradox in the  formulation discussed by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Gibbs himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' In this formulation, we are talking about an  inexplicable (paradoxical) jump in the entropy of mixing of two ideal gases during the transition  from mixing different to mixing identical gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  It is shown that the entropy of mixing of different ideal gases and the entropy of mixing  of identical ideal gases are different (non-identical) functions of the same gas parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' That,  called a paradoxical jump in the entropy of mixing, is not a jump in the value of some function,  but is the difference in the values of various functions, on condition that the variables and  parameters on which these functions depend remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Those who were looking for an  explanation of the original Gibbs paradox did not notice this and tried to solve an unsolvable  falsely posed problem: to find a parameter that change during the transition from different to  identical gases caused the difference in the values of non-identical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Introduction  The Gibbs Paradox is one of the most mysterious physical paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' It has been known  for over a hundred years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' It was explained or discussed by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Gibbs, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Planck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' van der  Waals, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Lorentz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Sommerfeld, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Schrödinger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Bridgman and other well-known  physicists (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' [1–6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Various explanations to this paradox have long been outlined in  textbooks (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' [2,3,7–9]), but papers devoted to it appear again and again (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=', [5,10–  19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  There are several formulations (versions, kinds) of the Gibbs paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The formulation,  which we, following the example of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' van Lith ([16]), call the original Gibbs paradox, arises  when considering in the framework of classical thermodynamics the question of the value of the  entropy of mixing of two ideal gases separated an initially by impenetrable partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  2  By reasoning, there is the following conclusion that the entropy of mixing of various ideal  gases with the same temperatures and pressures is not equal to zero and does not depend on the  properties of the gases being mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Also, by reasoning, there is an obtained conclusion that the  entropy of mixing of identical ideal gases with the same temperatures and pressures is equal to  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Thus, if we assume that there is a mixing of gases that are increasingly similar in properties  and are with the same temperatures and pressures, then the entropy of mixing remains constant  as long as there remains any, even vanishingly small, difference between the mixed gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' When  a transition to identical ideal gases occurs, the entropy mixing turns to zero by a jump, and the  magnitude of the jump does not depend on what and how much different these gases were.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' This  behavior of the entropy of mixing is paradoxical: a quantity that does not depend on the degree  of difference in properties of the gases suddenly vanishes when the difference in the properties  disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' This formulation is also called the thermodynamic version of the Gibbs paradox  [13,17], the Gibbs paradox of the second kind [14], the discontinuity puzzle [18], the first Gibbs  paradox [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  The jump in the entropy of mixing looks especially paradoxical because the entropy of  mixing, like the entropy of gases, is a function of the properties of gases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=', a quantity that  depends on other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The value of a function cannot change if the quantities on which it  depends have not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Consequently, the change in the entropy of mixing in the transition  from mixing different to mixing identical gases must be the result of a change in the value of  some variable or parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' But in the case of the specified jump in the entropy of mixing, it is  not clear which quantity changes to cause such a change in this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Therefore, those who  were looking for an explanation for this behavior of the entropy of mixing tried to find this  quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  For more than 100 years, many explanations of the Gibbs paradox in this formulation have  been proposed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=', [2–4, 7, 11, 13, 17]), but there is no one that would not cause  reasonable objections, and which could be called the final or generally accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  A common feature of all works devoted to this formulation of the Gibbs paradox is that  their authors did not take into account the fact that the conclusion about a paradoxical jump in  the entropy of mixing is not based on processing of empirical data, but arises as a result of  reasoning based on some premises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' They did not analyze this reasoning and did not notice that  the conclusion about a paradoxical jump in the entropy of mixing is made by comparing the  values of non-identical functions, which are erroneously considered identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' To demonstrate  this, we derive a number of equations and consider the question: why do the values of the  entropy of mixing of different and identical ideal gases differ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  3  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Derivation of equations for the entropy of mixing of ideal gases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' initially  separated by a partition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' and obtaining a conclusion about a paradoxical  jump  When deriving equations for the entropy of mixing of different and identical ideal gases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  we will use as initial equations the well-known equations for the entropy of an ideal gas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' systems  of ideal gases and change in entropy with change in the state of a thermodynamic system:  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6  \uf02b  \uf02b  \uf03d  i  i  i  i  Vi  i  i  i  i  i  s  n  V  R  T  c  n  T  V  n  S  0  ln  ln  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (1)  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  2  2  2  2  1  1  1  1  2  2  2  1  1  1  T  V  n  S  T  V  n  S  T  V  n  T  V  n  S b  \uf02b  \uf03d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (2)  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  2  2  1  1  2  1  T  V  n  S  T  V  n  S  T  V  n  n  S m  \uf02b  \uf03d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (3)  I  II  S  S  S  \uf02d  \uf03d  \uf044  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (4)  i  i  i  i  RT  n  V  p  \uf03d  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (5)  where  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  i  i  i  i  T  V  n  S  is the entropy of  in  moles of the ith ideal gas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' its volume is  iV  and temperature  is  iT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  Vi  c  is the molar heat capacity of the ith ideal gas at a constant volume, which depends on  the nature of the gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' R is universal gas constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  is0  is a constant, which depends on the nature  of the gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  )  ,  ,  ,  ,  ,  (  2  2  2  1  1  1  T  V  n  T  V  n  Sb  is the entropy of a system consisting of two ideal gases  separated by an impermeable partition, the quantities of which are  1n  and  2  n  moles, the volumes  are  1  V  and  2  V , and the temperatures are 1T  и  2T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  )  ,  ,  ,  (  2  1  T  V  n  n  Sm  is the entropy of a mixture of  1n  and  2  n  moles of different ideal gases, its volume is V and temperature is T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' ΔS is change in the  entropy of the system during its transition from initial state I to final state II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  IS  is the entropy of  the system in the initial state,  II  S  is the entropy of the system in the final state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ip  is the pressure  of the ith ideal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  The equation (3) expresses the Gibbs’ theorem on the entropy of a mixture of ideal gases  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' [7,8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  The equations (1)–(5) or equivalent equations are used by other authors who considered  the Gibbs paradox (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' [1–3,7–10,12,16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  Function designations in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (1)–(3) contain variables on which these functions depend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  In what follows, the designations of various functions will also indicate the variables on which  they depend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The values related to systems of different gases will be denoted by superscripts d,  and the values related to systems of identical ideal gases will be denoted by superscripts i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  4  In order not to be distracted by details that have nothing to do with the paradoxical jump  in the entropy of mixing, we consider the mixing of ideal gases with the same temperatures and  pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Since the gases are ideal, the temperature of the system after mixing will be equal to  the temperature in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Note: Gibbs considered the case of mixing ideal gases with  the same temperatures, pressures and volumes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Many authors consider the case of mixing  one and one mole of ideal gases with the same temperatures, pressures and volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' We will  consider a more general case — mixing  1n  and  2  n  moles of ideal gases that have the same initial  temperatures and pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  Suppose that  1n  and  2  n  moles of different ideal gases 1 and 2, that temperatures are equal  to T, and the volumes are  1  V  and  2  V , are separated by an impermeable partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The next  equation for the entropy of this system follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (1) and (2):  02  2  01  1  2  2  2  1  1  1  2  2  1  1  2  2  1  1  ln  ln  ln  )  (  )  ,  ,  ,  ,  (  s  n  s  n  n  V  n  n  V  n  R  T  c  n  c  n  T  V  n  V  n  S  V  V  d  b  \uf02b  \uf02b  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6  \uf02b  \uf02b  \uf02b  \uf03d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (6)  After removing the partition, a mixture of  1n  and  2  n  moles of ideal gases 1 and 2 of  volume  2  1  V  V \uf02b  is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The next equation for the entropy of this mixture follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (1) and (3):  02  2  01  1  2  2  1  2  1  2  1  1  2  2  1  1  2  1  2  1  ln  ln  ln  )  (  )  ,  ,  ,  (  s  n  s  n  n  V  V  n  n  V  V  n  R  T  c  n  c  n  T  V  V  n  n  S  V  V  m  \uf02b  \uf02b  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6  \uf02b  \uf02b  \uf02b  \uf02b  \uf02b  \uf03d  \uf02b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (7)  The next equation for the entropy of mixing of different ideal gases 1 and 2 with equal  initial temperatures follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (4), (6), (7), taking into account that  )  ,  ,  ,  ,  (  2  2  1  1  T  V  n  V  n  S  S  d  b  d  I \uf03d  and  )  ,  ,  ,  (  2  1  2  1  T  V  V  n  n  S  S  m  d  II  \uf02b  \uf03d  :  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6  \uf02b  \uf02d  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6  \uf02b  \uf02b  \uf02b  \uf03d  \uf044  2  2  2  1  1  1  2  2  1  2  1  2  1  1  2  2  1  1  ln  ln  ln  ln  )  ,  ,  ,  (  n  V  n  n  V  n  R  n  V  V  n  n  V  V  n  R  V  n  V  n  S d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (8)  If ideal gases have the same temperature and pressure, then it follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (5):  2  1  2  1  2  2  1  1  n  n  V  V  n  V  n  V  \uf02b  \uf02b  \uf03d  \uf03d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (9)  The equation (8) can be converted to:  )]  ln  ln  (  )  ln(  )  [(  ln  ln  ln  )  (  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  2  2  1  1  2  1  2  1  2  2  2  1  1  1  2  1  2  1  2  1  2  2  1  1  n  n  n  n  n  n  n  n  R  n  V  n  n  V  n  R  n  n  V  V  n  n  R  V  n  V  n  S d  \uf02b  \uf02d  \uf02b  \uf02b  \uf02b  \uf02b  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6  \uf02b  \uf02d  \uf02b  \uf02b  \uf02b  \uf03d  \uf044  (10)  From (9) and (10) follows the equation for the entropy of mixing of different ideal gases  with the same temperatures and pressures:  5  )]  ln  ln  (  )  ln(  )  [(  )  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (  2  2  1  1  2  1  2  1  2  1  n  n  n  n  n  n  n  n  R  n  n  S d  \uf02b  \uf02d  \uf02b  \uf02b  \uf03d  \uf044  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (11)  If also  n  n  n  \uf03d  \uf03d  2  1  , then from (11) it follows:  2  ln  2  )  ,  (  2  1  Rn  n  n  S d  \uf03d  \uf044  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (12)  If also  1  2  1  \uf03d  \uf03d n  n  , then from (11) it follows:  2  ln  2  )  ,  (  2  1  R  n  n  S d  \uf03d  \uf044  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (13)  Now let us consider the case of mixing of identical ideal gases with the same temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  Suppose,  1n  and  2  n  moles of identical gases 3 and 3, that temperatures are equal to T, and  the volumes are  1  V  and  2  V , are separated by an impenetrable partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The next equation for the  entropy of this system follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (1) and (2):  03  2  1  2  2  2  1  1  1  3  2  1  2  2  1  1  )  (  ln  ln  ln  )  (  )  ,  ,  ,  (  s  n  n  n  V  n  n  V  n  R  T  c  n  n  V  n  V  n  S  V  i  b  \uf02b  \uf02b  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6  \uf02b  \uf02b  \uf02b  \uf03d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (14)  After removing the partition,  2  1  n  n \uf02b  moles of pure ideal gas 3 of volume  2  1  V  V \uf02b  are  formed, that entropy according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (1) is:  03  2  1  2  1  2  1  2  1  3  2  1  2  1  2  1  3  )  (  ln  )  (  ln  )  (  )  ,  ,  (  s  n  n  n  n  V  V  n  n  R  T  c  n  n  T  V  V  n  n  S  V  \uf02b  \uf02b  \uf02b  \uf02b  \uf02b  \uf02b  \uf02b  \uf03d  \uf02b  \uf02b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (15)  The next equation for the entropy of mixing of identical ideal gases with equal initial  temperatures follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (4), (14), (15), taking into account that  )  ,  ,  ,  ,  (  2  2  1  1  T  V  n  V  n  S  S  i  b  i  I \uf03d  and  )  ,  ,  (  2  1  2  1  3  T  V  V  n  n  S  S i  II  \uf02b  \uf02b  \uf03d  :  \uf0f7\uf0f7  \uf0f8  \uf0f6  \uf0e7\uf0e7  \uf0e8  \uf0e6  \uf02b  \uf02d  \uf02b  \uf02b  \uf02b  \uf03d  \uf044  2  2  2  1  1  1  2  1  2  1  2  1  2  2  1  1  ln  ln  ln  )  (  )  ,  ,  ,  (  n  V  n  n  V  n  R  n  n  V  V  n  n  R  V  n  V  n  S i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (16)  From (16) and (11) follows the equation for the entropy of mixing of identical ideal gases  with the same temperatures and pressures:  0  )  ,  (  2  1  \uf03d  \uf044  n  n  S i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (17)  Based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (11) and (17), we obtain a conclusion about the paradoxical behavior of the  entropy of mixing in the transition from mixing different to mixing identical gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The entropy  of mixing  1n  and  2  n moles of different ideal gases with the same temperatures and pressures  )  ,  (  2  1 n  n  S d  \uf044  , according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (11), is not equal to zero, it depends only on the quantities of gases  1n  and  2  n  and does not depend on gases properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The entropy of mixing  1n and  2  n  moles of  identical ideal gases with the same temperatures and pressures  )  ,  (  2  1 n  n  S i  \uf044  , according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  (17), is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Let us assume that the quantities of mixed gases do not change, and the  6  properties of gases 1 and 2 approach the properties of gas 3 and, in the limit, gases 1 and 2  become identical to gas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The entropy of mixing, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (11), remains constant as  long as the gases are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' When gases 1 and 2 become identical to gas 3, their entropy of  mixing, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (17), is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Thus, during the transition from mixing different  ideal gases to mixing identical ideal gases, the entropy of mixing turns to zero by a jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  In the same way, by comparing the values of  )  ,  (  2  1 n  n  S i  \uf044  and  )  ,  (  2  1 n  n  S d  \uf044  for cases of  mixing different and identical ideal gases with the same temperatures and pressures, other  authors also obtain a conclusion on the paradoxical jump in the entropy of mixing (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' [1–  3,7–10,12,16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The conclusion about the paradoxical jump in the entropy of mixing seems  absolutely reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' This conclusion follows from a comparison of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (17) with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (11)–(13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (11)–(13), (17) are obtained on the basis of generally accepted initial equations by  mathematical derivations given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The correctness of the derivation of each of these  equations can be easily verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' All the initial equations and all the assumptions used in  obtaining each equation are given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' When obtaining each intermediate equation, it is  indicated from which initial and intermediate equations it follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Identification of the error, the consequence of which is the conclusion  about the paradoxical jump in the entropy of mixing  Consider the question: what causes the difference in the values of the entropy of mixing  of the same quantities of different and identical ideal gases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=', the difference in the values of  the functions  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S d  \uf044  and  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S i  \uf044  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  To find the answer to this question, we obtain a equation for the difference in values  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S d  \uf044  and  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S i  \uf044  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' It follows from (10) and (16):  )]  ln  ln  (  )  ln(  )  [(  )  ,  ,  ,  (  )  ,  ,  ,  (  2  2  1  1  2  1  2  1  2  2  1  1  2  2  1  1  n  n  n  n  n  n  n  n  R  V  n  V  n  S  V  n  V  n  S  i  d  \uf02b  \uf02d  \uf02b  \uf02b  \uf03d  \uf044  \uf02d  \uf044  (18)  It follows from (18):  )]  ln  ln  (  )  ln(  )  [(  )  ,  ,  ,  (  )  ,  ,  ,  (  2  2  1  1  2  1  2  1  2  2  1  1  i  2  2  1  1  n  n  n  n  n  n  n  n  R  V  n  V  n  S  V  n  V  n  S d  \uf02b  \uf02d  \uf02b  \uf02b  \uf02b  \uf044  \uf03d  \uf044  (19)  It follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (19) that the function called the entropy of mixing of different ideal  gases  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S d  \uf044  is equal to the sum of two functions: the function called the entropy of  mixing  of  identical  ideal  gases  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S i  \uf044  and  the  function  )]  ln  ln  (  )  ln(  )  ([  2  2  1  1  2  1  2  1  n  n  n  n  n  n  n  n  R  \uf02b  \uf02d  \uf02b  \uf02b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Therefore, the function called the entropy of  mixing of different ideal gases  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S d  \uf044  is not identical to the function called the  7  entropy of mixing of identical ideal gases  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S i  \uf044  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' As a result, these functions have  different values even if the variables and parameters on which these functions depend, namely  1n ,  2  n ,  1  V ,  2  V , have the same values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The value  )]  ln  ln  (  )  ln(  )  ([  2  2  1  1  2  1  2  1  n  n  n  n  n  n  n  n  R  \uf02b  \uf02d  \uf02b  \uf02b  called the magnitude of the jump in the entropy of mixing ideal gases during the transition from  mixing different to mixing identical gases, is not a change in the value of some function, but is  the difference in the values of various functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' The difference of this value from zero is due  precisely to the difference in the functions  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S d  \uf044  and  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S i  \uf044  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  However, the authors who, following Gibbs, draw the conclusion about a paradoxical  jump in the entropy of mixing to zero in the transition from mixing different to mixing identical  gases, erroneously believe that the quantities  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S d  \uf044  , that is expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (10),  and  )  ,  ,  ,  (  2  2  1  1  V  n  V  n  S i  \uf044  , that is expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' (16), are different values of the same function  — the entropy of mixing of ideal gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' But a function cannot have different values if the values  of the parameters and variables on that it depends do not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Therefore, those who are  looking for the reason for this jump face the problem of finding the parameter of gases, due to  which,  when  changing  from  different  to  identical  gases,  the  function  )]  ln  ln  (  )  ln(  )  ([  2  2  1  1  2  1  2  1  n  n  n  n  n  n  n  n  R  \uf02b  \uf02d  \uf02b  \uf02b  turns to zero, on condition that  1  n and  2  n  do not  change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' This problem arises due to the erroneous identification of non-identical functions and  is unsolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Therefore, no one has solved it for more than a hundred years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Conclusions  The appearance of original Gibbs paradox in the formulation of the conclusion on a  paradoxical jump in the entropy of mixing of ideal gases during the transition from mixing of  different gases to mixing of identical gases is due to the erroneous identification of non-identical  functions: the entropy of mixing of different ideal gases and the entropy of mixing of identical  ideal gases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' the first of which is the sum of the second and the function  )]  ln  ln  (  )  ln(  )  ([  2  2  1  1  2  1  2  1  n  n  n  n  n  n  n  n  R  \uf02b  \uf02d  \uf02b  \uf02b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' This error entails a completely unsolvable  problem: to find an parameter of ideal gas different from  1  n  and  2  n , changing when passing  from  different  gases  to  identical  ones  and  causing  the  function  )]  ln  ln  (  )  ln(  )  ([  2  2  1  1  2  1  2  1  n  n  n  n  n  n  n  n  R  \uf02b  \uf02d  \uf02b  \uf02b  turns to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  If this error is not made and it is recognized that the function called the entropy of mixing  of different ideal gases is not identical to the function called the entropy of mixing of identical  8  ideal gases, then the difference in their values will become impossible to interpret as a change  (jump) in the value of some function, and the original formulation Gibbs paradox will not arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  Acknowledgements  The author is grateful to professor Oleksandr Andriiko, senior lecturer Vasiliy  Pikhorovich and the late associate professor Viktor Haidey for helpful discussions and  comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='  References  [1] Gibbs J W 1928 The Collected Works Vol 1 Thermodynamics (New York: Longmans)  [2] Planck M 1903 Treatise on Thermodynamics (New York: Longmans)  [3] Sommerfeld A 1956 Thermodynamics and Statistical Mechanics (New York: Academic  Press)  [4] Khaytun S D 2010 The History of the Gibbs Paradox 3th edition (Moscow: KomKniga) (in  Russian)  [5] Gibbs paradox and its resolutions / Compiled by S-K Lin [Internet];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' 2009 Nov 7 [cited 2022  Dec 12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' [about 1 screen].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Available from: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='mdpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='org/lin/entropy/gibbs-  paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content='htm  [6] Darrigol O 2018 The Gibbs Paradox: Early History and Solutions Entropy 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' 443  [7] Bazarov I P 1991 Thermodynamics 4th edition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' rev and enl (Moscow: Vysshaya Shkola) (in  Russian)  [8] Zemansky M W and Dittman R H 1997Heat and Thermodynamics (7th edition) (New York:  McGraw-Hill)  [9] Kondepudi D and Prigogine I 1998 Modern thermodynamics: From heat engines to  dissipative structures (New York: Wiley)  [10] Gelfer Ya M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Luboshitz V L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' and Podgoretskii M I 1975 Gibbs Paradox and Identity of  Particles in Quantum Mechanics (Nauka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Moscow) (in Russian)  [11] Denbigh K G and Redhead M L G 1989 Gibbs’ paradox and non-uniform convergence  Synthese 81 283–312  [12] Urusov V S Gibbs paradox and symmetrization of a multicomponent system 2007 Dokl  Phys Chem 417 337–41  [13] Ainsworth P M 2012 The Gibbs Paradox and the Definition of Entropy in Statistical  Mechanics Philos Sci 79 542–60  9  [14] Peters H 2014 Demonstration and resolution of the Gibbs paradox of the first kind Eur J  Phys 35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' 015023  [15] Murashita Y and Ueda M 2017Gibbs Paradox Revisited from the Fluctuation Theorem with  Absolute Irreversibility Phys Rev Lett 118 060601  [16] van Lith J 2018 The Gibbs Paradox Lessons from Thermodynamics Entropy 20 328  [17] Dieks D 2018 The Gibbs Paradox and Particle Individuality Entropy 20 466  [18] Saunders S 2018 The Gibbs Paradox Entropy 20 552  [19] Swendsen R H 2018Probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' Entropy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
+page_content=' and Gibbs’ Paradox(es) Entropy 20 450' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfwvlY/content/2301.00653v1.pdf'}
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+Acoustic circular dichroism in a three-dimensional chiral metamaterial
+Qing Tong,1 Jensen Li,2 and Shubo Wang1, ∗
+1Department of Physics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China
+2Department of Physics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China
+(Dated: January 9, 2023)
+Circular dichroism (CD) is an intriguing chiroptical phenomenon associated with the interaction
+of chiral structures with circularly polarized lights. Although the CD effect has been extensively
+studied in optics, it has not yet been demonstrated in acoustic systems.
+Here, we demonstrate
+the acoustic CD effect in a three-dimensional chiral metamaterial supporting circularly polarized
+transverse sound.
+We find that the effect is negligible in the lossy metamaterial possessing C4
+rotational symmetry but can be strongly enhanced in the C2-symmetric system with inhomogeneous
+loss. The phenomena can be understood based on the properties of the metamaterial’s complex
+band structure and the quality factors of its eigenmodes. We show that the enhanced CD in the
+C2-symmetric system is attributed to the polarization bandgaps and the non-Hermitian exceptional
+points appearing near the Brillouin-zone center and boundaries.
+The results contribute to the
+understanding of chiral sound-matter interactions and can find applications in acoustic sensing of
+chiral structures and sound manipulations based on its vector properties.
+I.
+INTRODUCTION
+Chiral structures have novel properties deriving from
+the mirror-symmetry breaking [1, 2] and are extensively
+employed to realize polarization conversion [3, 4], un-
+usual optical forces [5–7], and synthetic gauge fields[8].
+The interaction between chiral structures and chiral light,
+i.e., light carrying spin and/or orbital angular momentum
+(OAM), can give rise to circular dichroism (CD) [2, 9, 10]
+and helical (or vortical) dichroism [11, 12], correspond-
+ing to the differential absorption of lights with opposite
+chirality. The CD effect has been investigated in vari-
+ous optical structures, ranging from bilayer chiral struc-
+tures [13, 14], nonplanar three-dimensional (3D) chiral
+structures [15], to gyroid structures [16, 17]. Recent re-
+search has uncovered the subtle relations between CD
+and the Ohmic dissipation of meta-atoms [18] as well as
+the bound states in the continuum [19], enabling a pro-
+found understanding of chiral light-matter interactions.
+The CD effect has been widely applied to analyze molecu-
+lar structures [20, 21] and to achieve chiral discrimination
+[22, 23].
+It is well known that sound can carry OAM in the
+form of vortices [24–26]. The acoustic OAM can induce
+chiral sound-matter interactions and give rise to intrigu-
+ing phenomena such as acoustic geometric phases [27]
+and the acoustic orbital Hall effect [28, 29].
+The chi-
+ral sound-matter interactions can enable rich manipula-
+tions of sound vortices, including asymmetric transmis-
+sion/reflection [30], reversal of orbital angular momen-
+tum [31], and acoustic helical dichroism [32].
+In ad-
+dition, the chiral sound-matter interactions can be ap-
+plied to manipulate matter, leading to acoustic levitation
+[33, 34], acoustic tweezers [35–37], and acoustic torque
+[38, 39]. While airborne sound is a longitudinal wave,
+∗ shubwang@cityu.edu.hk
+it was shown that inhomogeneous sound fields can carry
+nonzero acoustic spin density characterized by rotating
+velocity vector fields [40, 41]. Remarkably, spin-1 trans-
+verse sound can emerge in a micropolar metamaterial
+supporting synthetic shear forces in air [42]. In contrast
+to the conventional longitudinal sound, the transverse
+sound carries full vector properties similar to electro-
+magnetic waves.
+In particular, it can carry both spin
+and OAM with intriguing spin-orbit interactions. Explo-
+ration of this new type of sound and its counter-intuitive
+properties can generate new functionalities for acoustic
+applications.
+Here, for the first time, we demonstrate the acoustic
+CD effect in a 3D chiral metamaterial that supports cir-
+cularly polarized transverse sound. Using full-wave nu-
+merical simulations, we calculate the absorption of left-
+handed circularly polarized (LCP) and right-handed cir-
+cularly polarized (RCP) sound in the lossy metamaterial.
+We find that the CD effect strongly depends on the ro-
+tational symmetry of the metamaterial. The CD effect
+is negligible in the metamaterial with homogeneous loss
+satisfying the C4 rotational symmetry.
+In contrast, it
+is strongly enhanced in the metamaterial with inhomo-
+geneous loss satisfying the C2 rotational symmetry. By
+studying the complex band structure of the metamate-
+rial and the quality factors of its eigenmodes, we find that
+these properties originate from the polarization bandgaps
+and the non-Hermitian exceptional points (EPs) of the
+metamaterial.
+We organize the article as follows. In Section II, we
+introduce the acoustic chiral metamaterial and discuss
+its eigenmode properties. In Section III, we show the nu-
+merical results for the absorption of the LCP and RCP
+sound in two types of lossy metamaterial obeying the
+C4 and C2 rotational symmetry, respectively. Section IV
+presents the complex band structures of the lossy meta-
+materials, where we discuss the polarization bandgaps
+and the EPs to understand the CD effect. We draw the
+conclusion in Section V.
+arXiv:2301.02526v1  [physics.app-ph]  6 Jan 2023
+
+2
+II.
+THE CHIRAL METAMATERIAL
+We consider a 3D metamaterial with the cubical unit
+cell shown in Figure 1(a). The unit cell comprises three
+chiral resonators mutually connected by tubes, and it
+obeys the C4 rotational symmetry with respect to the
+x, y, and z axes.
+Figure 1(b) shows a half of the chi-
+ral resonator, where the radius is R = 5 cm, and the
+height is h = 1 cm. The orange blades are connected
+to the center post, and the gray blades are connected to
+the outer shell of the resonator. We assume that air is
+filled inside the resonator and all air-material interfaces
+are hard boundaries. In the considered range of frequen-
+cies, each resonator support subwavelength resonances.
+These resonances endow the metamaterial with intrigu-
+ing macroscopic acoustic properties.
+We first calculate the band structure of the metama-
+terial by using a finite-element package COMSOL Mul-
+tiphysics.
+The result is shown in Figure 1(c) for the
+wavevector in z direction. The 4th to 9th bands (count-
+ing from bottom) correspond to the eigenmodes domi-
+nated by the dipole resonances of the chiral resonators.
+Here, we focus on the bands labeled as B4, B6, B7, and
+B9. The eigenmodes of these bands are circularly polar-
+ized transverse sound [42], and their pressure fields are
+shown in Fig. 1(d) for ka/π = 0.2. As seen, the fields
+represent acoustic dipoles oscillating in a direction per-
+pendicular to the axis of the resonators. The arrowed cir-
+cles denote the rotation direction of the eigen fields. The
+red and blue arrowed circles correspond to the right circu-
+larly polarized (RCP) states and left circularly polarized
+(LCP) states, respectively. The velocity field averaged
+over the unit cell also circulates in the same direction.
+The collective motion of the acoustic dipoles gives rise
+to circularly-polarized transverse sound macroscopically.
+The transverse sound of the bands B4 and B6 (B7 and
+B9) have opposite handedness and carry opposite spin
+angular momentum.
+To demonstrate the emergence of circularly polarized
+transverse sound, we consider the metamaterial with 15
+unit cells in z direction, as shown in Fig. 2(a). Peri-
+odic boundary conditions are applied in x and y direc-
+tions. To excite the system, we set four input ports at
+the four tubes on the left side of the unit in Fig. 2(a)
+with the phases 0, 0.5π, π, and 1.5π, respectively. The
+azimuthal gradient of the phase decides the handedness
+(LCP or RCP) of the excited circularly-polarized trans-
+verse sound.
+In addition, we set another four output
+ports on the right side of the metamaterial to determine
+the transmission. We consider two frequencies, 0.625 kHz
+and 0.79 kHz corresponding to the frequencies marked by
+the dashed lines in Fig. 1(c), where four eigenstates (two
+are of LCP and two are of RCP) can be excited. To visu-
+alize the transverse sounds, we average the velocity field
+over each unit cell and plot the averaged velocity vec-
+tors (denoted by the red and blue arrows) for the 15 unit
+cells in Fig. 2(b)-(e). We observe that the velocity vec-
+tors indeed rotate in the xy plane, corresponding to spin
+a
+h
+R
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+ B4
+ B6
+ B7
+ B9
+ka/π
+Frequency (kHz)
+0.790 
+0.625
+＋
+＋
+－
+－
+(a)
+(c)
+(b)
+(d)
+＋
+Z
+B4
+Z
+B6
+B7
+Z
+Z
+B9
+－
+x
+y
+z
+x
+y
+z
+FIG. 1.
+(a) Unit cell of the 3D acoustic metamaterial. (b)
+Internal structure of the resonators. The geometry parame-
+ters are R = 5 cm, h = 1cm, a = 12.1 cm. (c) Band structure
+of the metamaterial. B4, B6, B7, and B9 denote the lowest
+four bands with circularly polarized eigenstates. (d) Pressure
+fields at ka/π = 0.2 for B4, B6, B7, and B9. The blue and red
+arrows denote the circulating direction of the eigen pressure
+fields for the LCP and RCP states, respectively.
+angular momentum in the longitudinal direction (i.e., z
+direction). The arrowed yellow circles denote the tem-
+poral evolution of the velocity vectors, with the arrows
+indicating the circulation direction. At either frequency,
+the two transverse sounds carry opposite spins, allowing
+the exploration of their different absorption inside the
+metamaterial.
+III.
+ACOUSTIC CIRCULAR DICHROISM
+We now consider the metamaterial with loss to inves-
+tigate the CD effect of the transverse sound. As shown
+in Fig.
+3(a), we employ a three-layer sandwich struc-
+ture: the bottom and top layers are lossless, while the
+middle layer (highlighted in blue) contains loss. The loss
+is introduced into the unit cells by adding an imaginary
+part to the sound speed v(1 + iα) with α characterizing
+the loss strength. We apply ports to excite the system
+(same as in Fig. 2) from the bottom of the lattice. The
+excited transverse sound propagates through the middle
+lossy layer and is measured in the top layer to determine
+its transmission. This directly maps to the usual config-
+uration of optical CD, where an optical structure is sand-
+
+3
+x
+y
+z
+(b)
+(a)
+(d)
+(c)
+(e)
+FIG. 2. (a) The metamaterial with 15 units in z direction and
+periodic in x and y directions. The transverse sound is excited
+by ports on the left end of the metamaterial. (b-e) Averaged
+velocity vectors in the metamaterial at the frequency f =
+0.625 kHz (b,c) and 0.79 kHz (d,e). The yellow circles with
+the arrow show the circulating direction of the velocity field
+on the xy−plane.
+wiched by air and reflection/transmission is measured in
+air. We apply COMSOL to simulate the system and cal-
+culate the reflection (R±) and transmission (T±) of the
+incident sound. The absorptions can then be determined
+as:
+A± = 1 − R± − T±,
+(1)
+∆A = |A+ − A−| ,
+(2)
+where “+” (“−”) denotes the RCP (LCP) state and ∆A
+is the differential absorption of the RCP and LCP sounds.
+We first consider the case with loss uniformly added to
+all the chiral resonators in the middle blue-colored layer.
+Figure 3(b) shows the numerical results of the reflections
+(solid lines) and transmissions (dashed lines). The inset
+shows the unit cell with the blue-colored regions contain-
+ing loss α = 0.006 , which has C4 rotational symmetry.
+As seen, the reflection R+ is nearly identical to R−, and
+there is a tiny difference between the transmissions T+
+and T−.
+The absorptions calculated using Eq.(1) are
+shown in Fig. 3(c) as the solid blue and red lines. We
+notice that the absorptions of LCP and RCP sounds are
+almost equal. As a result, the CD effect is negligible in
+this case with homogeneous loss. Figure 3(d) shows the
+absorption of the LCP and RCP sound as a function of
+the loss strength α at f = 0.65 kHz. We notice that the
+absorption A+ and A− are almost identical. The differ-
+ential absorption ∆A is less than 0.01 with the maximum
+appears at α = 0.006, as shown by the orange dashed line
+which has been multiplied by a factor of 10.
+The optical CD effect strongly depends on the rota-
+tional symmetry of the structures [43, 44]. To explore
+this symmetry dependence for acoustic CD, we break the
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+0.5
+1.0
+Frequency (kHz)
+ R+
+ R-
+ T+
+ T-
+  Reflection / 
+Transmission
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+0.4
+0.8
+ 
+ A+
+ A-
+Frequency (kHz)
+Absorption 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+Absorption 
+α
+ A+
+ A-
+ 10 |A+ - A-|
+(a)
+(b)
+(c)
+(d)
+x
+y
+z
+  Reflection / 
+Transmission
+FIG. 3.
+(a) The 3-layer metamaterial with loss uniformly
+added to the middle layer (colored in yellow). (b) Reflection,
+transmission, and (c) absorption as a function of frequency
+for the RCP (+) and LCP (-) sounds with α = 0.006. The
+inset in (b) shows the regions containing loss (blue colored),
+which satisfies the C4 symmetry. (d) Absorption of the LCP
+and RCP sounds as a function of the loss α at 0.65 kHz. In
+(c) and (d), the absorption difference (∆A) is multiplied by
+ten for easy visualization.
+C4 symmetry of the metamaterial by selectively adding
+loss to the unit cells. As shown in the inset of Fig. 4(a),
+we only add loss to the side resonator (i.e., two oppos-
+ing half resonators highlighted in blue) with the center
+axis in x-direction, reducing the symmetry of the meta-
+material from C4 to C2. We numerically calculated the
+transmission and reflection of this C2 system, and the re-
+sults are shown in Fig. 4(a) for loss α = 0.1. We observe
+large differences in the transmissions and reflections of
+the LCP and RCP sound in the frequency range [0.600
+kHz, 0.825 kHz], corresponding to the considered bands
+in Fig. 1(c). The absorptions calculated with Eq. (1) are
+shown in Fig. 4(b). As noticed, there is a significant dif-
+ference between the absorption of RCP sound (A+) and
+the absorption of RCP sound (A−). The differential ab-
+sorption ∆A (denoted by the orange dashed line) is much
+larger than the case of Fig. 3 and has two local maxima
+of about 0.5 appearing at 0.65 kHz and 0.80 kHz. This
+demonstrates the strong CD phenomena in the C2 meta-
+material. We also investigate the dependence of the CD
+on the loss magnitude α, and the results are shown in
+
+000004
+Fig. 4(c) for f = 0.65 kHz. The trends of A+ and A−
+are similar to those of the C4 system, but the absorp-
+tion difference ∆A (denoted by the dashed orange line)
+is much larger with a maximum value of 0.46 at α = 0.3
+(marked by the dashed line).
+The CD characterizes the different absorption of LCP
+and RCP sounds at the same frequency. We note that
+the LCP and RCP sounds have different wavelengths in-
+side the chiral metamaterial at the same frequency due
+to their different dispersions.
+It is thus interesting to
+compare their absorption for the same wavelength (or
+wavenumber) inside the metamaterial. Figure 4(d) shows
+the absorption of the transverse sound corresponding to
+the four bands B4, B6, B7, and B9 in Fig. 1(c). We
+only consider the range of 0 ≤ ka/π ≤ 0.2 where the ef-
+fective wavelength is well defined, and the excited state
+is RCP for band B4 and LCP for band B7 due to their
+negative group velocity in this range. As seen, the RCP
+sounds (corresponding to the solid and dashed red lines)
+generally have larger absorption compared with the LCP
+sounds (corresponding to the solid and dashed blue lines).
+This indicates that a strong CD effect also happens to
+the circularly polarized transverse sounds with the same
+wavelength (but not necessarily the same frequency).
+To intuitively understand the different absorption of
+LCP and RCP sounds, we then study the averaged ve-
+locity fields (averaged over one unit cell) in the 3-layer
+metamaterial with C2 symmetry.
+Figure 5(a) shows a
+side view of the metamaterial, where loss is added to the
+middle layers consisting of 5 unit cells (the blue color
+marks the resonators containing loss). The red and blue
+helical curves in the bottom layer denote the temporal
+trajectories of the velocity vectors of the incident RCP
+and LCP sounds, respectively. The helical curves in the
+upper layer denote the temporal trajectories of the ve-
+locity vectors of the transmitted sounds, which are in
+general elliptically polarized due to the coupling between
+the LCP and RCP sounds in the C2 absorptive layer. Fig-
+ure 5(b) and (c) show the numerical results of the trans-
+mitted velocity fields under the incidence of the RCP
+and LCP sounds, respectively, for f = 0.635 kHz and
+α = 0.1.
+The larger arrowed circles denote the time-
+evolution trajectories of the incident velocity fields, while
+the smaller ellipses denote the time-evolution trajecto-
+ries of the transmitted velocity fields. Figure 5(d) shows
+a comparison between the transmitted velocity fields un-
+der the incidence of LCP and RCP sounds (corresponding
+to a zoom-in of the results in Fig. 5(b) and (c)), which
+are different in both amplitude and ellipticity. Similar
+property also exists in the reflected fields.
+IV.
+COMPLEX BAND STRUCTURE AND
+EXCEPTIONAL POINTS
+We investigate the complex band structures of the sys-
+tems to uncover the origins of the acoustic CD and the
+different properties of the C2 and C4 systems. Figure 6(a)
+(a)
+(b)
+(c)
+(d)
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+0.5
+1.0
+Frequency (kHz)
+ R+
+ R-
+ T+
+ T-
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+0.5
+1.0
+ A+
+ A-
+ |A+ - A-|
+Frequency (kHz)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.5
+1.0
+α = 0.3
+α
++
+ A+
+ A-
+ |A  - A-|
+Absorption 
+0.00
+0.05
+0.10
+0.15
+0.20
+0.6
+0.8
+1.0
+0.4
+ A+-B4    
+ A+-B6    
+ A--B7    
+ A--B9
+Absorption
+ka/π
+  Reflection / 
+Transmission
+Absorption 
+FIG. 4. (a) Reflection, transmission, and (b) absorption of the
+RCP (+) and LCP (-) sounds for α = 0.1. The inset in (a)
+shows the regions with loss (blue colored), which satisfies the
+C2 symmetry. (c) Absorption of the LCP and RCP sounds as
+a function of the loss parameter α at 0.65 kHz. (d) Absorption
+of the LCP and RCP sounds as a function of the normalized
+wavenumber ka/π for the bands B4, B6, B7, and B9.
+and (b) show the real and imaginary parts of the complex
+band structure for the C4 system with loss α = 0.006
+(corresponding to the case of Fig. 3). The imaginary
+parts take positive values due to the time convention eiωt
+adopted in COMSOL. The insets (labeled as A, B, C,
+and D) on the right side show the zoom-ins of the bands
+enclosed by the dashed rectangles.
+The insets A and
+B depict the bands near the Brillouin zone centers and
+
+5
+RCP
+LCP
+(a)
+(b)
+(c)
+(d)
+𝑡
+𝑡
+𝑡
+𝑡
+x
+z
+x
+y
+Absorption layer
+FIG. 5. (a) Schematics of the CD effect in the acoustic meta-
+material. The red and blue helical curves denote the tempo-
+ral trajectories of the velocity vectors for the RCP and LCP
+sounds on the incident side (bottom) and the transmission
+side (top). (b, c) Larger (smaller) circles denote the evolu-
+tion trajectories of velocity field for the incident (transmitted)
+RCP and LCP sounds. The transmitted sounds are ellipti-
+cally polarized. (d) A zoom-in comparison of the transmitted
+velocity field’s trajectories under the incidence of RCP (red)
+and LCP (blue) sound. We set the frequency f = 0.635 kHz
+and loss α = 0.1.
+boundaries, respectively, for B4 and B6.
+Likewise, in-
+sets C and D show the bands closed to the Brillouin zone
+centers and boundaries, respectively, for B7 and B9. We
+notice that the band degeneracies are not affected by the
+loss due to the protection of the C4 symmetry.
+Figure 7 shows the complex band structure for the C2
+system with the same loss α = 0.006 for comparison with
+the C4 system. Interestingly, at the Brillouin zone cen-
+ter and boundaries, the real parts of the bands remain
+degenerate in a finite range of k values while the imagi-
+nary parts bifurcate in the same range, as shown in the
+insets on the right side.
+This indicates the emergence
+of non-Hermitian exceptional points [45–47]. Obviously,
+these EPs derive from the diabolic points of the origi-
+nal lossless system in Fig.
+1 (c).
+While the phenom-
+ena here is similar to the EPs spawn from Dirac points
+in two-dimensional photonic crystals [45], the underlying
+physical mechanism is different. The emergence of these
+EPs is attributed to the coupling and loss difference of
+the LCP and RCP transverse dipole modes induced by
+the breaking of C4 symmetry. We will elaborate on this
+point with an analytical model later.
+Figure 8(a) and (b) show the complex band structure
+for the C2 system with a larger loss α = 0.1, correspond-
+ing to the case of Fig. 4(a) and (b) with a much stronger
+CD effect. We notice that the EP features remain at the
+center and boundaries of the Brillouin zone. At the same
+time, small partial gaps appear at the frequencies of the
+EPs, as marked by the blue and red ribbons in the insets
+(a)
+(b)
+Numerical Data
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+0.95
+1.00
+1.05
+0.75
+0.76
+0.77
+-0.05 0.00
+0.05
+0.80
+0.81
+0.95
+1.00
+1.05
+0.64
+0.65
+-0.05 0.00
+0.05
+0.616
+0.624
+B
+A
+D
+B
+D
+ka/π
+Re (f) (kHz)
+ B4
+ B6
+ B7
+ B9
+Re (f) (kHz)
+ka/π
+C
+A
+B
+C
+D
+Fitting
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0
+1
+2
+3
+4
+5
+0.95
+1.00
+1.05
+4.5
+4.6
+-0.05 0.00
+0.05
+4.80
+4.85
+0.95
+1.00
+1.05
+3.9
+-0.05 0.00 0.05
+3.68
+3.72
+D
+D
+C
+B
+ka/π
+Im (f) (Hz)
+ B4
+ B6
+ B7
+ B9
+Im (f) (Hz)
+ka/π
+A
+B
+Numerical Data
+Fitting
+A
+B
+C
+D
+FIG. 6. The real (a) and imaginary (b) parts of the complex
+band structure for the C4 system at α = 0.006. Insets on the
+right side show the zoom-ins of the bands near the zone center
+and boundaries, corresponding to the dashed rectangles in (a)
+and (b). The solid lines in the insets denote the analytical
+fitting results.
+of Fig. 8(a). At the frequencies of the blue-ribbon (red-
+ribbon) region, only LCP (RCP) sound can propagate
+through the metamaterial [16, 47]. Thus, at the frequen-
+cies f = 0.62 kHz and f = 0.76 kHz, the RCP sound
+cannot propagate through the metamaterial. Similarly,
+at the frequencies of f = 0.65 kHz and f = 0.80 kHz, the
+LCP sound cannot propagate through the metamaterial.
+However, this does not necessarily indicate a large differ-
+ence in the reflection of LCP and RCP sounds at these
+frequencies due to the non-Hermitian nature of the meta-
+material. Whether strong reflection will appear at the
+partial polarization gaps depends on the damping of the
+corresponding eigenmodes. In the following, we will show
+that the eigenmodes’ damping property strongly affects
+the reflection and the CD effect.
+
+6
+(a)
+(b)
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+0.99
+1.00
+1.01
+0.758
+0.760
+-0.04
+0.00
+0.04
+0.800
+0.805
+0.96
+1.00
+1.04
+0.64
+0.65
+-0.04
+0.00
+0.04
+0.615
+0.620
+C
+D
+B
+D
+D
+B
+C
+A
+ka/π
+Re (f) (kHz)
+ka/π
+ B4
+ B6
+ B7
+ B9
+Re (f) (kHz)
+B
+A
+Numerical Data
+Fitting
+-1
+0
+1
+0
+1
+2
+0.99
+1.00
+1.01
+1.0
+1.5
+-0.04
+0.00
+0.04
+0
+1
+2
+0.96
+1.00
+1.04
+0.0
+1.5
+-0.04
+0.00
+0.04
+0.8
+1.6
+D
+B
+B
+C
+D
+B
+ka/π
+Im (f) (Hz)
+ B4 
+ B6
+ B7 
+ B9
+Im (f) (Hz)
+ka/π
+A
+C
+D
+A
+Numerical Data
+Fitting
+FIG. 7. The real (a) and imaginary (b) parts of complex band
+structure for the C2 system at α = 0.006. The insets on the
+right side show the zoom-ins of the bands near the zone center
+and boundaries, corresponding to the dashed rectangles in (a)
+and shaded rectangles in (b).
+The solid lines in the insets
+denote the analytical fitting results.
+To understand the damping of the eigenmodes, we in-
+vestigate the modes’ quality factor Q corresponding to
+the bands B4, B6, B7, and B9, for both the C4 and C2
+systems. The quality factor is calculated as Q =
+Re(f)
+2Im(f)
+[48]. The results are shown in Fig. 9(a) and (b) as a
+function of the real part of the eigenfrequency. We note
+that the eigenmode of each band can be either LCP or
+RCP, depending on the sign of its group velocity with
+respect to the phase velocity. Consequently, the quality
+factor Q of each band can be divided into two parts for
+the LCP (“−”) and RCP (“+”) states, respectively. As
+shown in Fig. 9(a), all eigenmodes of the C4 system have
+approximately the same quality factor. This explains the
+negligible CD effect in the C4 system with homogenous
+loss. In contrast, the quality factors of the LCP and RCP
+modes in the C2 system have a large difference, particu-
+(b)
+(a)
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+0.9
+1.0
+1.1
+0.75
+0.78
+-0.2
+0.0
+0.2
+0.78
+0.80
+0.8
+1.0
+1.2
+0.64
+0.68
+-0.2
+0.0
+0.2
+0.62
+0.63
+ka/π
+ka/π
+Re (f) (kHz)
+Re (f) (kHz)
+C
+D
+D
+B
+A
+ B4
+ B6
+ B7
+ B9
+B
+A
+B
+C
+D
+Numerical Data
+Fitting
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0
+10
+20
+30
+40
+0.9
+1.0
+1.1
+16
+24
+-0.2
+0.0
+0.2
+0
+25
+0.8
+1.0
+1.2
+0
+20
+-0.2
+0.0
+0.2
+10
+20
+ka/π
+ka/π
+B
+A
+D
+D
+B
+Im (f) (Hz)
+ B4
+ B6
+ B7
+ B9
+Im (f) (Hz)
+C
+A
+B
+C
+D
+Numerical Data
+Fitting
+FIG. 8. The real (a) and imaginary (b) parts of complex band
+structures for the C2 system at α = 0.1. The right insets show
+the zoom-ins of the bands near the Brillouin zone centers and
+boundaries, corresponding to the dashed rectangles in (a) and
+shaded rectangles in (b). The ribbons in the insets denote
+the partial bandgaps. The solid lines in the insets denote the
+analytical fitting results.
+larly in the vicinity of the polarization bandgaps marked
+by the red ribbons. The large difference in quality fac-
+tor indicates a large difference in the damping of LCP
+and RCP modes and thus explains the strong CD effect
+near the polarization bandgaps, in agreement with the
+numerical results in Fig. 4(b). For the LCP and RCP
+modes near the blue-ribbon band gaps, their quality fac-
+tors are much smaller than the modes near the red-ribbon
+bands, and the difference of their quality factors are also
+much smaller. Therefore, both LCP and RCP sounds at
+the frequencies of the blue-ribbon region are strongly ab-
+sorbed, and their reflections are small, leading to a weak
+CD effect, as confirmed by the numerical results in Fig.
+4(b).
+
+7
+(a)
+(b)
+0.6
+0.7
+0.8
+80
+82
+84
+Quality factor
+Re (f) (kHz)
+ Q4(+)  
+ Q4(-)  
+ Q6(+)  
+ Q6(-)
+ Q7(-)  
+ Q7(+)  
+ Q9(-)  
+ Q9(+)
+A
+0.6
+0.7
+0.8
+10
+100
+1000
+C
+D
+B
+0.803 kHz
+ Q4(+)
+ Q4(-)
+ Q6(+)
+ Q6(-)
+ Q7(-)
+ Q7(+)
+ Q9(-)
+ Q9(+)
+Quality factor
+Re (f) (kHz)
+0.645 kHz
+FIG. 9. The quality factor Q of the eigenmodes in the (a)
+C4 and (b) C2 systems, corresponding the cases of Fig.
+6
+and Fig. 8, respectively. The blue and red ribbons denote
+polarization bandgaps.
+To understand the emergence of the EPs in the C2
+system, we exploit an effective Hamiltonian to describe
+the coupling of the LCP and RCP modes near the Bril-
+louin zone center [45, 46, 49, 50]. As for the C4 system
+with homogenous loss, the effective Hamiltonian can be
+expressed as:
+HC4 =
+�
+ω0 − iγ
+(vR + ivI)k
+(vR + ivI)k
+ω0 − iγ
+�
+(3)
+which has the complex eigenvalues:
+ω = ω0 − iγ ± k(vR + ivI).
+(4)
+Here, ω0 is the eigenfrequency at k = 0, where the LCP
+and RCP modes are degenerate; vR and vI are the real
+and the imaginary parts of the complex group velocities,
+respectively; γ denotes the loss.
+In the C2 system, loss is selectively added to only one
+resonator in each unit cell. The breaking of C4 symmetry
+opens a gap at k = 0, which can be characterized by a
+perturbation term δ/2 in the Hamiltonian. The LCP and
+RCP modes at k = 0 now have different loss γ1 and γ2
+(γ1 ̸= γ2):
+HC2 =
+�
+ω0 − iγ1 + δ
+2
+(vR + ivI)k
+(vR + ivI)k
+ω0 − iγ2 − δ
+2
+�
+,
+(5)
+which has the complex eigenvalues:
+ω =ω0 − i(γ1 + γ2)
+2
+± 1
+2
+�
+[δ − i(γ1 − γ2)]2 − 4k2(vI − ivR)2.
+(6)
+These analytical expressions of the complex eigenval-
+ues in Eq.(4) and (6) are employed to fit the numeri-
+cal results for both the real and the imaginary parts.
+The fitting results are shown as the solid lines in the
+insets of Figs. 6-8, accordingly. We notice good quanti-
+tative agreements between the analytical and numerical
+results. In particular, the effective Hamiltonian correctly
+captures the EP features in the C2 systems. The fitting
+parameters for both C4 and C2 systems with different
+losses are summarized in Table I. We note that the mode
+damping parameters γ1,2 take negative values due to the
+time convention eiωt adopted in COMSOL.
+TABLE I. Fitting parameters for C4 and C2 systems
+System
+α
+ω0
+γ
+vR
+vI
+Inset
+C4
+0.006
+617.61
+-3.71
+2.96 0.018
+A (Fig. 6)
+803.5
+-4.82
+4.97
+0.03
+C (Fig. 6)
+C2
+α
+ω0
+γ1
+γ2
+vR
+vI
+δ
+Inset
+0.006
+617.12 -1.53
+-0.33 2.96 0.0011 0.036 A (Fig. 7)
+803.66 -0.01
+-2.30 4.99 -0.013 0.0075 C (Fig. 7)
+0.1
+622.12 -24.87 -5.13 2.98
+0.03
+4.77
+A (Fig. 8)
+800.40 -0.22 -39.12 4.98 -0.21
+4.22
+C (Fig. 8)
+The above effective Hamiltonians well explain the
+emergence of the EPs and the enhancement of CD by
+the EPs. In the C4 system, the LCP mode of the B4
+band and the RCP mode of the B6 band are orthogonal
+at k = 0 with vanished coupling. The damping of the
+LCP and RCP modes at the same excitation frequency
+are approximately equal due to homogeneous loss added
+to all resonators of the unit cell.
+Thus, their quality
+factors are almost equal (corresponding to the results in
+Fig. 9(a)). In the C2 system, the inhomogeneous loss
+breaks the C4 rotational symmetry and induces coupling
+between the original LCP and RCP modes at k = 0,
+which gives rise to the polarization bandgaps. In addi-
+tion, the two modes have different dampings due to the
+inhomogeneous material loss. These together give rise to
+the EPs and the bifurcation of the imaginary parts of the
+eigenfrequencies, leading to enlarged damping contrast
+of the LCP and RCP modes at the same excitation fre-
+quency and thus larger difference in their quality factors
+(corresponding to the results in Fig. 9(b)). Therefore,
+the strong CD effect in the C2 system is attributed to
+both the polarization bandgaps and the EPs.
+V.
+CONCLUSION
+In conclusion, we demonstrate the acoustic CD effect
+in a 3D chiral metamaterial supporting circularly polar-
+
+8
+ized transverse sound. We have investigated the effect
+in two types of systems with C4 and C2 rotational sym-
+metry, respectively. In the C4 system with loss homo-
+geneously added to all resonators of the unit cell, we
+observe a negligible acoustic CD effect.
+On the other
+hand, by selectively adding loss to part of the unit cell,
+reducing the system’s rotational symmetry from C4 to
+C2, the CD effect is enhanced strongly. With analysis of
+their complex band structures and quality factors of the
+eigenmodes, we uncover that the strong acoustic CD in
+the C2 system is attributed to polarization bandgaps and
+the emergence of non-Hermitian EPs. The polarization
+bandgaps induce selective transmission and absorption
+of the circularly polarized transverse sound with a par-
+ticular handedness. The EPs give rise to bifurcations of
+the imaginary parts of the eigen frequencies. These to-
+gether enhance the CD effect in the C2 system. It will be
+interesting to experimentally demonstrate the discussed
+phenomena. The metamaterial structures can be fabri-
+cated using 3D printing. Loss can be introduced into the
+structures by adding sponges. The transverse sound can
+be excited by using an array of speakers, and the reflec-
+tion/refraction can be measured with a microphone. The
+acoustic CD effect can find applications in sound manip-
+ulations based on its vector degrees of freedom and in
+acoustic sensing of chiral structures.
+The results con-
+tribute to the understanding of chiral sound-matter in-
+teractions in metamaterials and phononic crystals.
+ACKNOWLEDGMENTS
+The work described in this paper was supported by
+grants from the Research Grants Council of the Hong
+Kong Special Administrative Region, China (No. CityU
+21302018 and No. C6013-18G).
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diff --git a/edE0T4oBgHgl3EQfogFH/content/tmp_files/load_file.txt b/edE0T4oBgHgl3EQfogFH/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a98c92ee4cf6c22f9e9bde5a48b4ec943e6617f0
--- /dev/null
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@@ -0,0 +1,1085 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf,len=1084
+page_content='Acoustic circular dichroism in a three-dimensional chiral metamaterial  Qing Tong,1 Jensen Li,2 and Shubo Wang1, ∗  1Department of Physics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China  2Department of Physics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China  (Dated: January 9, 2023)  Circular dichroism (CD) is an intriguing chiroptical phenomenon associated with the interaction  of chiral structures with circularly polarized lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Although the CD effect has been extensively  studied in optics, it has not yet been demonstrated in acoustic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Here, we demonstrate  the acoustic CD effect in a three-dimensional chiral metamaterial supporting circularly polarized  transverse sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  We find that the effect is negligible in the lossy metamaterial possessing C4  rotational symmetry but can be strongly enhanced in the C2-symmetric system with inhomogeneous  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The phenomena can be understood based on the properties of the metamaterial’s complex  band structure and the quality factors of its eigenmodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We show that the enhanced CD in the  C2-symmetric system is attributed to the polarization bandgaps and the non-Hermitian exceptional  points appearing near the Brillouin-zone center and boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The results contribute to the  understanding of chiral sound-matter interactions and can find applications in acoustic sensing of  chiral structures and sound manipulations based on its vector properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  INTRODUCTION  Chiral structures have novel properties deriving from  the mirror-symmetry breaking [1, 2] and are extensively  employed to realize polarization conversion [3, 4], un-  usual optical forces [5–7], and synthetic gauge fields[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The interaction between chiral structures and chiral light,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=', light carrying spin and/or orbital angular momentum  (OAM), can give rise to circular dichroism (CD) [2, 9, 10]  and helical (or vortical) dichroism [11, 12], correspond-  ing to the differential absorption of lights with opposite  chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The CD effect has been investigated in vari-  ous optical structures, ranging from bilayer chiral struc-  tures [13, 14], nonplanar three-dimensional (3D) chiral  structures [15], to gyroid structures [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Recent re-  search has uncovered the subtle relations between CD  and the Ohmic dissipation of meta-atoms [18] as well as  the bound states in the continuum [19], enabling a pro-  found understanding of chiral light-matter interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The CD effect has been widely applied to analyze molecu-  lar structures [20, 21] and to achieve chiral discrimination  [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  It is well known that sound can carry OAM in the  form of vortices [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The acoustic OAM can induce  chiral sound-matter interactions and give rise to intrigu-  ing phenomena such as acoustic geometric phases [27]  and the acoustic orbital Hall effect [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The chi-  ral sound-matter interactions can enable rich manipula-  tions of sound vortices, including asymmetric transmis-  sion/reflection [30], reversal of orbital angular momen-  tum [31], and acoustic helical dichroism [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  In ad-  dition, the chiral sound-matter interactions can be ap-  plied to manipulate matter, leading to acoustic levitation  [33, 34], acoustic tweezers [35–37], and acoustic torque  [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' While airborne sound is a longitudinal wave,  ∗ shubwang@cityu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='hk  it was shown that inhomogeneous sound fields can carry  nonzero acoustic spin density characterized by rotating  velocity vector fields [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Remarkably, spin-1 trans-  verse sound can emerge in a micropolar metamaterial  supporting synthetic shear forces in air [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In contrast  to the conventional longitudinal sound, the transverse  sound carries full vector properties similar to electro-  magnetic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  In particular, it can carry both spin  and OAM with intriguing spin-orbit interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Explo-  ration of this new type of sound and its counter-intuitive  properties can generate new functionalities for acoustic  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Here, for the first time, we demonstrate the acoustic  CD effect in a 3D chiral metamaterial that supports cir-  cularly polarized transverse sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Using full-wave nu-  merical simulations, we calculate the absorption of left-  handed circularly polarized (LCP) and right-handed cir-  cularly polarized (RCP) sound in the lossy metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  We find that the CD effect strongly depends on the ro-  tational symmetry of the metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The CD effect  is negligible in the metamaterial with homogeneous loss  satisfying the C4 rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  In contrast, it  is strongly enhanced in the metamaterial with inhomo-  geneous loss satisfying the C2 rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' By  studying the complex band structure of the metamate-  rial and the quality factors of its eigenmodes, we find that  these properties originate from the polarization bandgaps  and the non-Hermitian exceptional points (EPs) of the  metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  We organize the article as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In Section II, we  introduce the acoustic chiral metamaterial and discuss  its eigenmode properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In Section III, we show the nu-  merical results for the absorption of the LCP and RCP  sound in two types of lossy metamaterial obeying the  C4 and C2 rotational symmetry, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Section IV  presents the complex band structures of the lossy meta-  materials, where we discuss the polarization bandgaps  and the EPs to understand the CD effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We draw the  conclusion in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='02526v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='app-ph]  6 Jan 2023  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  THE CHIRAL METAMATERIAL  We consider a 3D metamaterial with the cubical unit  cell shown in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The unit cell comprises three  chiral resonators mutually connected by tubes, and it  obeys the C4 rotational symmetry with respect to the  x, y, and z axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Figure 1(b) shows a half of the chi-  ral resonator, where the radius is R = 5 cm, and the  height is h = 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The orange blades are connected  to the center post, and the gray blades are connected to  the outer shell of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We assume that air is  filled inside the resonator and all air-material interfaces  are hard boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In the considered range of frequen-  cies, each resonator support subwavelength resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  These resonances endow the metamaterial with intrigu-  ing macroscopic acoustic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  We first calculate the band structure of the metama-  terial by using a finite-element package COMSOL Mul-  tiphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The result is shown in Figure 1(c) for the  wavevector in z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The 4th to 9th bands (count-  ing from bottom) correspond to the eigenmodes domi-  nated by the dipole resonances of the chiral resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Here, we focus on the bands labeled as B4, B6, B7, and  B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The eigenmodes of these bands are circularly polar-  ized transverse sound [42], and their pressure fields are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 1(d) for ka/π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' As seen, the fields  represent acoustic dipoles oscillating in a direction per-  pendicular to the axis of the resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The arrowed cir-  cles denote the rotation direction of the eigen fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The  red and blue arrowed circles correspond to the right circu-  larly polarized (RCP) states and left circularly polarized  (LCP) states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The velocity field averaged  over the unit cell also circulates in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The collective motion of the acoustic dipoles gives rise  to circularly-polarized transverse sound macroscopically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The transverse sound of the bands B4 and B6 (B7 and  B9) have opposite handedness and carry opposite spin  angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  To demonstrate the emergence of circularly polarized  transverse sound, we consider the metamaterial with 15  unit cells in z direction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Peri-  odic boundary conditions are applied in x and y direc-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' To excite the system, we set four input ports at  the four tubes on the left side of the unit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 2(a)  with the phases 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5π, π, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The  azimuthal gradient of the phase decides the handedness  (LCP or RCP) of the excited circularly-polarized trans-  verse sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  In addition, we set another four output  ports on the right side of the metamaterial to determine  the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We consider two frequencies, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='625 kHz  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='79 kHz corresponding to the frequencies marked by  the dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 1(c), where four eigenstates (two  are of LCP and two are of RCP) can be excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' To visu-  alize the transverse sounds, we average the velocity field  over each unit cell and plot the averaged velocity vec-  tors (denoted by the red and blue arrows) for the 15 unit  cells in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 2(b)-(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We observe that the velocity vec-  tors indeed rotate in the xy plane, corresponding to spin  a  h  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  B4  B6  B7  B9  ka/π  Frequency (kHz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='790  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='625  ＋  ＋  －  －  (a)  (c)  (b)  (d)  ＋  Z  B4  Z  B6  B7  Z  Z  B9  －  x  y  z  x  y  z  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  (a) Unit cell of the 3D acoustic metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (b)  Internal structure of the resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The geometry parame-  ters are R = 5 cm, h = 1cm, a = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (c) Band structure  of the metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' B4, B6, B7, and B9 denote the lowest  four bands with circularly polarized eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (d) Pressure  fields at ka/π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='2 for B4, B6, B7, and B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The blue and red  arrows denote the circulating direction of the eigen pressure  fields for the LCP and RCP states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  angular momentum in the longitudinal direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=', z  direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The arrowed yellow circles denote the tem-  poral evolution of the velocity vectors, with the arrows  indicating the circulation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' At either frequency,  the two transverse sounds carry opposite spins, allowing  the exploration of their different absorption inside the  metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  ACOUSTIC CIRCULAR DICHROISM  We now consider the metamaterial with loss to inves-  tigate the CD effect of the transverse sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  3(a), we employ a three-layer sandwich struc-  ture: the bottom and top layers are lossless, while the  middle layer (highlighted in blue) contains loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The loss  is introduced into the unit cells by adding an imaginary  part to the sound speed v(1 + iα) with α characterizing  the loss strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We apply ports to excite the system  (same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 2) from the bottom of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The  excited transverse sound propagates through the middle  lossy layer and is measured in the top layer to determine  its transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' This directly maps to the usual config-  uration of optical CD, where an optical structure is sand-  3  x  y  z  (b)  (a)  (d)  (c)  (e)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (a) The metamaterial with 15 units in z direction and  periodic in x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The transverse sound is excited  by ports on the left end of the metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (b-e) Averaged  velocity vectors in the metamaterial at the frequency f =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='625 kHz (b,c) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='79 kHz (d,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The yellow circles with  the arrow show the circulating direction of the velocity field  on the xy−plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  wiched by air and reflection/transmission is measured in  air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We apply COMSOL to simulate the system and cal-  culate the reflection (R±) and transmission (T±) of the  incident sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The absorptions can then be determined  as:  A± = 1 − R± − T±,  (1)  ∆A = |A+ − A−| ,  (2)  where “+” (“−”) denotes the RCP (LCP) state and ∆A  is the differential absorption of the RCP and LCP sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  We first consider the case with loss uniformly added to  all the chiral resonators in the middle blue-colored layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Figure 3(b) shows the numerical results of the reflections  (solid lines) and transmissions (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The inset  shows the unit cell with the blue-colored regions contain-  ing loss α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='006 , which has C4 rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  As seen, the reflection R+ is nearly identical to R−, and  there is a tiny difference between the transmissions T+  and T−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The absorptions calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (1) are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 3(c) as the solid blue and red lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We  notice that the absorptions of LCP and RCP sounds are  almost equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' As a result, the CD effect is negligible in  this case with homogeneous loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Figure 3(d) shows the  absorption of the LCP and RCP sound as a function of  the loss strength α at f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='65 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We notice that the  absorption A+ and A− are almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The differ-  ential absorption ∆A is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='01 with the maximum  appears at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='006, as shown by the orange dashed line  which has been multiplied by a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The optical CD effect strongly depends on the rota-  tional symmetry of the structures [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' To explore  this symmetry dependence for acoustic CD, we break the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  Frequency (kHz)  R+  R-  T+  T-  Reflection /  Transmission  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  A+  A-  Frequency (kHz)  Absorption  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  Absorption  α  A+  A-  10 |A+ - A-|  (a)  (b)  (c)  (d)  x  y  z  Reflection /  Transmission  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  (a) The 3-layer metamaterial with loss uniformly  added to the middle layer (colored in yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (b) Reflection,  transmission, and (c) absorption as a function of frequency  for the RCP (+) and LCP (-) sounds with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The  inset in (b) shows the regions containing loss (blue colored),  which satisfies the C4 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (d) Absorption of the LCP  and RCP sounds as a function of the loss α at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='65 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In  (c) and (d), the absorption difference (∆A) is multiplied by  ten for easy visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  C4 symmetry of the metamaterial by selectively adding  loss to the unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' As shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 4(a),  we only add loss to the side resonator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=', two oppos-  ing half resonators highlighted in blue) with the center  axis in x-direction, reducing the symmetry of the meta-  material from C4 to C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We numerically calculated the  transmission and reflection of this C2 system, and the re-  sults are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 4(a) for loss α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We observe  large differences in the transmissions and reflections of  the LCP and RCP sound in the frequency range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='600  kHz, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='825 kHz], corresponding to the considered bands  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The absorptions calculated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (1) are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' As noticed, there is a significant dif-  ference between the absorption of RCP sound (A+) and  the absorption of RCP sound (A−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The differential ab-  sorption ∆A (denoted by the orange dashed line) is much  larger than the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 3 and has two local maxima  of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5 appearing at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='65 kHz and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='80 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' This  demonstrates the strong CD phenomena in the C2 meta-  material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We also investigate the dependence of the CD  on the loss magnitude α, and the results are shown in  000004  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 4(c) for f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='65 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The trends of A+ and A−  are similar to those of the C4 system, but the absorp-  tion difference ∆A (denoted by the dashed orange line)  is much larger with a maximum value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='46 at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='3  (marked by the dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The CD characterizes the different absorption of LCP  and RCP sounds at the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We note that  the LCP and RCP sounds have different wavelengths in-  side the chiral metamaterial at the same frequency due  to their different dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  It is thus interesting to  compare their absorption for the same wavelength (or  wavenumber) inside the metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Figure 4(d) shows  the absorption of the transverse sound corresponding to  the four bands B4, B6, B7, and B9 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We  only consider the range of 0 ≤ ka/π ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='2 where the ef-  fective wavelength is well defined, and the excited state  is RCP for band B4 and LCP for band B7 due to their  negative group velocity in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' As seen, the RCP  sounds (corresponding to the solid and dashed red lines)  generally have larger absorption compared with the LCP  sounds (corresponding to the solid and dashed blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  This indicates that a strong CD effect also happens to  the circularly polarized transverse sounds with the same  wavelength (but not necessarily the same frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  To intuitively understand the different absorption of  LCP and RCP sounds, we then study the averaged ve-  locity fields (averaged over one unit cell) in the 3-layer  metamaterial with C2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Figure 5(a) shows a  side view of the metamaterial, where loss is added to the  middle layers consisting of 5 unit cells (the blue color  marks the resonators containing loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The red and blue  helical curves in the bottom layer denote the temporal  trajectories of the velocity vectors of the incident RCP  and LCP sounds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The helical curves in the  upper layer denote the temporal trajectories of the ve-  locity vectors of the transmitted sounds, which are in  general elliptically polarized due to the coupling between  the LCP and RCP sounds in the C2 absorptive layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Fig-  ure 5(b) and (c) show the numerical results of the trans-  mitted velocity fields under the incidence of the RCP  and LCP sounds, respectively, for f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='635 kHz and  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The larger arrowed circles denote the time-  evolution trajectories of the incident velocity fields, while  the smaller ellipses denote the time-evolution trajecto-  ries of the transmitted velocity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Figure 5(d) shows  a comparison between the transmitted velocity fields un-  der the incidence of LCP and RCP sounds (corresponding  to a zoom-in of the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 5(b) and (c)), which  are different in both amplitude and ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Similar  property also exists in the reflected fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  COMPLEX BAND STRUCTURE AND  EXCEPTIONAL POINTS  We investigate the complex band structures of the sys-  tems to uncover the origins of the acoustic CD and the  different properties of the C2 and C4 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Figure 6(a)  (a)  (b)  (c)  (d)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  Frequency (kHz)  R+  R-  T+  T-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  A+  A-  |A+ - A-|  Frequency (kHz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='3  α  +  A+  A-  |A  - A-|  Absorption  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='4  A+-B4  A+-B6  A--B7  A--B9  Absorption  ka/π  Reflection /  Transmission  Absorption  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (a) Reflection, transmission, and (b) absorption of the  RCP (+) and LCP (-) sounds for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The inset in (a)  shows the regions with loss (blue colored), which satisfies the  C2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (c) Absorption of the LCP and RCP sounds as  a function of the loss parameter α at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='65 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (d) Absorption  of the LCP and RCP sounds as a function of the normalized  wavenumber ka/π for the bands B4, B6, B7, and B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  and (b) show the real and imaginary parts of the complex  band structure for the C4 system with loss α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='006  (corresponding to the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The imaginary  parts take positive values due to the time convention eiωt  adopted in COMSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The insets (labeled as A, B, C,  and D) on the right side show the zoom-ins of the bands  enclosed by the dashed rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The insets A and  B depict the bands near the Brillouin zone centers and  5  RCP  LCP  (a)  (b)  (c)  (d)  𝑡  𝑡  𝑡  𝑡  x  z  x  y  Absorption layer  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (a) Schematics of the CD effect in the acoustic meta-  material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The red and blue helical curves denote the tempo-  ral trajectories of the velocity vectors for the RCP and LCP  sounds on the incident side (bottom) and the transmission  side (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (b, c) Larger (smaller) circles denote the evolu-  tion trajectories of velocity field for the incident (transmitted)  RCP and LCP sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The transmitted sounds are ellipti-  cally polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (d) A zoom-in comparison of the transmitted  velocity field’s trajectories under the incidence of RCP (red)  and LCP (blue) sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We set the frequency f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='635 kHz  and loss α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  boundaries, respectively, for B4 and B6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Likewise, in-  sets C and D show the bands closed to the Brillouin zone  centers and boundaries, respectively, for B7 and B9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We  notice that the band degeneracies are not affected by the  loss due to the protection of the C4 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Figure 7 shows the complex band structure for the C2  system with the same loss α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='006 for comparison with  the C4 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Interestingly, at the Brillouin zone cen-  ter and boundaries, the real parts of the bands remain  degenerate in a finite range of k values while the imagi-  nary parts bifurcate in the same range, as shown in the  insets on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  This indicates the emergence  of non-Hermitian exceptional points [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Obviously,  these EPs derive from the diabolic points of the origi-  nal lossless system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  While the phenom-  ena here is similar to the EPs spawn from Dirac points  in two-dimensional photonic crystals [45], the underlying  physical mechanism is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The emergence of these  EPs is attributed to the coupling and loss difference of  the LCP and RCP transverse dipole modes induced by  the breaking of C4 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We will elaborate on this  point with an analytical model later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Figure 8(a) and (b) show the complex band structure  for the C2 system with a larger loss α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='1, correspond-  ing to the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 4(a) and (b) with a much stronger  CD effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We notice that the EP features remain at the  center and boundaries of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' At the same  time, small partial gaps appear at the frequencies of the  EPs, as marked by the blue and red ribbons in the insets  (a)  (b)  Numerical Data  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content='624  B  A  D  B  D  ka/π  Re (f) (kHz)  B4  B6  B7  B9  Re (f) (kHz)  ka/π  C  A  B  C  D  Fitting  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content='72  D  D  C  B  ka/π  Im (f) (Hz)  B4  B6  B7  B9  Im (f) (Hz)  ka/π  A  B  Numerical Data  Fitting  A  B  C  D  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The real (a) and imaginary (b) parts of the complex  band structure for the C4 system at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Insets on the  right side show the zoom-ins of the bands near the zone center  and boundaries, corresponding to the dashed rectangles in (a)  and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The solid lines in the insets denote the analytical  fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' At the frequencies of the blue-ribbon (red-  ribbon) region, only LCP (RCP) sound can propagate  through the metamaterial [16, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Thus, at the frequen-  cies f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='62 kHz and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='76 kHz, the RCP sound  cannot propagate through the metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Similarly,  at the frequencies of f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='65 kHz and f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='80 kHz, the  LCP sound cannot propagate through the metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  However, this does not necessarily indicate a large differ-  ence in the reflection of LCP and RCP sounds at these  frequencies due to the non-Hermitian nature of the meta-  material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Whether strong reflection will appear at the  partial polarization gaps depends on the damping of the  corresponding eigenmodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In the following, we will show  that the eigenmodes’ damping property strongly affects  the reflection and the CD effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content='620  C  D  B  D  D  B  C  A  ka/π  Re (f) (kHz)  ka/π  B4  B6  B7  B9  Re (f) (kHz)  B  A  Numerical Data  Fitting  1  0  1  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content='6  D  B  B  C  D  B  ka/π  Im (f) (Hz)  B4  B6  B7  B9  Im (f) (Hz)  ka/π  A  C  D  A  Numerical Data  Fitting  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The real (a) and imaginary (b) parts of complex band  structure for the C2 system at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The insets on the  right side show the zoom-ins of the bands near the zone center  and boundaries, corresponding to the dashed rectangles in (a)  and shaded rectangles in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The solid lines in the insets  denote the analytical fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  To understand the damping of the eigenmodes, we in-  vestigate the modes’ quality factor Q corresponding to  the bands B4, B6, B7, and B9, for both the C4 and C2  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The quality factor is calculated as Q =  Re(f)  2Im(f)  [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 9(a) and (b) as a  function of the real part of the eigenfrequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We note  that the eigenmode of each band can be either LCP or  RCP, depending on the sign of its group velocity with  respect to the phase velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Consequently, the quality  factor Q of each band can be divided into two parts for  the LCP (“−”) and RCP (“+”) states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' As  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 9(a), all eigenmodes of the C4 system have  approximately the same quality factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' This explains the  negligible CD effect in the C4 system with homogenous  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In contrast, the quality factors of the LCP and RCP  modes in the C2 system have a large difference, particu-  (b)  (a)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content='63  ka/π  ka/π  Re (f) (kHz)  Re (f) (kHz)  C  D  D  B  A  B4  B6  B7  B9  B  A  B  C  D  Numerical Data  Fitting  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content='0  0  10  20  30  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content='2  0  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content='2  10  20  ka/π  ka/π  B  A  D  D  B  Im (f) (Hz)  B4  B6  B7  B9  Im (f) (Hz)  C  A  B  C  D  Numerical Data  Fitting  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The real (a) and imaginary (b) parts of complex band  structures for the C2 system at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The right insets show  the zoom-ins of the bands near the Brillouin zone centers and  boundaries, corresponding to the dashed rectangles in (a) and  shaded rectangles in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The ribbons in the insets denote  the partial bandgaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The solid lines in the insets denote the  analytical fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  larly in the vicinity of the polarization bandgaps marked  by the red ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The large difference in quality fac-  tor indicates a large difference in the damping of LCP  and RCP modes and thus explains the strong CD effect  near the polarization bandgaps, in agreement with the  numerical results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' For the LCP and RCP  modes near the blue-ribbon band gaps, their quality fac-  tors are much smaller than the modes near the red-ribbon  bands, and the difference of their quality factors are also  much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Therefore, both LCP and RCP sounds at  the frequencies of the blue-ribbon region are strongly ab-  sorbed, and their reflections are small, leading to a weak  CD effect, as confirmed by the numerical results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  7  (a)  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  80  82  84  Quality factor  Re (f) (kHz)  Q4(+)  Q4(-)  Q6(+)  Q6(-)  Q7(-)  Q7(+)  Q9(-)  Q9(+)  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='8  10  100  1000  C  D  B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='803 kHz  Q4(+)  Q4(-)  Q6(+)  Q6(-)  Q7(-)  Q7(+)  Q9(-)  Q9(+)  Quality factor  Re (f) (kHz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='645 kHz  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The quality factor Q of the eigenmodes in the (a)  C4 and (b) C2 systems, corresponding the cases of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  6  and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The blue and red ribbons denote  polarization bandgaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  To understand the emergence of the EPs in the C2  system, we exploit an effective Hamiltonian to describe  the coupling of the LCP and RCP modes near the Bril-  louin zone center [45, 46, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' As for the C4 system  with homogenous loss, the effective Hamiltonian can be  expressed as:  HC4 =  �  ω0 − iγ  (vR + ivI)k  (vR + ivI)k  ω0 − iγ  �  (3)  which has the complex eigenvalues:  ω = ω0 − iγ ± k(vR + ivI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  (4)  Here, ω0 is the eigenfrequency at k = 0, where the LCP  and RCP modes are degenerate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' vR and vI are the real  and the imaginary parts of the complex group velocities,  respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' γ denotes the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  In the C2 system, loss is selectively added to only one  resonator in each unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The breaking of C4 symmetry  opens a gap at k = 0, which can be characterized by a  perturbation term δ/2 in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The LCP and  RCP modes at k = 0 now have different loss γ1 and γ2  (γ1 ̸= γ2):  HC2 =  �  ω0 − iγ1 + δ  2  (vR + ivI)k  (vR + ivI)k  ω0 − iγ2 − δ  2  �  ,  (5)  which has the complex eigenvalues:  ω =ω0 − i(γ1 + γ2)  2  ± 1  2  �  [δ − i(γ1 − γ2)]2 − 4k2(vI − ivR)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  (6)  These analytical expressions of the complex eigenval-  ues in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' (4) and (6) are employed to fit the numeri-  cal results for both the real and the imaginary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The fitting results are shown as the solid lines in the  insets of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 6-8, accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We notice good quanti-  tative agreements between the analytical and numerical  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In particular, the effective Hamiltonian correctly  captures the EP features in the C2 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The fitting  parameters for both C4 and C2 systems with different  losses are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We note that the mode  damping parameters γ1,2 take negative values due to the  time convention eiωt adopted in COMSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Fitting parameters for C4 and C2 systems  System  α  ω0  γ  vR  vI  Inset  C4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='006  617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='61  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='71  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='018  A (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 6)  803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='82  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='03  C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 6)  C2  α  ω0  γ1  γ2  vR  vI  δ  Inset  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='006  617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='12 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='036 A (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 7)  803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='66 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='01  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='99 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='0075 C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 7)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='1  622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='12 -24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='87 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='03  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='77  A (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 8)  800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='40 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='22 -39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='98 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='22  C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 8)  The above effective Hamiltonians well explain the  emergence of the EPs and the enhancement of CD by  the EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In the C4 system, the LCP mode of the B4  band and the RCP mode of the B6 band are orthogonal  at k = 0 with vanished coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The damping of the  LCP and RCP modes at the same excitation frequency  are approximately equal due to homogeneous loss added  to all resonators of the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  Thus, their quality  factors are almost equal (corresponding to the results in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 9(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In the C2 system, the inhomogeneous loss  breaks the C4 rotational symmetry and induces coupling  between the original LCP and RCP modes at k = 0,  which gives rise to the polarization bandgaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In addi-  tion, the two modes have different dampings due to the  inhomogeneous material loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' These together give rise to  the EPs and the bifurcation of the imaginary parts of the  eigenfrequencies, leading to enlarged damping contrast  of the LCP and RCP modes at the same excitation fre-  quency and thus larger difference in their quality factors  (corresponding to the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' 9(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Therefore,  the strong CD effect in the C2 system is attributed to  both the polarization bandgaps and the EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  CONCLUSION  In conclusion, we demonstrate the acoustic CD effect  in a 3D chiral metamaterial supporting circularly polar-  8  ized transverse sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' We have investigated the effect  in two types of systems with C4 and C2 rotational sym-  metry, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' In the C4 system with loss homo-  geneously added to all resonators of the unit cell, we  observe a negligible acoustic CD effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  On the other  hand, by selectively adding loss to part of the unit cell,  reducing the system’s rotational symmetry from C4 to  C2, the CD effect is enhanced strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' With analysis of  their complex band structures and quality factors of the  eigenmodes, we uncover that the strong acoustic CD in  the C2 system is attributed to polarization bandgaps and  the emergence of non-Hermitian EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The polarization  bandgaps induce selective transmission and absorption  of the circularly polarized transverse sound with a par-  ticular handedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The EPs give rise to bifurcations of  the imaginary parts of the eigen frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' These to-  gether enhance the CD effect in the C2 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' It will be  interesting to experimentally demonstrate the discussed  phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The metamaterial structures can be fabri-  cated using 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Loss can be introduced into the  structures by adding sponges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The transverse sound can  be excited by using an array of speakers, and the reflec-  tion/refraction can be measured with a microphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' The  acoustic CD effect can find applications in sound manip-  ulations based on its vector degrees of freedom and in  acoustic sensing of chiral structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  The results con-  tribute to the understanding of chiral sound-matter in-  teractions in metamaterials and phononic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The work described in this paper was supported by  grants from the Research Grants Council of the Hong  Kong Special Administrative Region, China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' CityU  21302018 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' C6013-18G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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+page_content=' Qiu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
+page_content=' Rho, Electromagnetic chirality: from  fundamentals to nontraditional chiroptical phenomena,  Light: Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQfogFH/content/2301.02526v1.pdf'}
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diff --git a/gNAyT4oBgHgl3EQfj_gw/content/tmp_files/2301.00423v1.pdf.txt b/gNAyT4oBgHgl3EQfj_gw/content/tmp_files/2301.00423v1.pdf.txt
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@@ -0,0 +1,2765 @@
+arXiv:2301.00423v1  [math.OC]  1 Jan 2023
+Difference-of-Convex Reformulation for Chance Constrained
+Programs
+Peng Wang∗
+Rujun Jiang†
+Qingyuan Kong‡
+Laura Balzano§
+January 3, 2023
+Abstract
+Chance constrained programming refers to a type of optimization problem with uncertain
+constraints that are satisfied with at least a prescribed probability level.
+In this work,
+we study the sample average approximation of a class of chance constrained programs, in
+which the objective function is a difference-of-convex (DC) function and the functions in the
+chance constraint are all convex. Utilizing the empirical quantile of the chance constraint,
+we reformulate it into a DC constrained DC program. We propose a proximal DC algorithm
+(pDCA) for solving the reformulation. Moreover, we prove the global convergence of the
+proposed algorithm based on the Kurdyka-�Lojasiewicz property and derive the iteration
+complexity for finding an approximate KKT point. We point out that the proposed pDCA
+and its associated analysis apply to any DC constrained DC programs, which may be of
+independent interests. To support and complement our theoretical development, we show
+via numerical experiments that the proposed approach is competitive with a host of existing
+approaches.
+1
+Introduction
+Chance constrained programming is a powerful modeling paradigm for optimization problems
+with uncertain parameters, which has found wide applications in diverse fields, such as finance
+[10, 54], power systems [8, 75], and supply chain [67, 42], to name a few; see, e.g., [36] and
+the references therein for more applications.
+In general, the chance constrained program is
+to minimize a targeted loss subject to the probability of violating uncertain constraints being
+within a prespecified risk level. In this work, we consider a chance constrained program of the
+form
+min
+x∈X {f(x) : P (ci(x, ξ) ≤ 0, i ∈ {1, . . . , m}) ≥ 1 − α} ,
+(1)
+∗Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, USA.
+(pengwa@umich.edu).
+†Corresponding author. School of Data Science, Fudan University, Shanghai, China. (rjjiang@fudan.edu.cn).
+‡School of Data Science, Fudan University, Shanghai, China. (qykong21@m.fudan.edu.cn).
+§Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, USA.
+(girasole@umich.edu.)
+1
+
+where the vector x ∈ Rn denotes the decision variables, the functions f : Rn → R and ci :
+Rn × Rd → R for all i ∈ {1, . . . , m} are real-valued, the set X ⊆ Rn is deterministic, and
+ξ ∈ Rd is a random vector with its probability distribution supported on some set Ξ ⊆ Rd,
+and α ∈ (0, 1) is a given risk parameter. This problem is known as a single chance constrained
+program if m = 1, and a joint chance constrained program otherwise. Throughout this paper,
+we make the following assumptions for this problem.
+Assumption 1. (a) The function f takes the form of f = g − h, where g and h are continuous
+and convex (possibly non-smooth) functions defined on an open set that contains X. The func-
+tion g is ρ-strongly convex for some ρ ≥ 01.
+(b) The set X is non-empty, closed, and convex.
+(c) The functions ci(x, ξ) : Rn × Ξ → R for i = 1, . . . , m are convex and continuously differen-
+tiable in x for every ξ ∈ Ξ.
+In general, Problem (1) is highly intractable due to the chance constraint.
+Indeed, the
+feasible region formed by the chance constraint may be non-convex despite that ci(x, ξ) for
+all i ∈ {1, . . . , m} are convex. For example, even if ci(x, ξ) for i ∈ {1, . . . , m} are all linear
+in x and X is a polyhedron, the resulting feasible region defined by the chance constraint
+may not be convex [50]. Moreover, it is generally impossible to compute the probability of
+satisfying the constraint for a given x ∈ X when the distribution of ξ is unknown. In this work,
+we study Problem (1) in the data-driven setting, i.e., a set of independently and identically
+distributed (i.i.d.) samples {ˆξi}N
+i=1 generated according to the distribution of ξ is available,
+while its distribution is unknown. Motivated by the recent work on chance constrained programs
+[1, 49, 55, 58], we consider a sample average approximation (SAA) of Problem (1) over the
+samples {ˆξi}N
+i=1, which takes the form of
+min
+x∈X
+�
+f(x) :
+1
+N
+N
+�
+i=1
+1{C(x, ˆξi)≤0} ≥ 1 − α
+�
+,
+(2)
+where C(x, ξ) := max {ci(x, ξ) : i = 1, . . . , m}, and where
+1A is the indicator of event A. We
+should mention that Problem (2) also includes the scenario where the distribution is finite
+and discrete, and each event appears with probability 1/N. In particular, it has been shown
+in [49, 55] that solving Problem (2) can return a good approximate solution of Problem (1).
+However, Problem (2) is hard to optimize due to the discreteness of the constraint.
+Note
+that we assume that the objective function f in Problem (2) is a difference-of-convex (DC)
+function and all the functions ci(·, ξ) in the constraint are convex in Assumption 1. Then, a
+natural question is whether we can utilize these structures to develop an effective algorithmic
+framework for solving Problem (2). In this work, we answer this question in the affirmative.
+Exploiting these structures, we reformulate Problem (2) into a DC constrained DC problem and
+propose a proximal DC algorithm for solving the reformulation. In the literature, the existing
+approaches for solving Problem (2) generally can only prove subsequential convergence and
+have no iteration complexity analysis. In contrast with these results, we not only prove the
+1If ρ = 0, g is a general convex function.
+2
+
+subsequential and entire convergence to a Karush-Kuhn-Tucker (KKT) point of the proposed
+algorithm but also derive the iteration complexity for finding an approximate KKT point.
+1.1
+Related Works
+We first review some popular methods for solving chance constrained programs and then briefly
+talk about some DC algorithms closely related to our work. Since the first appearance of chance
+constrained programs in [15, 16], various algorithms for solving chance constrained problems
+under different settings have been proposed in a substantial body of literature over the past
+years. One well-known approach for solving Problem (1) is to reformulate the chance constraint
+into a convex constraint when the distribution of ξ is available.
+For example, Henrion [25,
+Lemma 2.2] showed that the chance constraint can be reformulated into a second-order cone
+if C(x, ξ) = ⟨ξ, x⟩ − b, ξ has an elliptical symmetric distribution, and b is a scalar. We refer
+the reader to [24, 39, 12, 26, 61] for more results on the convexity of the feasible region formed
+by chance constraints. These convex reformulations generally require a special distribution on
+random vector ξ, such as Gaussian or log-concave distributions.
+However, in practice, sometimes only a few random samples from the distribution of ξ are
+available while the distribution of ξ is unknown. To handle this scenario, one popular approach
+is to consider the SAA of the problem (see Problem (2)), which is obtained by replacing the
+true distribution with an empirical distribution corresponding to random samples. Luedtke
+and Ahmed [49] showed that the SAA with a risk level smaller than the required risk level can
+obtain a solution satisfying a chance constraint with high probability under suitable conditions.
+Later, Pagnoncelli et al. [55] showed that a solution of the SAA problem converges to that
+of the original problem with probability approaching one as N goes to infinity. Despite the
+fact that it possesses nice convergence properties, the SAA problem (2) is generally difficult to
+optimize due to its discrete nature. To solve it, many different approaches have been proposed
+in the literature. For example, Ahmed and Shapiro [1] proposed a mixed-integer programming
+(MIP) reformulation for the SAA problem; see also [36, 49, 50, 64] and the references therein.
+Curtis et al. [19] proposed a sequential algorithm, which minimizes quadratic subproblems with
+linear cardinality constraints iteratively.
+Bai et al. [5] proposed an augmented Lagrangian
+decomposition method for solving Problem (1) when ξ has a finite discrete distribution and
+cj(·, ξ) for j = 1, . . . , m are all affine. Recently, Pe˜na-Ordieres et al. [58] proposed a smoothing
+non-linear approximation of Problem (2) based on the empirical quantile of the chance constraint
+and developed a Sℓ1QP-type trust-region method to solve the approximation problem. Using
+a similar idea, Shen et al. [65] proposed a neural network model to approximate the empirical
+quantile of the chance constraint and employed a simulated annealing algorithm for solving
+the approximation problem. In general, some methods, such as [65], are heuristic in nature,
+and some other works, such as [5, 19, 58], only establish subsequential convergence for their
+proposed methods and have no iteration complexity analysis.
+The scenario approximation
+approach proposed by Calafiore and Campi [11], Nemirovski and Shapiro [52] is another well-
+known sample-based approach for solving Problem (1). This approach is simple and easy to
+implement, but it suffers from the solution becoming more and more conservative as the sample
+3
+
+size increases.
+Another notable approach for solving Problem (1) is to consider its conservative and tractable
+approximations. Among these approximations, the most famous one is the condition value-at-
+risk (CVaR) approximation proposed by Nemirovski and Shapiro [53], which is based on a
+conservative and convex approximation of the indication function. In particular, Hong and Liu
+[28] proposed a gradient-based Monte Carlo method for solving the CVaR approximation. To
+avoid overly conservative solutions, Hong et al. [29] studied a DC approximation of the chance
+constraint and tackled it by solving a sequence of convex approximations.
+Xie and Ahmed
+[71] proposed a bicriteria approximation for solving chance constrained covering problems and
+proved a constant factor approximation guarantee. More recently, Jiang and Xie [32] proposed
+a convex approximation named ALSO-X that always outperforms the CVaR approximation
+when uncertain constraints are convex. There are also other approximations based on different
+techniques in the literature; see, e.g., [14, 23].
+DC constrained DC programs2 refer to optimization problems that minimize a DC function
+subject to constraints defined by DC functions. Such problems have been extensively studied
+in the literature for decades [60, 30, 40]. One of the most popular methods for solving DC
+programs is the DC algorithm and its variants, which solve a sequence of convex subproblems
+by linearizing the second component of DC functions [29, 47, 59]. Le Thi et al. [41] proposed a
+penalty method and a DC algorithm using slack variables and showed that every accumulation
+point of the generated sequence is a KKT point of the considered problem. Later, Pang et al. [56]
+studied the proximal linearized method for DC programs and showed that every accumulation
+point of the generated sequence is a Bouligand-stationary point. Recently, van Ackooij et al. [68]
+developed a proximal bundle method for addressing DC programs and analyzed its convergence
+under different settings. Lu et al. [48] proposed penalty and augmented Lagrangian methods for
+solving DC programs, and established strong convergence guarantees for the proposed methods.
+1.2
+Our Contributions
+In this work, we study the SAA of the chance constrained program when the distribution of
+ξ is unknown but a set of i.i.d.
+samples {ˆξi}N
+i=1 generated according to its distribution is
+available. First, we reformulate the SAA (i.e., Problem (2)) of the chance constrained program
+(i.e., Problem (1)) into a DC constrained DC program by utilizing Assumption 1 and the
+empirical quantile function of C(x, ξ) over the samples {ˆξi}N
+i=1. Second, we propose a proximal
+DC algorithm (pDCA) for solving the reformulation, which proceeds by solving a sequence of
+convex subproblems by linearizing the second component of the obtained DC functions and
+adding a proximal term to the objective function. In particular, we show that it is easy to
+compute the required subgradients by using the structure of the DC functions. Moreover, the
+obtained subproblem can be rewritten in a form that is suitable for off-the-shelf solvers. Finally,
+we analyze the convergence and iteration complexity of the proposed method. Specifically, we
+show that any accumulation point of the sequence generated by the proposed method is a
+2For simplicity, we also call it DC programs.
+4
+
+KKT point of the reformulated problem under a constraint qualification. Then, we establish
+the convergence and convergence rate of the entire sequence by using the Kurdyka-�Lojasiewicz
+(K�L) inequality with the associated exponent. Moreover, we further show that the obtained
+DC program is equivalent to a convex constrained problem with a concave objective, which
+is amenable to the Frank-Wolfe (FW) method. By further showing the equivalence between
+proximal DC iterations for solving the DC program and modified FW iterations for solving
+the equivalent problem, we derive the iteration complexity of the pDCA for computing an
+approximate KKT point. In particular, in contrast to the standard iteration complexity of the
+FW method O(1/
+√
+k) (see, e.g., [38]), the iteration complexity of our considered FW method
+is improved to O(1/k) by utilizing the DC structure, where k is the number of iterations.
+The rest of this paper is organized as follows. In Section 2, we reformulate Problem (2) into
+a DC constrained DC program and introduce the proposed pDCA. In Section 3, we analyze the
+convergence and iteration complexity of the proposed method. In Section 4, we discuss some
+extensions of our approach. In Section 5, we report the experimental results of the proposed
+method and other existing methods. We end the paper with some conclusions in Section 6.
+1.3
+Notation and Definitions
+Besides the notation introduced earlier, we shall use the following notation throughout the
+paper. We write matrices in bold capital letters like A, vectors in bold lower-case letters like a,
+and scalars in plain letters like a. Given a matrix A ∈ Rm×n, we use aij to denote its (i, j)-th
+element. Given a vector x ∈ Rn, we use ∥x∥ to denote its Euclidean norm, xi its i-th element,
+and x[M] its M-th smallest element. We use 1 and 0 to denote the all-one vector and all-zero
+vector, respectively.
+Next, we introduce some concepts in non-smooth analysis that will be needed in our sub-
+sequent development. The details can be found in [63]. Let ϕ : Rn → (−∞, ∞] be a given
+function. We say that the function ϕ is proper if dom(ϕ) := {x ∈ Rn : ϕ(x) < ∞} ̸= ∅. A
+vector s ∈ Rn is said to be a Fr´echet subgradient of ϕ at x ∈ dom(ϕ) if
+lim inf
+y→x,y̸=x
+ϕ(y) − ϕ(x) − ⟨s, y − x⟩
+∥y − x∥2
+≥ 0.
+(3)
+The set of vectors s ∈ Rn satisfying (3) is called the Fr´echet subdifferential of f at x ∈ dom(ϕ)
+and denoted by �∂ϕ(x). The limiting subdifferential, or simply the subdifferential, of ϕ at x ∈
+dom(ϕ) is defined as
+∂ϕ(x) =
+�
+s ∈ Rn : ∃xk → x, sk → v with ϕ(xk) → ϕ(x), sk ∈ �∂ϕ(xk)
+�
+.
+When ϕ is proper and convex, thanks to [63, Proposition 8.12], the limiting subdifferential of
+ϕ at x ∈ dom(ϕ) coincides with the classic subdifferential defined as
+∂ϕ(x) = {s ∈ Rn : ϕ(y) ≥ ϕ(x) + ⟨s, y − x⟩, for all y ∈ Rn} .
+(4)
+5
+
+For a non-empty set S ⊆ Rn, its indicator function δS : Rn → {0, +∞} is defined as
+δS(x) =
+
+
+
+0,
+if x ∈ S,
++∞,
+otherwise.
+Its normal cone (resp. regular normal cone) at x ∈ S is defined as NS(x) := ∂δS(x) (reps.
+�
+NS(x) := �∂δS(x)). Given a point x ∈ Rn, its distance to S is defined as dist(x, S) = infy∈S ∥x−
+y∥. We say that S is regular at one of its points x if it is locally closed and satisfies NS(x) =
+�
+NS(x). In addition, we say that a function ϕ is regular at x if ϕ(x) is finite and its epigraph
+epi(ϕ) is regular at (x, ϕ(x)). Suppose that ϕ is a convex function. The directional derivative
+of ϕ at x ∈ Rn in the direction d ∈ Rn is defined by
+ϕ′(x, d) = lim
+tց0
+ϕ(x + td) − ϕ(x)
+t
+.
+In particular, it holds that
+ϕ′(x, d) = sup {⟨s, d⟩ : s ∈ ∂ϕ(x)} .
+(5)
+We say that a set valued mapping F : Rn → Rm is outer semi-continuous if for any sequence
+such that xk → x∗, yk → y∗ and yk ∈ F(xk), we have y∗ ∈ F(x∗). We next introduce the K�L
+property with the associated exponent; see, e.g., [2, 3, 4, 37].
+Definition 1 (K�L property and exponent). Suppose that ϕ : Rn → (−∞, ∞] is proper and lower
+semicontinuous. The function ϕ is said to satisfy the K�L property at ¯x ∈ {x ∈ Rn : ∂ϕ(x) ̸= ∅}
+if there exist a constant η ∈ (0, ∞], a neighborhood U of ¯x, and a continuous concave function
+ψ : [0, η) → R+ with ψ(0) = 0, ψ being continuously differentiable on (0, η), and ψ′(s) > 0 for
+s ∈ (0, η) such that
+ψ′ (ϕ(x) − ϕ(¯x)) dist(0, ∂ϕ(x)) ≥ 1
+(6)
+for all x ∈ U satisfying ϕ(¯x) < ϕ(x) < ϕ(¯x) + η. In particular, if ψ(s) = cs1−θ for some c > 0
+and θ ∈ (0, 1), then ϕ is said to satisfy the K�L property at ¯x with exponent θ.
+It is worth mentioning that a wide range of functions arising in applications satisfy the K�L
+property, such as proper and lower semicontinuous semialgebraic functions [3].
+2
+A Proximal DC Algorithm for Chance Constrained Programs
+In this section, we first reformulate Problem (2) into a DC constrained DC program based on
+the empirical quantile. Then, we propose a pDCA for solving the reformulation. To proceed,
+we introduce some further notions that will be used in the sequel. Let
+C(x, ξ) := max {ci(x, ξ) : i = 1, . . . , m} .
+(7)
+Given a set of samples {ˆξi}N
+i=1, let
+�C(x) :=
+�
+C(x, ˆξ1), . . . , C(x, ˆξN)
+�
+∈ RN.
+(8)
+6
+
+We define the p-th empirical quantile of C(x, ξ) over the samples {ˆξi}N
+i=1 for a probability
+p ∈ (0, 1) by
+ˆQc(p) := inf
+�
+y ∈ R : 1
+N
+N
+�
+i=1
+1{C(x, ˆξi)≤y} ≥ p
+�
+.
+Throughout this section, let
+M := ⌈(1 − α)N⌉.
+(9)
+2.1
+DC Reformulation of the Chance Constraint
+In this subsection, we reformulate the sample-based chance constraint in Problem (2) into a DC
+constraint using the empirical quantile function of C(x, ξ) over the samples {ˆξi}N
+i=1. To begin,
+according to [69, Chapter 21.2], the (1 − α)-th empirical quantile of C(x, ξ) over the samples
+{ˆξi}N
+i=1 for α ∈ (0, 1) is
+ˆQc(1 − α) = �C[M](x),
+where �C[M](x) denotes the M-th smallest element of �C(x). This allows us to get an equivalent
+form of Problem (2) as follows:
+min
+x∈X
+�
+f(x) :
+�C[M](x) ≤ 0
+�
+.
+(10)
+We should mention that the same empirical quantile-based problem has been considered in the
+literature. For example, Pe˜na-Ordieres et al. [58], Shen et al. [65] considered approximations of
+the quantile constraint, and Cui et al. [17] simplified the value at risk constraint of the loss of
+a portfolio, which is exactly the quantile constraint, by introducing new variables. By contrast,
+we directly handle the empirical quantile constraint by reformulating it into a DC form. To
+simplify our development, we denote the above constraint set by
+ZM :=
+�
+x ∈ Rn : �C[M](x) ≤ 0
+�
+.
+(11)
+Remark that if M = N, this constraint requires C(x, ˆξi) ≤ 0 for all i ∈ [N]. This, together with
+Assumption 1(c) and (7), implies that the constraint set ZN is convex. For this case, Problem
+(10) minimizes a DC objective function subject to convex constraints, and many algorithms in
+the literature have been proposed to solve this problem; see, e.g., [60] and the references therein.
+To avoid this case, we assume that M ≤ N − 1 throughout this paper. Using the structure
+of the function �C(·) and the convexity of ci(·, ξ), we show that the above constraint is a DC
+constraint.
+Lemma 1. Suppose that M ≤ N − 1. Let
+G(x) :=
+N
+�
+i=M
+�C[i](x), H(x) :=
+N
+�
+i=M+1
+�C[i](x).
+(12)
+Then, G and H are both continuous and convex functions, and the chance constraint in (11) is
+equivalent to a DC constraint
+G(x) − H(x) ≤ 0.
+(13)
+7
+
+Proof. The continuity of G and H follows from (7), (8), and Assumption 1(c). Since H(x)
+denotes the sum of T largest components of �C(x), we rewrite it as
+H(x) = max
+�N−M
+�
+t=1
+�Cit(x) : 1 ≤ i1 < i2 < · · · < iN−M ≤ N
+�
+.
+(14)
+According to the convexity of cj(x, ˆξi) for all i = 1, . . . , N and j = 1, . . . , m due to Assumption
+1(c) and the fact that the pointwise maximum of convex functions is still convex ([27, Proposition
+2.1.2]), we see that C(x, ˆξi) for all i = 1, . . . , N are convex. This, together with (14), the fact
+that the sum of convex functions is convex, and the fact that the pointwise maximum of convex
+functions is still convex, implies that H(x) is convex. By the same argument, we show that
+G(x) is convex. Given z ∈ RN and M ≤ N − 1, we decompose z[M] as
+z[M] =
+N
+�
+i=M
+z[i] −
+N
+�
+i=M+1
+z[i], for all M = 1, . . . , N − 1.
+(15)
+This, together with (12), implies that �C[M](x) ≤ 0 is equivalent to (13).
+Consequently, using Lemma 1 and Assumption 1(a), Problem (10) can be cast as the fol-
+lowing DC constrained DC program:
+min
+x∈X
+f(x) := g(x) − h(x)
+s.t.
+G(x) − H(x) ≤ 0,
+(16)
+where g, h are continuous and convex and G, H defined in (12) are also continuous and convex.
+2.2
+A Proximal DC Algorithm for Chance Constrained Programs
+In this subsection, we propose a proximal DC algorithm for solving Problem (16). To begin, we
+define
+I := {(i1, i2, . . . , iN−M) : 1 ≤ i1 < i2 < · · · < iN−M ≤ N} ,
+(17)
+and denote the active index set of C(x, ˆξi) and H(x) in (12) respectively by
+Mi
+c(x) :=
+�
+j ∈ {1, . . . , m} : cj(x, ˆξi) = C(x, ˆξi)
+�
+,
+(18)
+MH(x) :=
+�
+I ∈ I :
+N−M
+�
+t=1
+�Cit(x) = H(x)
+�
+.
+(19)
+We then talk about how to compute an element in these two active sets, respectively. Specifically,
+for the former one, we compute the function values of cj(x, ˆξi) for all j = 1, . . . , m and obtain
+an element in the index set Mi
+c(x) by finding an index j∗ ∈ {1, . . . , m} such that cj∗(x, ˆξi) has
+the largest value. For the latter one, after we compute C(x, ˆξi) for all i = 1, . . . , N using (7),
+we obtain an element in the index set MH(x) by finding an index (i∗
+1, . . . , i∗
+N−M) ∈ I such that
+{C(x, ˆξi∗
+t )}T
+t=1 is the T largest elements in {C(x, ˆξi)}N
+i=1. Now, we specify how to compute the
+subgradient of H(x) efficiently by utilizing its structure.
+8
+
+Lemma 2. Let H be defined in (12). Given an x ∈ Rn, it holds that
+∂H(x) = conv
+�
+∪
+N−M
+�
+t=1
+∂ �Cit(x) : (i1, . . . , iN−M) ∈ MH(x)
+�
+,
+(20)
+where
+∂ �Ci(x) = conv
+�
+∪{∇cj(x, ˆξi)} : j ∈ Mi
+c(x)
+�
+(21)
+for all i = 1, . . . , N and conv(A) denotes the convex hull of the set A.
+Proof. It follows from (14) and the rule of calculating the subdifferential of the pointwise max-
+imum of convex functions ([27, Corollary 4.3.2]) that
+∂H(x) = conv
+�
+∪∂
+N−M
+�
+t=1
+�Cit(x) : (i1, . . . , iN−M) ∈ MH(x)
+�
+= conv
+�
+∪
+N−M
+�
+t=1
+∂ �Cit(x) : (i1, . . . , iN−M) ∈ MH(x)
+�
+,
+where the second equality follows from the continuity and the convexity of C(x, ˆξi) for all
+i = 1, . . . , N. Since �Ci(x) = C(x, ˆξi) = max{cj(x, ˆξi) : j = 1, . . . , m} for any i ∈ {1, . . . , N},
+using the rule of calculating the subdifferential of the pointwise maximum of convex functions
+again and Assumption 1(c), we obtain (21).
+Now, we are ready to propose a proximal DC algorithm for solving Problem (16). Specifically,
+suppose that an initial point x0 ∈ X satisfying G(x0) − H(x0) ≤ 0 is available. At the k-th
+iteration, we choose sk
+h ∈ ∂h(xk) and sk
+H ∈ ∂H(xk), and generate the next iterate xk+1 by
+solving the following convex subproblem
+xk+1 ∈ argmin
+x∈X
+g(x) − h(xk) − ⟨sk
+h, x − xk⟩ + β
+2 ∥x − xk∥2
+s.t.
+G(x) − H(xk) − ⟨sk
+H, x − xk⟩ ≤ 0,
+(22)
+where β ≥ 0 is a penalty parameter. As shown in Lemma 2, the subgradient sk
+H can be easily
+computed. However, Problem (22) is still not suitable for off-the-shelf solvers, because it is
+difficult to directly input G(x) defined in (12), which involves the sum of the N −M +1 largest
+components of �C(x, ξ), into solvers due to its combinatorial nature. To handle this issue, we
+reformulate Problem (22) into a form that is suitable for solvers by introducing an auxiliary
+variable z ∈ RN such that C(x, ˆξi) ≤ zi, for all i = 1, . . . , N. Note that
+N
+�
+i=M
+z[i] = max
+u∈Rn
+�
+⟨u, z⟩ : 0 ≤ u ≤ 1, 1T u = N − M + 1
+�
+.
+This is a linear program and its dual problem is
+min
+λ∈RN,µ∈R {⟨1, λ⟩ + (N − M + 1)µ : z − λ − µ1 ≤ 0, λ ≥ 0} .
+9
+
+Using the strong duality of linear programming, we rewrite Problem (22) as
+xk+1 =
+argmin
+x∈X,z∈RN,λ∈RN,µ∈R
+g(x) − h(xk) − ⟨sk
+h, x − xk⟩ + β
+2 ∥x − xk∥2
+s.t.
+⟨1, λ⟩ + (N − M + 1)µ − H(xk) − ⟨sk
+H, x − xk⟩ ≤ 0,
+z − λ − µ1 ≤ 0, λ ≥ 0,
+cj(x, ˆξi) − zi ≤ 0, ∀ i =, 1 . . . , N, j = 1, . . . , m.
+(23)
+We remark that we can also eliminate the auxiliary variable z ∈ RN by combining cj(x, ˆξi)−zi ≤
+0 for i = 1, . . . , N, j = 1, . . . , m and z−λ−µ1 ≤ 0 together and obtain cj(x, ˆξi)−λi−µ ≤ 0 for
+i = 1, . . . , N, j = 1, . . . , m. We summarize the proposed proximal DC algorithm in Algorithm
+1.
+Algorithm 1 A Proximal DC Algorithm for Chance Constrained Programs
+1: Input: data samples {ˆξi}N
+i=1, feasible point x0, β ≥ 0.
+2: for k = 0, 1, . . . do
+3:
+take any sk
+h ∈ ∂h(xk) and sk
+H ∈ ∂H(xk)
+4:
+solve Problem (23) to obtain an xk+1
+5:
+if a termination criterion is met then
+6:
+stop and return xk+1
+7:
+end if
+8: end for
+Before studying its convergence, we make some remarks on Algorithm 1. First, Algorithm
+1 is closely related to sequential convex programming methods in [47, 73]. However, different
+from them, we fully exploit the structure of the DC function and reformulate the subproblem
+into a form that is suitable for off-the-shelf solvers. Moreover, we should also mention that our
+DC approach differs from that in [29], since their DC approach is based on the DC approxima-
+tion of the indicator function, while ours directly handles the empirical quantile of the chance
+constraint. Second, a key issue in our implementation is how to choose a feasible initial point
+x0. A common approach is to solve a convex approximation of Problem (2) such as CVaR in
+[53] to generate a feasible point. Third, the penalty parameter β can be updated in an adaptive
+manner as long as it is non-increasing and positive. In our numerical experiments, we observe
+that this adaptive scheme may empirically accelerate the convergence of the pDCA. Finally, the
+subproblem (23) is easy to solve in some cases. Specifically, it is observed that the functions
+cj(·, ξ) for all j = 1, . . . , m in many practical applications take the linear form; see, e.g., [50, 35].
+Based on this observation, suppose that in (16) X is a polyhedron and
+g(x) = aT
+0 x, cj(x, ξ) = aT
+j x + bT
+j ξ, for all j = 1, . . . , m.
+(24)
+Then, substituting (24) into (23) with β = 0 (resp. β > 0) yields a linear (resp. quadratic)
+program with (m+2)N +1 linear constraints (without considering the linear constraints in X).
+We can solve it easily by inputting it into off-the-shelf linear (resp. quadratic) programming
+10
+
+solvers, such as MOSEK, Gurobi, and CPLEX. In addition, suppose that in (16) X is a polyhedron
+and
+g(x) = xT Ax + aT
+0 x, cj(x, ξ) = aT
+j x + bT
+j ξ, for all j = 1, . . . , m,
+(25)
+where A ∈ Rn×n is a symmetric matrix. The resulting subproblem (23) is a quadratic program
+when β ≥ 0.
+3
+Convergence and Iteration Complexity Analysis
+In this section, we study the convergence properties of Algorithm 1. Towards this end, we first
+show the subsequential convergence of the sequence {xk} generated by Algorithm 1 to a KKT
+point of Problem (16) under some constraint qualification. Second, we prove convergence of
+the entire sequence {xk} if in addition the K�L property holds for a specially designed potential
+function. Finally, we analyze the iteration complexity of Algorithm 1. We point out that the
+proposed algorithm and its convergence in fact apply to Problem (16) with G(x) and H(x) being
+general convex functions defined on an open set that contains X, which takes the form of general
+DC constrained DC programs. An extension to multiple DC constraints will be discussed in
+Section 4.2.
+Before we proceed, we introduce some further notation, assumptions, and definitions that
+will be used throughout this section. To begin, we specify the convex constraints in the set X
+as follows:
+X =
+�
+x ∈ Rn : aT
+i x + bi = 0, i ∈ E, ωi(x) ≤ 0, i ∈ I
+�
+,
+(26)
+where ai ∈ Rn and bi ∈ R for all i ∈ E, ωi : Rn → R for all i ∈ I are convex and continuously
+differentiable functions, and E and I are finite sets of indices. We denote the active set of the
+inequality constraints at x ∈ X by
+A(x) := {i ∈ I : ωi(x) = 0} ,
+(27)
+and the feasible set of Problem (16) by
+¯
+X := {x ∈ X : G(x) − H(x) ≤ 0} .
+We now introduce a generalized version of the Mangasarian-Fromovitz constraint qualification
+(MFCQ), which is a widely used assumption on the algebraic description of the feasible set
+of constrained problems that ensures that the KKT conditions hold at any local minimum
+([47, 72]).
+Assumption 2 (Generalized MFCQ). The generalized MFCQ holds for all x ∈ ¯
+X, i.e., there
+exists y ∈ X such that for G(x) = H(x), we have
+G(y) − H(x) − ⟨sH, y − x⟩ < 0, for all sH ∈ ∂H(x),
+(28)
+⟨∇ωi(x), y − x⟩ < 0, for all i ∈ A(x).
+(29)
+11
+
+We next introduce the definition of KKT points for Problem (16).
+Definition 2 (KKT points of Problem (16)). We say that x ∈ ¯
+X is a KKT point of Problem
+(16) if there exists λ ∈ R+ such that (x, λ) satisfies λ (G(x) − H(x)) = 0 and
+0 ∈ ∂g(x) − ∂h(x) + λ (∂G(x) − ∂H(x)) + NX (x).
+Note that every local minimizer of Problem (16) is a KKT point under the generalized
+MFCQ. More precisely, suppose that x∗ ∈ ¯
+X is a local minimizer of Problem (16), P = {x :
+aT x + bi = 0, i ∈ E} is a polyhedron, and there exists d ∈ TP(x∗) for x∗ ∈ X satisfying
+G(x∗) = H(x∗) and ω(x∗) such that
+G′(x∗, d) <
+inf
+s∈∂H(x∗) sT d,
+and
+ω′
+i(x∗, d) = ωi(x∗)T d < 0, i ∈ E.
+(30)
+Then, there exists λ∗ ∈ R+ such that x∗ is a KKT point of Problem (16). This result is a direct
+consequence of [47, Theorem 2.1]. It is worth noting that (30) holds if the generalized MFCQ
+holds at x∗; see, e.g., Remark (b) of [47, Theorem 2.1].
+3.1
+Subsequential Convergence to a KKT Point
+In this subsection, our goal is to show that any accumulation point of the sequence {xk}
+generated by Algorithm 1 is a KKT point of Problem (16).
+Lemma 3. Suppose that Assumptions 1 holds, the function f is given in Problem (16), and the
+level set
+�
+x ∈ ¯
+X : f(x) ≤ f(x0)
+�
+is bounded. Let {xk} be the sequence generated by Algorithm
+1 with ρ + β > 0. Then, the following statements hold:
+(i) It holds for all k ≥ 0 that xk ∈ ¯
+X and
+f(xk+1) − f(xk) ≤ −ρ + β
+2
+∥xk+1 − xk∥2.
+(31)
+(ii) The sequence {xk} ⊆ ¯
+X is bounded.
+(iii) It holds that
+lim
+k→∞ ∥xk+1 − xk∥ = 0.
+(32)
+Proof. (i) According to the feasibility of xk+1 to Problem (22), sk
+H ∈ ∂H(xk), and the convexity
+of H, we have xk+1 ∈ X and
+G(xk+1) ≤ H(xk) + ⟨sk
+H, xk+1 − xk⟩ ≤ H(xk+1).
+(33)
+This implies xk+1 ∈ ¯
+X for all k ≥ 0. Moreover, it follows from the optimality of xk+1 for
+Problem (22) and the ρ-strongly convexity of g that for all k ≥ 0,
+g(xk+1) − h(xk) − ⟨sk
+h, xk+1 − xk⟩ + ρ + β
+2
+∥xk+1 − xk∥2 ≤ g(xk) − h(xk).
+12
+
+This, together with the convexity of h and sk
+h ∈ ∂h(xk), yields that for all k ≥ 0,
+g(xk+1) − h(xk+1) + ρ + β
+2
+∥xk+1 − xk∥2 ≤ g(xk) − h(xk),
+which is equivalent to (31).
+(ii) According to (31), the function value f(xk) is monotonically decreasing and thus we
+have f(xk+1) ≤ f(x0) for all k ≥ 1.
+This, together with the level-boundness of the set
+�
+x ∈ X c : f(x) ≤ f(x0)
+�
+, implies that {xk} is bounded.
+(iii) The boundedness of the sequence {xk}, together with continuity of f implies that
+{f(xk)} is bounded from below. Using this and the fact that {f(xk)} is monotonically de-
+creasing, we obtain that there exists some f ∗ such that f(xk) → f ∗.
+It follows from (31)
+that
+α + β
+2
+∞
+�
+k=0
+∥xk+1 − xk∥2 ≤ f(x0) − lim
+k→∞f(xk+1) = f(x0) − f ∗ < ∞.
+This implies (32).
+Armed with the above lemma, we are ready to show the subsequential convergence of the
+sequence {xk} generated by Algorithm 1 to a KKT point of Problem (16).
+Theorem 1. Suppose that Assumptions 1 and 2 hold, the function f is given in Problem (16),
+and the level set
+�
+x ∈ ¯
+X : f(x) ≤ f(x0)
+�
+is bounded. Let {xk} be the sequence generated by
+Algorithm 1 with ρ + β > 0. Then, any accumulation point of {xk} is a KKT point of Problem
+(16).
+Proof. According to (i) in Lemma 3, it holds that xk ∈ ¯
+X for all k ≥ 0. This, together with the
+generalized MFCQ in Assumption 2 and the equivalence between the Slater condition and the
+MFCQ by [18, Exercise 2.3.3(b)], yields that there exists x ∈ X such that for any sk
+H ∈ ∂H(xk),
+G(x) − H(xk) − ⟨sk
+H, x − xk⟩ < 0, ωi(x) < 0, ∀ i ∈ A(x).
+(34)
+This is exactly the Slater condition for Problem (22). According to this, (26), and [62, Corollary
+28.2.1, Theorem 28.3], there exists a Lagrange multiplier λk ∈ R for all k ≥ 0 such that the
+following KKT system holds:
+
+
+
+
+
+
+
+
+
+G(xk+1) − H(xk) − ⟨sk
+H, xk+1 − xk⟩ ≤ 0, xk+1 ∈ X,
+λk �
+G(xk+1) − H(xk) − ⟨sk
+H, xk+1 − xk⟩
+�
+= 0, λk ≥ 0,
+0 ∈ ∂g(xk+1) − sk
+h + β(xk+1 − xk) + λk �
+∂G(xk+1) − sk
+H
+�
++ NX (xk+1).
+(35)
+It follows from (ii) of Lemma 3 that {xk} is bounded. Let x∗ be an accumulation point of {xk}
+such that there exists a subsequence {xki} with limi→∞ xki = x∗. We claim that the sequence
+{λk} is bounded. Passing to a further subsequence if necessary, we assume without loss of
+generality that limi→∞ λki = λ∗. According to (32) in Lemma 3, we have limi→∞(xki+1−xki) =
+0. Using this fact, the outer semi-continuity of ∂g, ∂h, ∂G, ∂H [63, Definition 5.4, Proposition
+13
+
+8.7], and sk
+h ∈ ∂h(xk), sk
+H ∈ ∂H(xk), we have upon passing to the limit as i goes to infinity in
+(35) with k = ki that sk
+h → s∗
+h ∈ ∂h(x∗) and sk
+H → s∗
+H ∈ ∂H(x∗), and thus
+0 ∈ ∂g(x∗) − ∂h(x∗) + λ∗ (∂G(x∗) − ∂H(x∗)) + NX (x∗).
+(36)
+On the other hand, using (35) and (32) with k = ki and the boundedness of ∂H(x∗), letting
+i → ∞, we have
+G(x∗) ≤ H(x∗), λ∗ (G(x∗) − H(x∗)) = 0.
+(37)
+Moreover, since λk ≥ 0 and xk ∈ ¯
+X for all k ≥ 0, we have λ∗ ≥ 0 and x∗ ∈ ¯
+X. This, together
+with (36), (37), and Definition 2, implies that x∗ is a KKT point of Problem (16).
+The rest of the proof is devoted to proving that {λk} is bounded. Without loss of generality,
+we assume that {ai : i ∈ E} is linearly independent, since otherwise we can obtain the same
+results by eliminating the redundant linear equalities. It follows from [63, Theorem 6.14] for
+any x ∈ X that
+NX (x) =
+��
+i∈E
+uiai +
+�
+i∈I
+vi∇ωi(x) : vi ≥ 0, for i ∈ A(x), vi = 0, for i /∈ A(x)
+�
+.
+This, together with (35), yields that there exist uk
+i for i ∈ E, vk
+i ≥ 0 for i ∈ A(xk+1), and vk
+i = 0
+for i /∈ A(xk+1) such that
+0 ∈ ∂g(xk+1) − sk
+h + β(xk+1 − xk) + λk �
+∂G(xk+1) − sk+1
+H
+�
++
+�
+i∈E
+uk
+i ai +
+�
+i∈I
+vk
+i ∇ωi(xk+1).
+(38)
+Then, let
+ρk :=
+�
+(λk)2 +
+�
+i∈E
+(uk
+i )2 +
+�
+i∈I
+(vk
+i )2, τ k := λk
+ρk , µk
+i := uk
+i
+ρk , νk
+i := vk
+i
+ρk .
+Suppose to the contrary that {λk} is unbounded.
+This implies that ρk is also unbounded.
+Then, there exists a subsequence {λkj} such that |λkj| → ∞ as j goes to infinity. Passing
+to a further subsequence if necessary, suppose that there exist τ ∗ ∈ R+, µ∗
+i , ν∗
+i ∈ R+, x∗, and
+s∗
+H ∈ ∂H(x∗) such that limj→∞ τ kj = τ ∗, limj→∞ µkj
+i
+= µ∗
+i , limj→∞ νkj
+i
+= ν∗
+i , limj→∞ xkj = x∗,
+and limj→∞ skj
+H = s∗
+H, where skj
+H ∈ ∂H(xkj), due to λk ≥ 0, v∗
+i ≥ 0 for i ∈ I, the boundness
+of {τ k}, {µk}, {νk}, {xk}, and ∂H(xk), and the outer semi-continuity of ∂H. Then, dividing
+both sides of (38) by |ρkj|, letting j → ∞, and using (32), the outer semi-continuity of ∂g and
+∂h, and the boundness of ∂g(x∗), ∂h(x∗), and {xk}, we have
+0 ∈ τ ∗ (∂G(x∗) − s∗
+H) +
+�
+i∈E
+µ∗
+i ai +
+�
+i∈I
+ν∗
+i ∇ωi(x∗).
+(39)
+Using the definitions of τ ∗, µ∗, and ν∗, we further have
+(τ ∗)2 + ∥µ∗∥2 + ∥ν∗∥2 = 1,
+(40)
+14
+
+(Case 1) Suppose that τ ∗ = 0. Due to (39), we have
+0 =
+�
+i∈E
+µ∗
+i ai +
+�
+i∈I
+ν∗
+i ∇ωi(x∗).
+(41)
+According to Assumption 2, there exists y ∈ X such that ⟨∇ωi(x∗), y−x∗⟩ < 0 for all i ∈ A(x∗).
+Moreover, since A(xk) ⊆ A(x∗) when k is sufficiently large, we have i /∈ A(xk) if i /∈ A(x∗).
+Therefore, we have νk
+i = 0 for all i /∈ A(xk) as k → ∞, which implies ν∗
+i = 0 for i /∈ A(x∗).
+Then, taking inner products with y − x∗ on both sides of (41) yields
+0 =
+�
+i∈A(x∗)
+ν∗
+i ⟨∇ωi(x∗), y − x∗⟩,
+where the equality follows from ⟨ai, y − x∗⟩ = 0 for i ∈ E and ν∗
+i = 0 for i /∈ A(x∗). This,
+together with ⟨∇ωi(x∗), y−x∗⟩ < 0 for all i ∈ A(x∗), gives ν∗
+i = 0 for all i ∈ A(x∗). Substituting
+this and ν∗
+i = 0 for i /∈ A(x∗) into (41), we have 0 = �
+i∈E µ∗
+i ai. Noting that we assume that
+{ai : i ∈ E} is linearly independent, we have µ∗
+i = 0 for all i ∈ E. Therefore, ν∗
+i = 0 for all i ∈ I
+and µ∗
+i = 0 for all i ∈ E. This contradicts (40).
+(Case 2) Suppose that τ ∗ > 0. We first consider the case of G(x∗) < H(x∗). It follows from
+the second line of (35) with k = kj, j → ∞, and (32) that limj→∞ λkj = 0. This implies τ ∗ = 0,
+which contradict (40). We then must have G(x∗) = H(x∗). This, together with the convexity
+of G and (28) in Assumption 2, yields that there exists y ∈ X such that
+⟨¯sG − s∗
+H, y − x∗⟩ ≤ G(y) − G(x∗) − ⟨s∗
+H, y − x∗⟩ = G(y) − H(x∗) − ⟨s∗
+H, y − x∗⟩ < 0, (42)
+where ¯sG is an arbitrary subgradient of G at x. According to (39), there exists s∗
+G ∈ ∂G(x∗)
+such that
+0 = τ ∗ (s∗
+G − s∗
+H) +
+�
+i∈E
+µ∗
+i ai +
+�
+i∈I
+ν∗
+i ∇ωi(x∗).
+(43)
+Taking inner products with y − x∗ on both sides yields
+0 = τ ∗⟨s∗
+G − s∗
+H, y − x∗⟩ +
+�
+i∈A(x∗)
+ν∗
+i ⟨∇ωi(x∗), y − x∗⟩.
+Note that ν∗
+i ≥ 0 due to vk
+i ≥ 0 for all i ∈ I. This, together with (29) at x∗ and (42), implies
+τ ∗ = 0, which is a contradiction. We prove the claim.
+3.2
+Convergence of the Entire Sequence to a KKT Point
+In this subsection, we employ the analysis framework proposed in [2, 4] based on the K�L property
+to study the sequential convergence of Algorithm 1 for β > 0. Our first step is to show that the
+sequence generated by Algorithm 1 satisfies sufficient decrease and relative error conditions with
+respect to a potential function. Motivated by the potential functions constructed in [46, 73], we
+construct the following potential function
+ϕ(x, y, z) := g(x) − ⟨x, y⟩ + h∗(y) + δ ¯F (·)≤0(x, z) + δX (x),
+(44)
+15
+
+where
+¯F(x, z) := G(x) − ⟨x, z⟩ + H∗(z).
+(45)
+Then, we characterize the subdifferential of δ ¯F (·)≤0(x, z) using its structure and the convexity
+of G and H. We point out that this characterization holds for G and H being arbitrary proper
+closed convex functions.
+Lemma 4. Suppose that Assumption 2 holds and (x, z) satisfies ¯F(x, z) ≤ 0. It holds that the
+function δ ¯F (·) is regular and
+∂δ ¯F (·)≤0(x, z) =
+��
+λ(s1 − z)
+λ(−x + s2)
+�
+: s1 ∈ ∂G(x), s2 ∈ ∂H∗(z), λ ≥ 0, λ ¯F(x, z) = 0
+�
+.
+(46)
+Proof. To begin, let
+S :=
+�
+(x, z) : ¯F(x, z) ≤ 0
+�
+.
+In particular, we write S = ¯F −1(R−). Because G and H∗ are both convex functions, then ¯F is
+locally Lipschitz continuous. It follows from [63, Definition 9.1] that ¯F : Rn×Rn → R is a strictly
+continuous function since it is locally Lipschitz continuous. We obtain that y ∈ NR−( ¯F(x, z))
+is equivalent to y ≥ 0, y ¯F(x, z) = 0. Suppose that the only vector y ∈ NR−( ¯F(x, z)) with
+0 ∈ ∂(y ¯F)(x, z) is y = 0. Moreover, R− is regular at ¯F(x, z) and ¯F is regular due to convexity
+of G and H∗. These, together with [63, Corollary 10.50], imply that S is regular, which further
+implies δ ¯F(·) is regular by [63, Exercise 8.14], and (46) holds.
+The rest of our proof is to show that the only vector y ∈ NR−( ¯F(x, z)) with 0 ∈ ∂(y ¯F)(x, z)
+is y = 0. Suppose that this is not the case. Then, there exists y > 0 such that y ∈ NR−( ¯F(x, z))
+with 0 ∈ ∂(y ¯F)(x, z), which implies
+�
+(x, z) :
+¯F(x, z) = 0, 0 ∈ ∂ ¯F(x, z)
+�
+̸= ∅.
+(47)
+Let (x, z) be such that ¯F(x, z) = 0 and 0 ∈ ∂ ¯F(x, z). It follows from ¯F(x, z) = 0 and (45)
+that
+G(x) − ⟨x, z⟩ + H∗(z) = 0.
+(48)
+Using [3, Proposition 2.1] and 0 ∈ ∂ ¯F(x, z) with (45), we have
+z ∈ ∂G(x), x ∈ ∂H∗(z).
+In particular, x ∈ ∂H∗(z) holds if and only if z ∈ ∂H(x) because H is a pointwise maximum of
+continuous and convex functions, and thus closed and convex, which implies H∗(z) + H(x) =
+⟨x, z⟩ according to Young’s inequality. Using this and (48), we have G(x) = H(x). It follows
+from z ∈ ∂G(x) and the convexity of G that
+G(y) ≥ G(x) + ⟨z, y − x⟩, for all y ∈ Rn.
+Plugging z ∈ ∂H(x) and G(x) = H(x) into the above inequality yields that for z ∈ ∂H(x),
+G(y) ≥ H(x) + ⟨z, y − x⟩, for all y ∈ Rn,
+which contradicts (28) in Assumption 2. We prove the claim.
+16
+
+Now, we are ready to show that the sequence {(xk, sk
+h, sk
+H)} generated by Algorithm 1
+satisfies the sufficient decrease and relative error conditions mentioned earlier.
+Lemma 5. Suppose that Assumptions 1 and 2 hold.
+Let {(xk+1, sk
+h, sk
+H)} be the sequence
+generated by Algorithm 1 with ρ + β > 0. Then, the following statements hold:
+(i) [Sufficient Decrease] The sequence {(xk+1, sk
+h, sk
+H)} is bounded. It holds for all k ≥ 1 that
+ϕ(xk+1, sk
+h, sk
+H) − ϕ(xk, sk−1
+h
+, sk−1
+H
+) ≤ −ρ + β
+2
+∥xk+1 − xk∥2.
+(ii) [Relative Error] There exists a constant κ > 0 such that for all k ≥ 0,
+dist
+�
+0, ∂ϕ(xk+1, sk
+h, sk
+H)
+�
+≤ κ∥xk+1 − xk∥.
+Proof. (i) It follows from (i) in Lemma 3 that {xk} ⊆ ¯
+X is bounded. This, together with the
+fact that h and H are convex, implies that {(sk
+h, sk
+H)} is bounded. Therefore, the sequence
+{(xk+1, sk
+h, sk
+H)} is bounded. According to (45), we have for all k ≥ 0,
+¯F(xk+1, sk
+H) = G(xk+1) − ⟨xk+1, sk
+H⟩ + H∗(sk
+H)
+= G(xk+1) + H∗(sk
+H) − ⟨xk, sk
+H⟩ − ⟨xk+1 − xk, sk
+H⟩
+= G(xk+1) − H(xk) − ⟨xk+1 − xk, sk
+H⟩ ≤ 0,
+(49)
+where the last equality follows from H(xk) + H∗(sk
+H) = ⟨xk, sk
+H⟩ due to Young’s inequality and
+sk
+H ∈ ∂H(xk), and the inequality is due to the constraint in (22). Moreover, it follows from
+(22) and the ρ-strongly convexity of g that for all k ≥ 0,
+g(xk+1) − ⟨sk
+h, xk+1 − xk⟩ + ρ + β
+2
+∥xk+1 − xk∥2 ≤ g(xk).
+(50)
+This, together with (49) and xk ∈ X, implies for all k ≥ 1,
+ϕ(xk+1, sk
+h, sk
+H) = g(xk+1) − ⟨xk+1, sk
+h⟩ + h∗(sk
+h)
+≤ g(xk) − ⟨sk
+h, xk⟩ − ρ + β
+2
+∥xk+1 − xk∥2 + h∗(sk
+h)
+= g(xk) − h(xk) − ρ + β
+2
+∥xk+1 − xk∥2
+≤ g(xk) − ⟨xk, sk−1
+h
+⟩ + h∗(sk−1
+h
+) − ρ + β
+2
+∥xk+1 − xk∥2
+= ϕ(xk, sk−1
+h
+, sk−1
+H
+) − ρ + β
+2
+∥xk+1 − xk∥2,
+where the first inequality uses (50), the second equality follows from h(xk) + h∗(sk
+h) = ⟨xk, sk
+h⟩
+due to sk
+h ∈ ∂h(xk) and Young’s inequality, the second inequality follows from h(xk)+h∗(sk−1
+h
+) ≥
+⟨xk, sk−1
+h
+⟩ due to Young’s inequality, and the last equality is due to xk ∈ X, (44), and (49).
+(ii) According to [63, Exercise 8.8(c)], we compute
+∂ϕ(x, y, z) = ∂
+�
+g(x) + h∗(y) + δ ¯F (·)≤0(x, z) + δX (x)
+�
++
+
+
+−y
+−x
+0
+
+
+=
+�
+
+
+
+
+
+
+
+∂g(x) − y + λ(∂G(x) − z) + NX(x)
+−x + ∂h∗(y)
+λ(−x + ∂H ∗ (z))
+
+ : λ ≥ 0, λ ¯F(x, z) = 0
+
+
+
+
+
+,
+17
+
+where the second equality uses [63, Corollary 10.9] and the fact that g, h∗ and δX are regular
+due to the convexity and δ ¯F (·) is regular due to Lemma 4. Therefore, we have
+∂ϕ(xk+1, sk
+h, sk
+H) =
+�
+
+
+
+
+
+
+
+∂g(xk+1) − sk
+h + λ(∂G(xk+1) − sk
+H) + NX (xk+1)
+−xk+1 + ∂h∗(sk
+h)
+λ(−xk+1 + ∂H∗(sk
+H))
+
+ :
+λ ≥ 0, λ ¯F(xk+1, sk
+H) = 0
+�
+.
+(51)
+It follows from Assumption 2 that the KKT system (35) holds for Problem (22). Then we have
+λk ≥ 0 and
+λk ¯F(xk+1, sk
+H) = λk �
+G(xk+1) − ⟨xk+1, sk
+H⟩ + H∗(sk
+H)
+�
+= λk �
+G(xk+1) − H(xk) − ⟨xk+1 − xk, sk
+H⟩
+�
+= 0,
+(52)
+where the first equality uses (45), the second equality follows from H(xk) + H∗(sk
+H) = ⟨xk, sk
+H⟩
+due to sk
+H ∈ ∂H(xk) and Young’s inequality, and the last equality follows from the second line
+in (35). It follows from the last line in (35) that
+β(xk − xk+1) ∈ ∂g(xk+1) − sk
+h + λk �
+∂G(xk+1) − sk
+H
+�
++ NX (xk+1).
+This, together with (49), (51), (52) with λk ≥ 0, sk
+h ∈ ∂h(xk), sk
+H ∈ ∂H(xk), and the fact that
+y ∈ ∂ψ(x) if and only if x ∈ ∂ψ∗(y) provided that ψ is a proper closed convex function, yields
+that
+
+
+β(xk − xk+1)
+xk − xk+1
+λk(xk − xk+1)
+
+ ∈ ∂ϕ(xk+1, sk
+h, sk
+H)
+This implies
+dist
+�
+0, ∂ϕ(xk+1, sk
+h, sk
+H)
+�
+≤ (β + 1 + λk)∥xk+1 − xk∥,
+where λk ≥ 0 is bounded in (35) according to the proof of Theorem 1. Then, we complete the
+proof.
+Since g, h, G, and H are continuous and convex functions and X is a closed and convex
+set, we can verify that ϕ is a K�L function with exponent θ ∈ [0, 1) according to [9, Theorem 3].
+Using Lemma 5 and the analysis in [2, 3, 4, 9, 46, 73], we shall prove the sequential convergence
+and the convergence rate of the sequence {xk} generated by Algorithm 1. The proof is rather
+standard and thus we omit it. We refer the reader to [2, 46] for the detailed arguments.
+Theorem 2. Suppose that Assumptions 1 and 2 hold, the function f is given in Problem (16),
+and the level set
+�
+x ∈ X c : f(x) ≤ f(x0)
+�
+is bounded. Then, the sequence {xk} generated by
+Algorithm 1 with ρ + β > 0 converges to a KKT point x∗ of Problem (16). Let θ ∈ [0, 1) denote
+the K�L exponent of ϕ in (44). There exists an integer k∗ ≥ 1 such that the following statements
+18
+
+hold:
+(i) If θ = 0, then {xk} converges finitely, i.e., xk = x∗ for all k ≥ k∗.
+(ii) If θ ∈ (0, 1/2], then {xk} converges linearly, i.e., there exist c > 0 and q ∈ (0, 1) such that
+for all k ≥ k∗,
+∥xk − x∗∥ ≤ cqk.
+(iii) If θ ∈ (1/2, 1), then {xk} converges sublinearly, i.e., there exist c > 0 such that for all
+k ≥ k∗,
+∥xk − x∗∥ ≤ ck− 1−θ
+2θ−1 .
+It follows from Theorem 2 that the proximal DC algorithm achieves linear convergence when
+the K�L exponent θ = 1/2. Therefore, an interesting future direction is to investigate under what
+conditions the K�L exponent of Problem (16) is 1/2; see, e.g., [43, 33, 34, 45, 70, 76].
+3.3
+Iteration Complexity for Computing an Approximate KKT Point
+In this subsection, we analyze the iteration complexity of Algorithm 1 for computing an ap-
+proximate KKT point of Problem (16). Motivated by the analysis framework in [74] for DC
+constrained DC programs with all functions being differentiable, we connect Algorithm 1 to a
+variant of the Frank-Wolfe (FW) method.
+To simplify notation, let
+w := (x, s, t),
+q(w) := s − h(x),
+Q(w) := t − H(x),
+and
+W := {w : x ∈ X, g(x) ≤ s, G(x) ≤ t} .
+In particular, we should mention that q and Q are both concave functions and W is a convex
+set. We rewrite Problem (16) as follows by introducing auxiliary variables s, t ∈ R:
+min
+x∈X,s∈R,t∈Rs − h(x)
+s.t.
+g(x) ≤ s, G(x) ≤ t, t − H(x) ≤ 0.
+(53)
+We further express Problem (53) as
+min
+w∈W q(w)
+s.t. Q(w) ≤ 0,
+(54)
+Based on the above setup, we directly show the equivalence between the proximal DC iterations
+in (22) and a variant of FW iterations applied to Problem (54).
+Lemma 6. Suppose that Assumptions 1 and 2 hold. The proximal DC iterations in (22) with
+β ≥ 0 is equivalent to the following variant of FW iterations:
+wk+1 ∈ argmin
+w∈W
+q(wk) + ⟨sk
+q, w − wk⟩ + β
+2 ∥w − wk∥2
+T
+s.t.
+Q(wk) + ⟨sk
+Q, w − wk⟩ ≤ 0,
+(55)
+19
+
+where sk
+q = (−sk
+h, 1, 0), sk
+h ∈ ∂h(xk), sk
+Q = (−sk
+H, 0, 1), sk
+H ∈ ∂H(xk), and we define ∥z∥T =
+��n
+i=1 z2
+i for any z ∈ Rn+2.
+Proof. The proof follows directly from the definitions of W, q(w), Q(w), and the fact that any
+optimal solution of (55) must satisfy sk = g(xk) and s = g(x).
+We next use the equivalent expression (54) to give an equivalent characterization of KKT
+points (see Definition 2) of Problem (16) under the generalized MFCQ in Assumption 2.
+Lemma 7. Suppose that Assumptions 1 and 2 hold. Given ¯w ∈ W, sq ∈ ∂q( ¯w) with g(¯x) ≤
+¯s, G(¯x) ≤ ¯t, and sQ ∈ ∂Q( ¯w), suppose that
+⟨sq, w − ¯w⟩ + β
+2 ∥w − ¯w∥2
+T ≥ 0
+(56)
+for all w ∈ W satisfying Q( ¯w) + ⟨sQ, w − ¯w⟩ ≤ 0. Then, ¯x is a KKT point of Problem (16).
+Proof. According to the statement of the lemma and [6, Proposition 2.1.2], we obtain that
+¯w ∈ W is an optimal solution to the following convex problem:
+min
+w∈W
+⟨sq, w − ¯w⟩ + β
+2 ∥w − ¯w∥2
+T
+s.t.
+Q( ¯w) + ⟨sQ, w − ¯w⟩ ≤ 0.
+Note that ⟨sq, w − ¯w⟩ = −⟨sh, x − ¯x⟩ + (s − ¯s). Moreover, the optimal solution of (55) must
+satisfy s = g(x) and ¯s = g(¯x). Then we have
+⟨sq, w − ¯w⟩ + β
+2 ∥w − ¯w∥2
+T = g(x) − g(¯x) − ⟨sh, x − ¯x⟩ + β
+2 ∥x − ¯x∥2.
+(57)
+Thus ¯x is an optimal solution to the following convex problem:
+min
+x∈X
+g(x) − g(¯x) − ⟨sh, x − ¯x⟩ + β
+2 ∥x − ¯x∥2
+s.t.
+G(x) − H(¯x) − ⟨sH, x − ¯x⟩ ≤ 0,
+where sh ∈ ∂h(¯x) and sH ∈ ∂H(¯x). This, together with the Slater’s condition due to Assump-
+tion 2, implies that there exists λ ∈ R+ such that (¯x, λ) satisfies the KKT system in Definition
+2.
+Consequently, studying the iteration complexity of Algorithm 1 for computing an approxi-
+mate KKT point of Problem (16) is equivalent to that of the variant of the FW iterations (55)
+for computing a point satisfying (56). However, we cannot expect to achieve a solution that
+satisfies (56) in practice. Instead, we often obtain an approximate solution as shown in the next
+theorem, which can be seen as an approximation of a KKT point of Problem (16). The next
+theorem gives the iteration complexity for achieving an approximate solution.
+20
+
+Theorem 3. Suppose that Assumptions 1 and 2 hold. Let {xk} be the sequence generated by
+Algorithm 1. Then, there exists ℓ ∈ [k] such that
+⟨sq, w − wℓ⟩ + β
+2 ∥w − wℓ∥2
+T ≥ −1
+k
+�
+q(w0) − q∗�
+,
+(58)
+for all w ∈ W and Q(wl) + ⟨sl
+Q, w − wl⟩ ≤ 0, where q∗ ∈ R is the optimal value of Problem
+(54) and sl
+Q ∈ ∂Q(wl).
+Proof. According to Lemma 6, a sequence {wk} generated by iterations (55) satisfies wk =
+(xk, sk, tk) for all k ≥ 0. Since q is a concave function and sk
+q ∈ ∂q(wk), we have
+⟨sk
+q, wk − wk+1⟩ ≤ q(wk) − q(wk+1).
+Averaging the above inequality over k yields
+1
+k
+k
+�
+i=1
+⟨sk
+q, wk − wk+1⟩ ≤ 1
+k
+�
+q(w0) − q(wk+1)
+�
+≤ 1
+k
+�
+q(w0) − q∗�
+,
+where the last inequality follows from the fact that q∗ ∈ R is the optimal value of Problem (54).
+This implies that there exists an index ℓ ∈ {1, . . . , k} such that
+⟨sℓ
+q, wℓ − wℓ+1⟩ ≤ 1
+k
+�
+q(w0) − q∗�
+.
+(59)
+Moreover, it follows from the optimality wk+1 to Problem (55) that for all w ∈ W satisfying
+Q(wℓ) + ⟨sℓ
+Q, w − wℓ⟩ ≤ 0,
+⟨sℓ
+q, wℓ+1 − wℓ⟩ + β
+2 ∥wℓ+1 − wℓ∥2
+T ≤ ⟨sℓ
+q, w − wℓ⟩ + β
+2 ∥wℓ − w∥2
+T .
+This, together with (59), implies that it holds for all w ∈ W satisfying Q(wℓ)+⟨sℓ
+Q, w−wℓ⟩ ≤ 0
+that
+⟨sq, w − wℓ⟩ + β
+2 ∥w − wℓ∥2
+T ≥ ⟨sℓ
+q, wℓ+1 − wℓ⟩ + β
+2 ∥wℓ+1 − wℓ∥2
+T ≥ −1
+k
+�
+q(w0) − q∗�
+.
+We complete the proof.
+We remark that in contrast to Theorems 1 and 2 that require ρ + β > 0, Theorem 3 can
+be applied to analyze the case of ρ + β ≥ 0. It is worth noting that when β = 0, the standard
+iteration complexity of the FW method for general nonconvex problems is O(1/
+√
+k) (see, e.g.,
+[38]), but the iteration complexity of our proposed FW method is improved to O(1/k) as we
+construct a concave minimization surrogate using the DC structure.
+4
+Extensions
+In this section, we first discuss how to extend our approach to solve chance constrained problems
+with the chance constraint estimated by general non-parametric estimation. We then extend
+the proximal DC algorithm for solving Problem (16) with multiple DC constraints, which can
+be used to solve chance constraint programs with multiple chance constraints. Finally, we show
+that our technique can also be used to solve cardinality constrained optimization problems.
+21
+
+4.1
+Extension to L-Estimators of the Empirical Quantile
+In statistics, an L-estimator is a linear combination of order statistics of a sample drawn from
+the population distribution, which plays an important role in non-parametric estimation. The
+main advantage of L-estimators is that they are easy to calculate and often resistant to outliers.
+Due to this, L-estimators have been widely used in the literature; see, e.g., [17, 51, 66]. This
+naturally motivates us to apply the L-estimators to Problem (2).
+To proceed, we formally introduce L-estimators. Suppose that a set of samples {Xi}N
+i=1 is
+i.i.d. according to some unknown distribution FX. In general, L-estimators of the empirical
+quantile take the form �N
+i=1 wiX[i], where w ∈ ∆ :=
+�
+u ∈ RN : 0 ≤ u ≤ 1, 1T u = 1
+�
+.
+In
+statistics, there are many different L-estimators that outperform the empirical quantile in both
+theory and practice; see, e.g., [17, 21, 31, 69]. Then, we consider some typical L-estimators of
+the p empirical quantile for p ∈ (0, 1), i.e., X[M], where M = ⌈pN⌉. The first one is the weighted
+average at X[M−1] (see, e.g., [21, 31]) defined as
+L1 = (1 − g)X[M−1] + gX[M],
+where g = Np − M + 1. Another one is the kernel quantile estimator (see, e.g., [44, 57]) defined
+as
+L2 =
+N
+�
+i=1
+�� i/N
+(i−1)/N
+1
+hK
+�x − p
+h
+�
+dx
+�
+X[i],
+where h > 0 is a constant and K(t) is a kernel function satisfying
+� ∞
+−∞ K(t)dt = 1, K(t) ≥ 0,
+and K(−t) = K(t). It is worth noting that this kernel quantile estimator can be viewed as a
+smoothing version of the empirical quantile estimator.
+Now, we apply L-estimators to the SAA of the chance constrained program. Specifically,
+replacing the the empirical quantile �C[M](x) in Problem (10) with its L-estimator yields the
+following problem:
+min
+x∈X
+�
+f(x) :
+N
+�
+i=1
+wi �C[i](x) ≤ 0
+�
+,
+(60)
+where the weight w ∈ ∆ is given. It is worth pointing out that Problem (2) is actually a special
+case of Problem (60) by taking wM = 1 and wi = 0 for all i ̸= M. Then, we reformulate this
+problem into a DC constrained DC program. Before we proceed, let
+¯Z :=
+�
+x ∈ Rn :
+N
+�
+i=1
+wi �C[i](x) ≤ 0
+�
+.
+(61)
+Similar to Lemma 1, we can also express the above constraint as a DC constraint.
+Lemma 8. Let
+G(x) :=
+N
+�
+i=1
+wi
+N
+�
+j=i
+�C[j](x), H(x) :=
+N−1
+�
+i=1
+wi
+N
+�
+j=i+1
+�C[j](x),
+(62)
+22
+
+where w ∈ ∆.
+Then, G and H are both continuous and convex functions, and the chance
+constraint in ¯Z is equivalent to a DC constraint
+G(x) − H(x) ≤ 0.
+Proof. Using the argument in Lemma 1, we can show that �N
+j=i �C[j](x) for i = 1, . . . , N are
+convex functions. Since each of G and H in (62) is a positive weighted sum of convex functions,
+G and H are both convex functions. According to (15), we have for i = 1, . . . , N − 1,
+�C[i](x) =
+N
+�
+j=i
+�C[j](x) −
+N
+�
+j=i+1
+�C[j](x).
+This yields that
+N
+�
+i=1
+wi �C[i](x) =
+N−1
+�
+i=1
+wi �C[i](x) + wN �C[N](x) =
+N−1
+�
+i=1
+wi
+
+
+N
+�
+j=i
+�C[j](x) −
+N
+�
+j=i+1
+�C[j](x)
+
+ + wN �C[N](x)
+=
+N
+�
+i=1
+wi
+N
+�
+j=i
+�C[j](x) −
+N−1
+�
+i=1
+wi
+N
+�
+j=i+1
+�C[j](x) = G(x) − H(x).
+We then obtain a DC constrained DC program for L-estimators of the empirical quantile.
+Consequently, we can still apply the proposed pDCA for solving the resulting problem.
+4.2
+Extension to Multiple DC Constraints
+In this subsection, we consider that Problem (16) has multiple DC constraints
+Gi(x) − Hi(x) ≤ 0, for i = 1, . . . , K,
+(63)
+where Gi : Rn → R and Hi : Rn → R are continuous and convex functions. That is, we consider
+the problem
+min
+x∈X
+f(x) := g(x) − h(x)
+s.t.
+Gi(x) − Hi(x) ≤ 0, for i = 1, . . . , K.
+(64)
+We can still apply the proximal DC algorithm for solving this problem. Specifically, suppose
+that an initial point x0 ∈ X satisfying Gi(x0) − Hi(x0) ≤ 0, i = 1, . . . , K is available. At the
+k-th iteration, we choose sk
+h ∈ ∂h(xk) and sk
+Hi ∈ ∂Hi(xk) for i = 1, . . . , K, and generate the
+next iterate xk+1 by solving the following convex subproblem
+xk+1 ∈ argmin
+x∈X
+g(x) − h(xk) − ⟨sk
+h, x − xk⟩ + β
+2 ∥x − xk∥2
+s.t.
+Gi(x) − Hi(xk) − ⟨sk
+Hi, x − xk⟩ ≤ 0, for i = 1, . . . , K,
+(65)
+where β ≥ 0 is a penalty parameter. In particular, we can also prove subsequential convergence
+to a KKT point for the proximal DC algorithm by assuming the following generalized MFCQ:
+23
+
+Assumption 3 (Generalized MFCQ). The generalized MFCQ holds for Problem (64), i.e.,
+there exists y ∈ X such that for Gi(x) = Hi(x),
+Gi(y) − Hi(x) − ⟨sHi, y − x⟩ < 0, for all sHi ∈ ∂Hi(x), i = 1, . . . , K,
+⟨∇ωi(x), y − x⟩ < 0, for all i ∈ A(x).
+Using the similar argument in Section 3.1, we can obtain the following result:
+Corollary 1. Suppose that Assumptions 1 and 3 hold, the function f is given in Problem (64),
+X is of the form of (26), and the level set
+�
+x ∈ X : f(x) ≤ f(x0), Gi(x) − Hi(x) ≤ 0, for i = 1, . . . , K
+�
+is bounded. Let {xk} be the sequence generated by (65) with ρ+β > 0. Then, any accumulation
+point of {xk} is a KKT point of Problem (64).
+We remark that although it is pointed out in [56] that multiple DC constraints can be
+combined into a single nondifferentiable DC constraint using the max-function, it makes the
+constraint more complicated and thus a more difficult pDCA subproblem.
+4.3
+Extension to Cardinality Constrained Optimization Problems
+We consider the following cardinality constrained optimization problems:
+min
+x∈Rn {f(x) : ∥x∥0 ≤ K, x ∈ X} ,
+(66)
+where ∥x∥0 denotes the cardinality of the vector, and K is an interger satisfying 1 ≤ K ≤ N −1.
+This problem has found wide applications in diverse fields, such as quantitative finance [7, 22]
+and signal processing [13], to name a few. By introducing an auxiliary variable z ∈ Rn, the
+cardinality constraint ∥x∥0 ≤ K is equivalent to z[N−K] ≤ 0, zi = |xi| for all i = 1, . . . , n. This
+implies that we can rewrite Problem (66) as
+min
+x∈X,z∈Rn
+�
+f(x) : z[N−K] ≤ 0, xi − zi ≤ 0, −xi − zi ≤ 0, i = 1, . . . , n
+�
+.
+As in Lemma 1, we can further rewrite the constraint z[N−K] ≤ 0 into a DC constraint. Then,
+we can apply the proposed approach for solving the resulting problem.
+5
+Experimental Results
+In this section, we conduct experiments to study the performance of our proposed method
+on both synthetic and real data sets. For ease of reference, we denote our proposed method
+by pDCA (resp.
+DCA) when β > 0 (resp.
+β = 0) in Algorithm 1.
+We also compare our
+methods with some state-of-the-art methods, which are CVaR in [53], the bisection-based CVaR
+method3 (Bi-CVaR) in [5, Section 4.1], mixed-integer program (MIP) in [1], an augmented
+3The bisection based CVaR method is a heuristic approach that can improve the performance of CVaR.
+24
+
+Lagrangian decomposition method (ALDM) in [5], and a DC approximation-based successive
+convex approximation method (SCA) in [29]. Our codes are implemented in MATLAB 2022b.
+In particular, we use the optimization solver Gurobi (version 9.5.2) for solving linear, quadratic,
+and mixed integer subproblems. All the experiments are conducted on a Linux server with
+256GB RAM and 24-core AMD EPYC 7402 2.8GHz CPU.
+For pDCA, we update the penalty parameter β in an adaptive manner. That is, we set
+βk+1 = βk/4 for k = 0, 1, 2, . . . For pDCA on each data set, we explore two different settings of
+the regularization parameter β0, which will be specified later. We respectively denote them by
+pDCA-1 and pDCA-2. We set the parameters of the remaining methods as those provided in
+the corresponding papers. For the tested methods DCA, pDCA, Bi-CVaR, ALDM, and SCA,
+we use the point returned by CVaR as their starting point. In each test, we terminate the
+tested methods when |f k − f k+1|/ max{1, |f k+1|} ≤ 10−6, for k = 0, 1, 2, . . . , or the running
+time reaches 1800 seconds.4
+5.1
+VaR-Constrained Portfolio Selection Problem
+In this subsection, we study the VaR-constrained mean-variance portfolio selection problem,
+which aims to minimize the risk while pursuing a targeted level of returns with probability at
+least 1 − α. Let µ ∈ Rn and Σ ∈ Rn×n respectively denote expectation and covariance matrix
+of the returns of n risky assets, and γ ∈ R+ denote the risk aversion factor. By letting x ∈ Rn
++
+denote the allocation vector such that the weight of the i-th risky asset is xi for i ∈ [n], this
+problem is formulated as follows:
+min
+x∈Rn γxT Σx − µT x
+s.t.
+P
+�
+ξT x ≥ R
+�
+≥ 1 − α,
+n
+�
+i=1
+xi = 1, 0 ≤ xi ≤ u, i = 1, . . . , n,
+(67)
+where R ∈ R+ is a prespecified level on the return and u ∈ R+ is an upper bound on the
+weights.
+We use 2523 daily return data of 435 stocks included in Standard & Poor’s 500 Index between
+March 2006 and March 2016, which can be downloaded from https://sem.tongji.edu.cn/semch_data/faculty_cv/xjz/ccop.html.
+Following [5], we generate the data input by choosing n = 100, 200, 300, 400, respectively. For
+each n, we generate 5 instances from the daily return data set by randomly selecting n stocks
+from the 435 stocks and N = 3n samples ˆξℓ for all ℓ ∈ [N] from the 2523 daily return data.
+Then, we compute the sample mean µ and sample covariance matrix Σ using these data. We
+set the remaining parameters as follows: R = 0.02%, γ = 2, and u = 0.5. In the tests, we set
+the initial regularization parameter β0 of pDCA-1 and pDCA-2 as 0.1 and 1, respectively. In
+Table 1 and the other two tables below for the other two experiments, we use “fval” to denote
+the averaged returned objective value for the test problems, “time” the averaged CPU time (in
+seconds), and “prob” the empirical in-sample probability of the chance constraint, all of which
+4Since we only check the running time at the end of each iteration, the actual finishing time of an algorithm
+may be longer than this limit.
+25
+
+are averaged over 5 instances. We highlight the best values except those of MIP and CVaR
+for items “fval” and “time” since MIP is not suitable for large-scale data sets and the solution
+returned by CVaR is too conservative.
+Table 1: Comparison of different methods on the portfolio selection problem.
+(α,n)
+MIP
+CVaR
+Bi-CVaR
+DCA
+pDCA-1
+pDCA-2
+ALDM
+SCA
+�
+0.05
+100
+�
+fval
+-1.3550
+-1.1861
+-1.2592
+-1.2860
+-1.2897
+-1.3037
+-1.3221
+-1.2732
+time
+35.87
+0.1271
+1.868
+0.4603
+0.7387
+0.9553
+3.576
+0.8343
+prob
+0.9500
+0.9887
+0.9500
+0.9627
+0.9587
+0.9587
+0.9420*
+0.9593
+�
+0.05
+200
+�
+fval
+-1.3531
+-1.1914
+-1.2754
+-1.2950
+-1.2923
+-1.3066
+-1.3284
+-1.2787
+time
+1800
+0.3778
+5.013
+1.683
+1.808
+2.861
+9.901
+2.589
+prob
+0.9500
+0.9873
+0.9500
+0.9553
+0.9560
+0.9560
+0.9447*
+0.9580
+�
+0.05
+300
+�
+fval
+-1.3484
+-1.1830
+-1.2629
+-1.2935
+-1.2835
+-1.2934
+-1.3279
+-1.2525
+time
+1800
+0.9473
+12.26
+7.403
+6.188
+8.749
+19.59
+6.890
+prob
+0.9500
+0.9853
+0.9500
+0.9529
+0.9553
+0.9553
+0.9456*
+0.9584
+�
+0.05
+400
+�
+fval
+-1.3719
+-1.1939
+-1.2886
+-1.3143
+-1.3206
+-1.3291
+-1.3150
+-1.2775
+time
+1800
+1.861
+26.61
+20.07
+15.87
+16.46
+24.01
+16.26
+prob
+0.9502
+0.9860
+0.9500
+0.9547
+0.9512
+0.9512
+0.9467*
+0.9595
+�
+0.1
+100
+�
+fval
+-1.4429
+-1.2284
+-1.3781
+-1.3699
+-1.3761
+-1.3839
+-1.3545
+-1.3826
+time
+7.376
+0.1262
+1.875
+0.7790
+0.7084
+0.9591
+0.7826
+0.8081
+prob
+0.9000
+0.9687
+0.9007
+0.9140
+0.9113
+0.9113
+0.9093
+0.9153
+�
+0.1
+200
+�
+fval
+-1.4244
+-1.2371
+-1.3815
+-1.3772
+-1.3764
+-1.3934
+-1.3266
+-1.3827
+time
+1225
+0.3467
+5.093
+3.385
+3.040
+4.350
+0.3601
+3.582
+prob
+0.9000
+0.9620
+0.9007
+0.9087
+0.9127
+0.9127
+0.9193
+0.9103
+�
+0.1
+300
+�
+fval
+-1.4410
+-1.2284
+-1.3999
+-1.4015
+-1.3959
+-1.4052
+-1.3000
+-1.3899
+time
+1800
+0.9493
+12.32
+14.44
+11.43
+11.18
+0.8458
+11.16
+prob
+0.9000
+0.9633
+0.9000
+0.9053
+0.9042
+0.9042
+0.9353
+0.9107
+�
+0.1
+400
+�
+fval
+-1.4694
+-1.2467
+-1.4200
+-1.4352
+-1.4316
+-1.4262
+-1.3017
+-1.4190
+time
+1800
+1.833
+26.42
+31.05
+32.69
+27.70
+0.9201
+27.62
+prob
+0.9000
+0.9653
+0.9002
+0.9047
+0.9067
+0.9067
+0.9412
+0.9100
+“*” indicates that the computed probability is lower than the targeted level in Problem (2), which implies the
+returned solution is not feasible (the same for Table 2 and 3). The magnitude of fval is 10−2.
+We observe from Table 1 that although MIP achieves the lowest objective value, it is the
+most time-consuming. In addition, we observe that pDCA is slightly better than DCA and
+both pDCA and DCA generally outperform CVaR, Bi-CVaR, ALDM, and SCA in terms of
+the objective value. Table 1 also demonstrates that CVaR is the fastest method, while DCA
+and pDCA are comparable to the remaining ones. Finally, we also observe that the in-sample
+probabilities of DCA and pDCA are generally comparable to those of the other methods, except
+that ALDM fails to satisfy the chance constraint for α = 0.05 and sometimes is too conservative
+for α = 0.1.
+26
+
+5.2
+Probabilistic Transportation Problem with Convex Objective
+In this subsection, we consider a probabilistic version of the classical transportation problem,
+which has been widely studied in the literature; see, e.g., [5, 50]. This problem is to minimize
+the transportation cost of delivering products from n suppliers to m customers. The customer
+demands are random and the j-th customer’s demand is represented by a random variable ξj
+for each j ∈ {1, . . . , m}. The i-th supplier has a limited production capacity θi ∈ R+ for each
+i ∈ {1, . . . , n}. The cost of shipping a unit of product from supplier i ∈ {1, . . . , n} to customer
+j ∈ {1, . . . , m} is cij ∈ R+. Suppose that the shipment quantities are required to be determined
+before the customer demands are known. By letting xij denote the amount of shipment delivered
+from supplier i ∈ {1, . . . , n} to customer j ∈ {1, . . . , m}, this problem is formulated as
+min
+x∈Rn×m
+n
+�
+i=1
+m
+�
+j=1
+cijxij
+s.t.
+P
+� n
+�
+i=1
+xij ≥ ξj, j = 1, . . . , m
+�
+≥ 1 − α,
+m
+�
+j=1
+xij ≤ θi, xij ≥ 0, i = 1, . . . , n, j = 1, . . . , m.
+(68)
+Table 2: Comparison of different methods on the probabilistic transportation problem.
+(α,N)
+MIP
+CVaR
+Bi-CVaR
+DCA
+pDCA-1
+pDCA-2
+ALDM
+SCA
+�
+0.05
+500
+�
+fval
+4.2584
+4.3843
+4.3700
+4.3262
+4.3239
+4.3251
+4.7091
+4.1716
+time
+73.89
+1.796
+22.84
+3.681
+405.2
+503.1
+58.76
+6.697
+prob
+0.9500
+1.0000
+0.9504
+0.9500
+0.9500
+0.9500
+0.9504
+0.8180*
+�
+0.05
+1000
+�
+fval
+4.3655
+4.5423
+4.4931
+4.4445
+4.4435
+4.4467
+4.8644
+4.4447
+time
+543.0
+2.818
+44.35
+5.895
+2441
+1915
+50.63
+73.90
+prob
+0.9500
+0.9984
+0.9500
+0.9500
+0.9500
+0.9500
+0.9636
+0.9312*
+�
+0.05
+1500
+�
+fval
+4.3946
+4.6120
+4.5067
+4.4631
+4.4742
+4.4891
+4.8634
+4.5818
+time
+891.6
+4.34
+70.75
+12.66
+1925
+2002
+44.63
+261.5
+prob
+0.9500
+0.9980
+0.9504
+0.9500
+0.9500
+0.9500
+0.9787
+0.9508
+�
+0.05
+2000
+�
+fval
+4.4167
+4.6538
+4.5199
+4.4898
+4.5063
+4.5391
+4.8597
+4.5488
+time
+1535
+5.959
+95.60
+14.99
+2310
+2447
+46.52
+336.7
+prob
+0.9500
+0.9848
+0.9504
+0.9500
+0.9500
+0.9500
+0.9843
+0.9515
+�
+0.1
+500
+�
+fval
+4.1874
+4.3833
+4.3262
+4.2591
+4.2548
+4.2548
+4.7110
+4.3092
+time
+171.6
+1.626
+24.75
+4.521
+528.5
+591.7
+42.70
+65.16
+prob
+0.9000
+0.9916
+0.9000
+0.9000
+0.9000
+0.9000
+0.9812
+0.9008
+�
+0.1
+1000
+�
+fval
+4.2790
+4.5306
+4.3869
+4.3617
+4.3590
+4.3633
+4.8027
+4.4135
+time
+674.5
+2.928
+47.76
+9.151
+1944
+1921
+44.59
+164.868
+prob
+0.9000
+0.9684
+0.9002
+0.9000
+0.9000
+0.9000
+0.9682
+0.9028
+�
+0.1
+1500
+�
+fval
+4.3031
+4.5473
+4.3975
+4.3694
+4.3753
+4.3937
+4.7085
+4.4092
+time
+1673
+5.073
+74.30
+11.84
+1899
+1954
+46.92
+326.652
+prob
+0.9000
+0.9633
+0.9000
+0.9000
+0.9000
+0.9000
+0.9628
+0.9041
+�
+0.1
+2000
+�
+fval
+4.3212
+4.5638
+4.3998
+4.3805
+4.4010
+4.4280
+4.7992
+4.4406
+time
+1801
+5.982
+102.8
+14.08
+2217
+2190
+51.36
+507.0
+prob
+0.9000
+0.9636
+0.9001
+0.9000
+0.9000
+0.9000
+0.9630
+0.9110
+The magnitude of fval is 107.
+27
+
+In our experiments, we use the setting in Luedtke et al. [50] to generate parameters (θ, c, ˆξ),
+which is downloaded from http://homepages.cae.wisc.edu/~luedtkej/. In particular, we
+choose (n, m) = (40, 100) and N = 500, 1000, 1500, 2000. We set β0 = 1, 10 for pDCA-1 and
+pDCA-2, respectively. We report the experimental results in Table 2. We observe that DCA
+and pDCA in general can find significantly better solutions than CVaR and ALDM, and slightly
+better solutions than Bi-CVaR and SCA in terms of objective values. Meanwhile, we see that
+MIP returns either global optimal solutions or best objective values among all the algorithms
+in the time limit. We also observe that the CPU time of the DCA is less than Bi-CVaR and
+ALDM, much less than that of MIP and pDCA, and is slightly larger than that of CVaR.
+We should mention that pDCA is the most time-consuming among the tested methods, since
+it solves a quadratic programming subproblem in each iteration, while other methods solve a
+linear programming subproblem. Table 2 also indicates that the in-sample probabilities of DCA
+and pDCA are exactly the risk level 1 − α in all instances, while the in-sample probabilities of
+ALDM and SCA may be either too loose or too conservative.
+5.3
+Probabilistic Transportation Problem with Non-Convex Objective
+In this subsection, we consider a probabilistic version of the classical transportation problem
+with a non-convex objective function, which has been studied in [5, 20]. In particular, the setting
+of this problem is exactly the same as that in Section 5.2 except for the objective function. Here,
+we assume that the transportation cost from supplier i to customer j consists of the normal
+cost cijxij and cost discount aijx2
+ij (aij < 0). Consequently, this problem can be formulated as
+min
+x∈Rn×m
+n
+�
+i=1
+m
+�
+j=1
+cijxij + aijx2
+ij
+s.t.
+P
+� n
+�
+i=1
+xij ≥ ξj, j = 1, . . . , m
+�
+≥ 1 − α,
+m
+�
+j=1
+xij ≤ θi, xij ≥ 0, i = 1, . . . , n, j = 1, . . . , m,
+(69)
+In our test, we set aij = −cij/ (2θi) for all i, j, and the remaining setting is the same as that in
+the last section.
+Since the objective function of this problem is non-convex, CVaR and Bi-CVaR cannot
+handle this problem. Then, we only compare our proposed method with MIP, ALDM, and
+SCA. To generate a feasible initial point, we apply CVaR to solve Problem (69) without cost
+discount in the objective function. We report the experimental results in Table 3, which are
+similar to those of Table 2.
+We further point out that although MIP achieves the lowest
+objective value, it reaches the time limit for all the instances, which indicates the hardness of
+the additional non-convex term in the objective. In terms of objective values and running time,
+we observe that DCA generally outperforms pDCA, ALDM, and SCA in most of cases. The
+CPU time for DCA is similar to that of the convex case in Table 2. The reason is similar, i.e.,
+the subproblems of DCA are all linear programs like the convex cases in (68). We also observe
+that the in-sample probabilities of DCA and pDCA are generally closer to the risk level 1 − α
+than ALDM and SCA in all instances.
+28
+
+Table 3: Comparison of different methods on the probabilistic transportation problem with a
+non-convex objective function.
+(α,N)
+MIP
+DCA
+pDCA-1
+pDCA-2
+ALDM
+SCA
+�
+0.05
+500
+�
+fval
+3.5098
+3.6012
+3.5973
+3.5962
+4.0023
+3.4808
+time
+1805
+7.448
+340.7
+458.8
+267.6
+8.42
+prob
+0.9500
+0.9500
+0.9500
+0.9500
+0.9504
+0.8180*
+�
+0.05
+1000
+�
+fval
+3.5868
+3.6830
+3.6822
+3.7027
+4.1015
+3.6819
+time
+1803
+15.76
+1989
+1851
+178.6
+87.53
+prob
+0.9500
+0.9500
+0.9500
+0.9500
+0.9714
+0.9318*
+�
+0.05
+1500
+�
+fval
+3.6123
+3.6888
+3.7170
+3.7455
+3.9974
+3.7691
+time
+1803
+23.12
+1927
+1986
+186.5
+309.4
+prob
+0.9500
+0.9500
+0.9500
+0.9500
+0.9845
+0.9504
+�
+0.05
+2000
+�
+fval
+3.6237
+3.7133
+3.7575
+3.7882
+4.0842
+3.7481
+time
+1803
+33.04
+2381
+2381
+147.4
+412.9
+prob
+0.9500
+0.9500
+0.9500
+0.9500
+0.9845
+0.9505
+�
+0.1
+500
+�
+fval
+3.4581
+3.5473
+3.5436
+3.5421
+4.0195
+3.5784
+time
+1804
+8.845
+413.1
+405.3
+175.4
+67.68
+prob
+0.9000
+0.9000
+0.9000
+0.9000
+0.9904
+0.9016
+�
+0.1
+1000
+�
+fval
+3.5238
+3.6224
+3.6229
+3.6406
+3.9981
+3.6503
+time
+1802
+16.20
+1888
+1949
+151.1
+201.2
+prob
+0.9000
+0.9000
+0.9000
+0.9000
+0.9684
+0.9010
+�
+0.1
+1500
+�
+fval
+3.5427
+3.6231
+3.6482
+3.6779
+4.0223
+3.6499
+time
+1802
+25.45
+1896
+1976
+177.2
+401.5
+prob
+0.9000
+0.9000
+0.9000
+0.9000
+0.9629
+0.9007
+�
+0.1
+2000
+�
+fval
+3.5521
+3.6281
+3.6775
+3.7071
+4.0006
+3.6647
+time
+1802
+27.14
+2242
+2248
+156.4
+612.6
+prob
+0.9000
+0.9000
+0.9000
+0.9032
+0.9631
+0.9114
+The magnitude of fval is 107.
+6
+Conclusions
+In this paper, we proposed a new DC reformulation based on the empirical quantile for solving
+data-driven chance constrained programs and proposed a proximal DC algorithm to solve it. We
+proved the subsequential and sequential convergence to a KKT point of the proposed method
+and derived the iteration complexity for computing an approximate KKT point. We point out
+that our analysis holds for general DC constrained DC programs beyond those reformulated
+from chance constrained programs and can be extended to DC programs with multiple DC
+constraints. We also show possible extensions of our methods to L-estimators for quantile in
+chance constrained programs and cardinality constrained programs. Finally, we demonstrated
+the efficiency and efficacy of the proposed method via numerical experiments.
+29
+
+Acknowledgements
+We would like to thank Lai Tian (The Chinese University of Hong Kong) for the fruitful dis-
+cussion of the theoretical analysis of this work.
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diff --git a/gNAyT4oBgHgl3EQfj_gw/content/tmp_files/load_file.txt b/gNAyT4oBgHgl3EQfj_gw/content/tmp_files/load_file.txt
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@@ -0,0 +1,1869 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf,len=1868
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='00423v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='OC]  1 Jan 2023  Difference-of-Convex Reformulation for Chance Constrained  Programs  Peng Wang∗  Rujun Jiang†  Qingyuan Kong‡  Laura Balzano§  January 3, 2023  Abstract  Chance constrained programming refers to a type of optimization problem with uncertain  constraints that are satisfied with at least a prescribed probability level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  In this work,  we study the sample average approximation of a class of chance constrained programs, in  which the objective function is a difference-of-convex (DC) function and the functions in the  chance constraint are all convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Utilizing the empirical quantile of the chance constraint,  we reformulate it into a DC constrained DC program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We propose a proximal DC algorithm  (pDCA) for solving the reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Moreover, we prove the global convergence of the  proposed algorithm based on the Kurdyka-�Lojasiewicz property and derive the iteration  complexity for finding an approximate KKT point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We point out that the proposed pDCA  and its associated analysis apply to any DC constrained DC programs, which may be of  independent interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To support and complement our theoretical development, we show  via numerical experiments that the proposed approach is competitive with a host of existing  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  1  Introduction  Chance constrained programming is a powerful modeling paradigm for optimization problems  with uncertain parameters, which has found wide applications in diverse fields, such as finance  [10, 54], power systems [8, 75], and supply chain [67, 42], to name a few;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [36] and  the references therein for more applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  In general, the chance constrained program is  to minimize a targeted loss subject to the probability of violating uncertain constraints being  within a prespecified risk level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In this work, we consider a chance constrained program of the  form  min  x∈X {f(x) : P (ci(x, ξ) ≤ 0, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m}) ≥ 1 − α} ,  (1)  ∗Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (pengwa@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' School of Data Science, Fudan University, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' (rjjiang@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  ‡School of Data Science, Fudan University, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' (qykong21@m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  §Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (girasole@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=')  1  where the vector x ∈ Rn denotes the decision variables, the functions f : Rn → R and ci :  Rn × Rd → R for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m} are real-valued, the set X ⊆ Rn is deterministic, and  ξ ∈ Rd is a random vector with its probability distribution supported on some set Ξ ⊆ Rd,  and α ∈ (0, 1) is a given risk parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This problem is known as a single chance constrained  program if m = 1, and a joint chance constrained program otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Throughout this paper,  we make the following assumptions for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' (a) The function f takes the form of f = g − h, where g and h are continuous  and convex (possibly non-smooth) functions defined on an open set that contains X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The func-  tion g is ρ-strongly convex for some ρ ≥ 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (b) The set X is non-empty, closed, and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (c) The functions ci(x, ξ) : Rn × Ξ → R for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m are convex and continuously differen-  tiable in x for every ξ ∈ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  In general, Problem (1) is highly intractable due to the chance constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Indeed, the  feasible region formed by the chance constraint may be non-convex despite that ci(x, ξ) for  all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m} are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' For example, even if ci(x, ξ) for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m} are all linear  in x and X is a polyhedron, the resulting feasible region defined by the chance constraint  may not be convex [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Moreover, it is generally impossible to compute the probability of  satisfying the constraint for a given x ∈ X when the distribution of ξ is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In this work,  we study Problem (1) in the data-driven setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', a set of independently and identically  distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=') samples {ˆξi}N  i=1 generated according to the distribution of ξ is available,  while its distribution is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Motivated by the recent work on chance constrained programs  [1, 49, 55, 58], we consider a sample average approximation (SAA) of Problem (1) over the  samples {ˆξi}N  i=1, which takes the form of  min  x∈X  �  f(x) :  1  N  N  �  i=1  1{C(x, ˆξi)≤0} ≥ 1 − α  �  ,  (2)  where C(x, ξ) := max {ci(x, ξ) : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m}, and where  1A is the indicator of event A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We  should mention that Problem (2) also includes the scenario where the distribution is finite  and discrete, and each event appears with probability 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In particular, it has been shown  in [49, 55] that solving Problem (2) can return a good approximate solution of Problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  However, Problem (2) is hard to optimize due to the discreteness of the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Note  that we assume that the objective function f in Problem (2) is a difference-of-convex (DC)  function and all the functions ci(·, ξ) in the constraint are convex in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, a  natural question is whether we can utilize these structures to develop an effective algorithmic  framework for solving Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In this work, we answer this question in the affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Exploiting these structures, we reformulate Problem (2) into a DC constrained DC problem and  propose a proximal DC algorithm for solving the reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In the literature, the existing  approaches for solving Problem (2) generally can only prove subsequential convergence and  have no iteration complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In contrast with these results, we not only prove the  1If ρ = 0, g is a general convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  2  subsequential and entire convergence to a Karush-Kuhn-Tucker (KKT) point of the proposed  algorithm but also derive the iteration complexity for finding an approximate KKT point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1  Related Works  We first review some popular methods for solving chance constrained programs and then briefly  talk about some DC algorithms closely related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Since the first appearance of chance  constrained programs in [15, 16], various algorithms for solving chance constrained problems  under different settings have been proposed in a substantial body of literature over the past  years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' One well-known approach for solving Problem (1) is to reformulate the chance constraint  into a convex constraint when the distribution of ξ is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  For example, Henrion [25,  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2] showed that the chance constraint can be reformulated into a second-order cone  if C(x, ξ) = ⟨ξ, x⟩ − b, ξ has an elliptical symmetric distribution, and b is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We refer  the reader to [24, 39, 12, 26, 61] for more results on the convexity of the feasible region formed  by chance constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' These convex reformulations generally require a special distribution on  random vector ξ, such as Gaussian or log-concave distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  However, in practice, sometimes only a few random samples from the distribution of ξ are  available while the distribution of ξ is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To handle this scenario, one popular approach  is to consider the SAA of the problem (see Problem (2)), which is obtained by replacing the  true distribution with an empirical distribution corresponding to random samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Luedtke  and Ahmed [49] showed that the SAA with a risk level smaller than the required risk level can  obtain a solution satisfying a chance constraint with high probability under suitable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Later, Pagnoncelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [55] showed that a solution of the SAA problem converges to that  of the original problem with probability approaching one as N goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Despite the  fact that it possesses nice convergence properties, the SAA problem (2) is generally difficult to  optimize due to its discrete nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To solve it, many different approaches have been proposed  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' For example, Ahmed and Shapiro [1] proposed a mixed-integer programming  (MIP) reformulation for the SAA problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see also [36, 49, 50, 64] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [19] proposed a sequential algorithm, which minimizes quadratic subproblems with  linear cardinality constraints iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [5] proposed an augmented Lagrangian  decomposition method for solving Problem (1) when ξ has a finite discrete distribution and  cj(·, ξ) for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m are all affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Recently, Pe˜na-Ordieres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [58] proposed a smoothing  non-linear approximation of Problem (2) based on the empirical quantile of the chance constraint  and developed a Sℓ1QP-type trust-region method to solve the approximation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Using  a similar idea, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [65] proposed a neural network model to approximate the empirical  quantile of the chance constraint and employed a simulated annealing algorithm for solving  the approximation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In general, some methods, such as [65], are heuristic in nature,  and some other works, such as [5, 19, 58], only establish subsequential convergence for their  proposed methods and have no iteration complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  The scenario approximation  approach proposed by Calafiore and Campi [11], Nemirovski and Shapiro [52] is another well-  known sample-based approach for solving Problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This approach is simple and easy to  implement, but it suffers from the solution becoming more and more conservative as the sample  3  size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Another notable approach for solving Problem (1) is to consider its conservative and tractable  approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Among these approximations, the most famous one is the condition value-at-  risk (CVaR) approximation proposed by Nemirovski and Shapiro [53], which is based on a  conservative and convex approximation of the indication function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In particular, Hong and Liu  [28] proposed a gradient-based Monte Carlo method for solving the CVaR approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To  avoid overly conservative solutions, Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [29] studied a DC approximation of the chance  constraint and tackled it by solving a sequence of convex approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Xie and Ahmed  [71] proposed a bicriteria approximation for solving chance constrained covering problems and  proved a constant factor approximation guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' More recently, Jiang and Xie [32] proposed  a convex approximation named ALSO-X that always outperforms the CVaR approximation  when uncertain constraints are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' There are also other approximations based on different  techniques in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [14, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  DC constrained DC programs2 refer to optimization problems that minimize a DC function  subject to constraints defined by DC functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Such problems have been extensively studied  in the literature for decades [60, 30, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' One of the most popular methods for solving DC  programs is the DC algorithm and its variants, which solve a sequence of convex subproblems  by linearizing the second component of DC functions [29, 47, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Le Thi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [41] proposed a  penalty method and a DC algorithm using slack variables and showed that every accumulation  point of the generated sequence is a KKT point of the considered problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Later, Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [56]  studied the proximal linearized method for DC programs and showed that every accumulation  point of the generated sequence is a Bouligand-stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Recently, van Ackooij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [68]  developed a proximal bundle method for addressing DC programs and analyzed its convergence  under different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [48] proposed penalty and augmented Lagrangian methods for  solving DC programs, and established strong convergence guarantees for the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2  Our Contributions  In this work, we study the SAA of the chance constrained program when the distribution of  ξ is unknown but a set of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  samples {ˆξi}N  i=1 generated according to its distribution is  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' First, we reformulate the SAA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', Problem (2)) of the chance constrained program  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', Problem (1)) into a DC constrained DC program by utilizing Assumption 1 and the  empirical quantile function of C(x, ξ) over the samples {ˆξi}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Second, we propose a proximal  DC algorithm (pDCA) for solving the reformulation, which proceeds by solving a sequence of  convex subproblems by linearizing the second component of the obtained DC functions and  adding a proximal term to the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In particular, we show that it is easy to  compute the required subgradients by using the structure of the DC functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Moreover, the  obtained subproblem can be rewritten in a form that is suitable for off-the-shelf solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Finally,  we analyze the convergence and iteration complexity of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Specifically, we  show that any accumulation point of the sequence generated by the proposed method is a  2For simplicity, we also call it DC programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  4  KKT point of the reformulated problem under a constraint qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, we establish  the convergence and convergence rate of the entire sequence by using the Kurdyka-�Lojasiewicz  (K�L) inequality with the associated exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Moreover, we further show that the obtained  DC program is equivalent to a convex constrained problem with a concave objective, which  is amenable to the Frank-Wolfe (FW) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' By further showing the equivalence between  proximal DC iterations for solving the DC program and modified FW iterations for solving  the equivalent problem, we derive the iteration complexity of the pDCA for computing an  approximate KKT point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In particular, in contrast to the standard iteration complexity of the  FW method O(1/  √  k) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [38]), the iteration complexity of our considered FW method  is improved to O(1/k) by utilizing the DC structure, where k is the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In Section 2, we reformulate Problem (2) into  a DC constrained DC program and introduce the proposed pDCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In Section 3, we analyze the  convergence and iteration complexity of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In Section 4, we discuss some  extensions of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In Section 5, we report the experimental results of the proposed  method and other existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We end the paper with some conclusions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='3  Notation and Definitions  Besides the notation introduced earlier, we shall use the following notation throughout the  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We write matrices in bold capital letters like A, vectors in bold lower-case letters like a,  and scalars in plain letters like a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Given a matrix A ∈ Rm×n, we use aij to denote its (i, j)-th  element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Given a vector x ∈ Rn, we use ∥x∥ to denote its Euclidean norm, xi its i-th element,  and x[M] its M-th smallest element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We use 1 and 0 to denote the all-one vector and all-zero  vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Next, we introduce some concepts in non-smooth analysis that will be needed in our sub-  sequent development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The details can be found in [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let ϕ : Rn → (−∞, ∞] be a given  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We say that the function ϕ is proper if dom(ϕ) := {x ∈ Rn : ϕ(x) < ∞} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' A  vector s ∈ Rn is said to be a Fr´echet subgradient of ϕ at x ∈ dom(ϕ) if  lim inf  y→x,y̸=x  ϕ(y) − ϕ(x) − ⟨s, y − x⟩  ∥y − x∥2  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (3)  The set of vectors s ∈ Rn satisfying (3) is called the Fr´echet subdifferential of f at x ∈ dom(ϕ)  and denoted by �∂ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The limiting subdifferential, or simply the subdifferential, of ϕ at x ∈  dom(ϕ) is defined as  ∂ϕ(x) =  �  s ∈ Rn : ∃xk → x, sk → v with ϕ(xk) → ϕ(x), sk ∈ �∂ϕ(xk)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  When ϕ is proper and convex, thanks to [63, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='12], the limiting subdifferential of  ϕ at x ∈ dom(ϕ) coincides with the classic subdifferential defined as  ∂ϕ(x) = {s ∈ Rn : ϕ(y) ≥ ϕ(x) + ⟨s, y − x⟩, for all y ∈ Rn} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (4)  5  For a non-empty set S ⊆ Rn, its indicator function δS : Rn → {0, +∞} is defined as  δS(x) =  \uf8f1  \uf8f2  \uf8f3  0,  if x ∈ S,  +∞,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Its normal cone (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' regular normal cone) at x ∈ S is defined as NS(x) := ∂δS(x) (reps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  �  NS(x) := �∂δS(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Given a point x ∈ Rn, its distance to S is defined as dist(x, S) = infy∈S ∥x−  y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We say that S is regular at one of its points x if it is locally closed and satisfies NS(x) =  �  NS(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In addition, we say that a function ϕ is regular at x if ϕ(x) is finite and its epigraph  epi(ϕ) is regular at (x, ϕ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that ϕ is a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The directional derivative  of ϕ at x ∈ Rn in the direction d ∈ Rn is defined by  ϕ′(x, d) = lim  tց0  ϕ(x + td) − ϕ(x)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  In particular, it holds that  ϕ′(x, d) = sup {⟨s, d⟩ : s ∈ ∂ϕ(x)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (5)  We say that a set valued mapping F : Rn → Rm is outer semi-continuous if for any sequence  such that xk → x∗, yk → y∗ and yk ∈ F(xk), we have y∗ ∈ F(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We next introduce the K�L  property with the associated exponent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [2, 3, 4, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Definition 1 (K�L property and exponent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that ϕ : Rn → (−∞, ∞] is proper and lower  semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The function ϕ is said to satisfy the K�L property at ¯x ∈ {x ∈ Rn : ∂ϕ(x) ̸= ∅}  if there exist a constant η ∈ (0, ∞], a neighborhood U of ¯x, and a continuous concave function  ψ : [0, η) → R+ with ψ(0) = 0, ψ being continuously differentiable on (0, η), and ψ′(s) > 0 for  s ∈ (0, η) such that  ψ′ (ϕ(x) − ϕ(¯x)) dist(0, ∂ϕ(x)) ≥ 1  (6)  for all x ∈ U satisfying ϕ(¯x) < ϕ(x) < ϕ(¯x) + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In particular, if ψ(s) = cs1−θ for some c > 0  and θ ∈ (0, 1), then ϕ is said to satisfy the K�L property at ¯x with exponent θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  It is worth mentioning that a wide range of functions arising in applications satisfy the K�L  property, such as proper and lower semicontinuous semialgebraic functions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  2  A Proximal DC Algorithm for Chance Constrained Programs  In this section, we first reformulate Problem (2) into a DC constrained DC program based on  the empirical quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, we propose a pDCA for solving the reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To proceed,  we introduce some further notions that will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let  C(x, ξ) := max {ci(x, ξ) : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (7)  Given a set of samples {ˆξi}N  i=1, let  �C(x) :=  �  C(x, ˆξ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , C(x, ˆξN)  �  ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (8)  6  We define the p-th empirical quantile of C(x, ξ) over the samples {ˆξi}N  i=1 for a probability  p ∈ (0, 1) by  ˆQc(p) := inf  �  y ∈ R : 1  N  N  �  i=1  1{C(x, ˆξi)≤y} ≥ p  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Throughout this section, let  M := ⌈(1 − α)N⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (9)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1  DC Reformulation of the Chance Constraint  In this subsection, we reformulate the sample-based chance constraint in Problem (2) into a DC  constraint using the empirical quantile function of C(x, ξ) over the samples {ˆξi}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To begin,  according to [69, Chapter 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2], the (1 − α)-th empirical quantile of C(x, ξ) over the samples  {ˆξi}N  i=1 for α ∈ (0, 1) is  ˆQc(1 − α) = �C[M](x),  where �C[M](x) denotes the M-th smallest element of �C(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This allows us to get an equivalent  form of Problem (2) as follows:  min  x∈X  �  f(x) :  �C[M](x) ≤ 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (10)  We should mention that the same empirical quantile-based problem has been considered in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' For example, Pe˜na-Ordieres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [58], Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [65] considered approximations of  the quantile constraint, and Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [17] simplified the value at risk constraint of the loss of  a portfolio, which is exactly the quantile constraint, by introducing new variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' By contrast,  we directly handle the empirical quantile constraint by reformulating it into a DC form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To  simplify our development, we denote the above constraint set by  ZM :=  �  x ∈ Rn : �C[M](x) ≤ 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (11)  Remark that if M = N, this constraint requires C(x, ˆξi) ≤ 0 for all i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This, together with  Assumption 1(c) and (7), implies that the constraint set ZN is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' For this case, Problem  (10) minimizes a DC objective function subject to convex constraints, and many algorithms in  the literature have been proposed to solve this problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [60] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  To avoid this case, we assume that M ≤ N − 1 throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Using the structure  of the function �C(·) and the convexity of ci(·, ξ), we show that the above constraint is a DC  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that M ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let  G(x) :=  N  �  i=M  �C[i](x), H(x) :=  N  �  i=M+1  �C[i](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (12)  Then, G and H are both continuous and convex functions, and the chance constraint in (11) is  equivalent to a DC constraint  G(x) − H(x) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (13)  7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The continuity of G and H follows from (7), (8), and Assumption 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Since H(x)  denotes the sum of T largest components of �C(x), we rewrite it as  H(x) = max  �N−M  �  t=1  �Cit(x) : 1 ≤ i1 < i2 < · · · < iN−M ≤ N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (14)  According to the convexity of cj(x, ˆξi) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m due to Assumption  1(c) and the fact that the pointwise maximum of convex functions is still convex ([27, Proposition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2]), we see that C(x, ˆξi) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This, together with (14), the fact  that the sum of convex functions is convex, and the fact that the pointwise maximum of convex  functions is still convex, implies that H(x) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' By the same argument, we show that  G(x) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Given z ∈ RN and M ≤ N − 1, we decompose z[M] as  z[M] =  N  �  i=M  z[i] −  N  �  i=M+1  z[i], for all M = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (15)  This, together with (12), implies that �C[M](x) ≤ 0 is equivalent to (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Consequently, using Lemma 1 and Assumption 1(a), Problem (10) can be cast as the fol-  lowing DC constrained DC program:  min  x∈X  f(x) := g(x) − h(x)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  G(x) − H(x) ≤ 0,  (16)  where g, h are continuous and convex and G, H defined in (12) are also continuous and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2  A Proximal DC Algorithm for Chance Constrained Programs  In this subsection, we propose a proximal DC algorithm for solving Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To begin, we  define  I := {(i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , iN−M) : 1 ≤ i1 < i2 < · · · < iN−M ≤ N} ,  (17)  and denote the active index set of C(x, ˆξi) and H(x) in (12) respectively by  Mi  c(x) :=  �  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m} : cj(x, ˆξi) = C(x, ˆξi)  �  ,  (18)  MH(x) :=  �  I ∈ I :  N−M  �  t=1  �Cit(x) = H(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (19)  We then talk about how to compute an element in these two active sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Specifically,  for the former one, we compute the function values of cj(x, ˆξi) for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m and obtain  an element in the index set Mi  c(x) by finding an index j∗ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m} such that cj∗(x, ˆξi) has  the largest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' For the latter one, after we compute C(x, ˆξi) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N using (7),  we obtain an element in the index set MH(x) by finding an index (i∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , i∗  N−M) ∈ I such that  {C(x, ˆξi∗  t )}T  t=1 is the T largest elements in {C(x, ˆξi)}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Now, we specify how to compute the  subgradient of H(x) efficiently by utilizing its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  8  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let H be defined in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Given an x ∈ Rn, it holds that  ∂H(x) = conv  �  ∪  N−M  �  t=1  ∂ �Cit(x) : (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , iN−M) ∈ MH(x)  �  ,  (20)  where  ∂ �Ci(x) = conv  �  ∪{∇cj(x, ˆξi)} : j ∈ Mi  c(x)  �  (21)  for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N and conv(A) denotes the convex hull of the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It follows from (14) and the rule of calculating the subdifferential of the pointwise max-  imum of convex functions ([27, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2]) that  ∂H(x) = conv  �  ∪∂  N−M  �  t=1  �Cit(x) : (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , iN−M) ∈ MH(x)  �  = conv  �  ∪  N−M  �  t=1  ∂ �Cit(x) : (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , iN−M) ∈ MH(x)  �  ,  where the second equality follows from the continuity and the convexity of C(x, ˆξi) for all  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Since �Ci(x) = C(x, ˆξi) = max{cj(x, ˆξi) : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m} for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N},  using the rule of calculating the subdifferential of the pointwise maximum of convex functions  again and Assumption 1(c), we obtain (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Now, we are ready to propose a proximal DC algorithm for solving Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Specifically,  suppose that an initial point x0 ∈ X satisfying G(x0) − H(x0) ≤ 0 is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' At the k-th  iteration, we choose sk  h ∈ ∂h(xk) and sk  H ∈ ∂H(xk), and generate the next iterate xk+1 by  solving the following convex subproblem  xk+1 ∈ argmin  x∈X  g(x) − h(xk) − ⟨sk  h, x − xk⟩ + β  2 ∥x − xk∥2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  G(x) − H(xk) − ⟨sk  H, x − xk⟩ ≤ 0,  (22)  where β ≥ 0 is a penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' As shown in Lemma 2, the subgradient sk  H can be easily  computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' However, Problem (22) is still not suitable for off-the-shelf solvers, because it is  difficult to directly input G(x) defined in (12), which involves the sum of the N −M +1 largest  components of �C(x, ξ), into solvers due to its combinatorial nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To handle this issue, we  reformulate Problem (22) into a form that is suitable for solvers by introducing an auxiliary  variable z ∈ RN such that C(x, ˆξi) ≤ zi, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Note that  N  �  i=M  z[i] = max  u∈Rn  �  ⟨u, z⟩ : 0 ≤ u ≤ 1, 1T u = N − M + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This is a linear program and its dual problem is  min  λ∈RN,µ∈R {⟨1, λ⟩ + (N − M + 1)µ : z − λ − µ1 ≤ 0, λ ≥ 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  9  Using the strong duality of linear programming, we rewrite Problem (22) as  xk+1 =  argmin  x∈X,z∈RN,λ∈RN,µ∈R  g(x) − h(xk) − ⟨sk  h, x − xk⟩ + β  2 ∥x − xk∥2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  ⟨1, λ⟩ + (N − M + 1)µ − H(xk) − ⟨sk  H, x − xk⟩ ≤ 0,  z − λ − µ1 ≤ 0, λ ≥ 0,  cj(x, ˆξi) − zi ≤ 0, ∀ i =, 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (23)  We remark that we can also eliminate the auxiliary variable z ∈ RN by combining cj(x, ˆξi)−zi ≤  0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m and z−λ−µ1 ≤ 0 together and obtain cj(x, ˆξi)−λi−µ ≤ 0 for  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We summarize the proposed proximal DC algorithm in Algorithm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Algorithm 1 A Proximal DC Algorithm for Chance Constrained Programs  1: Input: data samples {ˆξi}N  i=1, feasible point x0, β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  2: for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' do  3:  take any sk  h ∈ ∂h(xk) and sk  H ∈ ∂H(xk)  4:  solve Problem (23) to obtain an xk+1  5:  if a termination criterion is met then  6:  stop and return xk+1  7:  end if  8: end for  Before studying its convergence, we make some remarks on Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' First, Algorithm  1 is closely related to sequential convex programming methods in [47, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' However, different  from them, we fully exploit the structure of the DC function and reformulate the subproblem  into a form that is suitable for off-the-shelf solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Moreover, we should also mention that our  DC approach differs from that in [29], since their DC approach is based on the DC approxima-  tion of the indicator function, while ours directly handles the empirical quantile of the chance  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Second, a key issue in our implementation is how to choose a feasible initial point  x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' A common approach is to solve a convex approximation of Problem (2) such as CVaR in  [53] to generate a feasible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Third, the penalty parameter β can be updated in an adaptive  manner as long as it is non-increasing and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In our numerical experiments, we observe  that this adaptive scheme may empirically accelerate the convergence of the pDCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Finally, the  subproblem (23) is easy to solve in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Specifically, it is observed that the functions  cj(·, ξ) for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m in many practical applications take the linear form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [50, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Based on this observation, suppose that in (16) X is a polyhedron and  g(x) = aT  0 x, cj(x, ξ) = aT  j x + bT  j ξ, for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (24)  Then, substituting (24) into (23) with β = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' β > 0) yields a linear (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' quadratic)  program with (m+2)N +1 linear constraints (without considering the linear constraints in X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We can solve it easily by inputting it into off-the-shelf linear (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' quadratic) programming  10  solvers, such as MOSEK, Gurobi, and CPLEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In addition, suppose that in (16) X is a polyhedron  and  g(x) = xT Ax + aT  0 x, cj(x, ξ) = aT  j x + bT  j ξ, for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m,  (25)  where A ∈ Rn×n is a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The resulting subproblem (23) is a quadratic program  when β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  3  Convergence and Iteration Complexity Analysis  In this section, we study the convergence properties of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Towards this end, we first  show the subsequential convergence of the sequence {xk} generated by Algorithm 1 to a KKT  point of Problem (16) under some constraint qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Second, we prove convergence of  the entire sequence {xk} if in addition the K�L property holds for a specially designed potential  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Finally, we analyze the iteration complexity of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We point out that the  proposed algorithm and its convergence in fact apply to Problem (16) with G(x) and H(x) being  general convex functions defined on an open set that contains X, which takes the form of general  DC constrained DC programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' An extension to multiple DC constraints will be discussed in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Before we proceed, we introduce some further notation, assumptions, and definitions that  will be used throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To begin, we specify the convex constraints in the set X  as follows:  X =  �  x ∈ Rn : aT  i x + bi = 0, i ∈ E, ωi(x) ≤ 0, i ∈ I  �  ,  (26)  where ai ∈ Rn and bi ∈ R for all i ∈ E, ωi : Rn → R for all i ∈ I are convex and continuously  differentiable functions, and E and I are finite sets of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We denote the active set of the  inequality constraints at x ∈ X by  A(x) := {i ∈ I : ωi(x) = 0} ,  (27)  and the feasible set of Problem (16) by  ¯  X := {x ∈ X : G(x) − H(x) ≤ 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We now introduce a generalized version of the Mangasarian-Fromovitz constraint qualification  (MFCQ), which is a widely used assumption on the algebraic description of the feasible set  of constrained problems that ensures that the KKT conditions hold at any local minimum  ([47, 72]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Assumption 2 (Generalized MFCQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The generalized MFCQ holds for all x ∈ ¯  X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', there  exists y ∈ X such that for G(x) = H(x), we have  G(y) − H(x) − ⟨sH, y − x⟩ < 0, for all sH ∈ ∂H(x),  (28)  ⟨∇ωi(x), y − x⟩ < 0, for all i ∈ A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (29)  11  We next introduce the definition of KKT points for Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Definition 2 (KKT points of Problem (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We say that x ∈ ¯  X is a KKT point of Problem  (16) if there exists λ ∈ R+ such that (x, λ) satisfies λ (G(x) − H(x)) = 0 and  0 ∈ ∂g(x) − ∂h(x) + λ (∂G(x) − ∂H(x)) + NX (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Note that every local minimizer of Problem (16) is a KKT point under the generalized  MFCQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' More precisely, suppose that x∗ ∈ ¯  X is a local minimizer of Problem (16), P = {x :  aT x + bi = 0, i ∈ E} is a polyhedron, and there exists d ∈ TP(x∗) for x∗ ∈ X satisfying  G(x∗) = H(x∗) and ω(x∗) such that  G′(x∗, d) <  inf  s∈∂H(x∗) sT d,  and  ω′  i(x∗, d) = ωi(x∗)T d < 0, i ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (30)  Then, there exists λ∗ ∈ R+ such that x∗ is a KKT point of Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This result is a direct  consequence of [47, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It is worth noting that (30) holds if the generalized MFCQ  holds at x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', Remark (b) of [47, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1  Subsequential Convergence to a KKT Point  In this subsection, our goal is to show that any accumulation point of the sequence {xk}  generated by Algorithm 1 is a KKT point of Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that Assumptions 1 holds, the function f is given in Problem (16), and the  level set  �  x ∈ ¯  X : f(x) ≤ f(x0)  �  is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let {xk} be the sequence generated by Algorithm  1 with ρ + β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, the following statements hold:  (i) It holds for all k ≥ 0 that xk ∈ ¯  X and  f(xk+1) − f(xk) ≤ −ρ + β  2  ∥xk+1 − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (31)  (ii) The sequence {xk} ⊆ ¯  X is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (iii) It holds that  lim  k→∞ ∥xk+1 − xk∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (32)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' (i) According to the feasibility of xk+1 to Problem (22), sk  H ∈ ∂H(xk), and the convexity  of H, we have xk+1 ∈ X and  G(xk+1) ≤ H(xk) + ⟨sk  H, xk+1 − xk⟩ ≤ H(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (33)  This implies xk+1 ∈ ¯  X for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Moreover, it follows from the optimality of xk+1 for  Problem (22) and the ρ-strongly convexity of g that for all k ≥ 0,  g(xk+1) − h(xk) − ⟨sk  h, xk+1 − xk⟩ + ρ + β  2  ∥xk+1 − xk∥2 ≤ g(xk) − h(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  12  This, together with the convexity of h and sk  h ∈ ∂h(xk), yields that for all k ≥ 0,  g(xk+1) − h(xk+1) + ρ + β  2  ∥xk+1 − xk∥2 ≤ g(xk) − h(xk),  which is equivalent to (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (ii) According to (31), the function value f(xk) is monotonically decreasing and thus we  have f(xk+1) ≤ f(x0) for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This, together with the level-boundness of the set  �  x ∈ X c : f(x) ≤ f(x0)  �  , implies that {xk} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (iii) The boundedness of the sequence {xk}, together with continuity of f implies that  {f(xk)} is bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Using this and the fact that {f(xk)} is monotonically de-  creasing, we obtain that there exists some f ∗ such that f(xk) → f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  It follows from (31)  that  α + β  2  ∞  �  k=0  ∥xk+1 − xk∥2 ≤ f(x0) − lim  k→∞f(xk+1) = f(x0) − f ∗ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This implies (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Armed with the above lemma, we are ready to show the subsequential convergence of the  sequence {xk} generated by Algorithm 1 to a KKT point of Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 2 hold, the function f is given in Problem (16),  and the level set  �  x ∈ ¯  X : f(x) ≤ f(x0)  �  is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let {xk} be the sequence generated by  Algorithm 1 with ρ + β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, any accumulation point of {xk} is a KKT point of Problem  (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' According to (i) in Lemma 3, it holds that xk ∈ ¯  X for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This, together with the  generalized MFCQ in Assumption 2 and the equivalence between the Slater condition and the  MFCQ by [18, Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='3(b)], yields that there exists x ∈ X such that for any sk  H ∈ ∂H(xk),  G(x) − H(xk) − ⟨sk  H, x − xk⟩ < 0, ωi(x) < 0, ∀ i ∈ A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (34)  This is exactly the Slater condition for Problem (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' According to this, (26), and [62, Corollary  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1, Theorem 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='3], there exists a Lagrange multiplier λk ∈ R for all k ≥ 0 such that the  following KKT system holds:  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  G(xk+1) − H(xk) − ⟨sk  H, xk+1 − xk⟩ ≤ 0, xk+1 ∈ X,  λk �  G(xk+1) − H(xk) − ⟨sk  H, xk+1 − xk⟩  �  = 0, λk ≥ 0,  0 ∈ ∂g(xk+1) − sk  h + β(xk+1 − xk) + λk �  ∂G(xk+1) − sk  H  �  + NX (xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (35)  It follows from (ii) of Lemma 3 that {xk} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let x∗ be an accumulation point of {xk}  such that there exists a subsequence {xki} with limi→∞ xki = x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We claim that the sequence  {λk} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Passing to a further subsequence if necessary, we assume without loss of  generality that limi→∞ λki = λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' According to (32) in Lemma 3, we have limi→∞(xki+1−xki) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Using this fact, the outer semi-continuity of ∂g, ∂h, ∂G, ∂H [63, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='4, Proposition  13  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='7], and sk  h ∈ ∂h(xk), sk  H ∈ ∂H(xk), we have upon passing to the limit as i goes to infinity in  (35) with k = ki that sk  h → s∗  h ∈ ∂h(x∗) and sk  H → s∗  H ∈ ∂H(x∗), and thus  0 ∈ ∂g(x∗) − ∂h(x∗) + λ∗ (∂G(x∗) − ∂H(x∗)) + NX (x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (36)  On the other hand, using (35) and (32) with k = ki and the boundedness of ∂H(x∗), letting  i → ∞, we have  G(x∗) ≤ H(x∗), λ∗ (G(x∗) − H(x∗)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (37)  Moreover, since λk ≥ 0 and xk ∈ ¯  X for all k ≥ 0, we have λ∗ ≥ 0 and x∗ ∈ ¯  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This, together  with (36), (37), and Definition 2, implies that x∗ is a KKT point of Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  The rest of the proof is devoted to proving that {λk} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Without loss of generality,  we assume that {ai : i ∈ E} is linearly independent, since otherwise we can obtain the same  results by eliminating the redundant linear equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It follows from [63, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='14] for  any x ∈ X that  NX (x) =  ��  i∈E  uiai +  �  i∈I  vi∇ωi(x) : vi ≥ 0, for i ∈ A(x), vi = 0, for i /∈ A(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This, together with (35), yields that there exist uk  i for i ∈ E, vk  i ≥ 0 for i ∈ A(xk+1), and vk  i = 0  for i /∈ A(xk+1) such that  0 ∈ ∂g(xk+1) − sk  h + β(xk+1 − xk) + λk �  ∂G(xk+1) − sk+1  H  �  +  �  i∈E  uk  i ai +  �  i∈I  vk  i ∇ωi(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (38)  Then, let  ρk :=  �  (λk)2 +  �  i∈E  (uk  i )2 +  �  i∈I  (vk  i )2, τ k := λk  ρk , µk  i := uk  i  ρk , νk  i := vk  i  ρk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Suppose to the contrary that {λk} is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This implies that ρk is also unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Then, there exists a subsequence {λkj} such that |λkj| → ∞ as j goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Passing  to a further subsequence if necessary, suppose that there exist τ ∗ ∈ R+, µ∗  i , ν∗  i ∈ R+, x∗, and  s∗  H ∈ ∂H(x∗) such that limj→∞ τ kj = τ ∗, limj→∞ µkj  i  = µ∗  i , limj→∞ νkj  i  = ν∗  i , limj→∞ xkj = x∗,  and limj→∞ skj  H = s∗  H, where skj  H ∈ ∂H(xkj), due to λk ≥ 0, v∗  i ≥ 0 for i ∈ I, the boundness  of {τ k}, {µk}, {νk}, {xk}, and ∂H(xk), and the outer semi-continuity of ∂H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, dividing  both sides of (38) by |ρkj|, letting j → ∞, and using (32), the outer semi-continuity of ∂g and  ∂h, and the boundness of ∂g(x∗), ∂h(x∗), and {xk}, we have  0 ∈ τ ∗ (∂G(x∗) − s∗  H) +  �  i∈E  µ∗  i ai +  �  i∈I  ν∗  i ∇ωi(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (39)  Using the definitions of τ ∗, µ∗, and ν∗, we further have  (τ ∗)2 + ∥µ∗∥2 + ∥ν∗∥2 = 1,  (40)  14  (Case 1) Suppose that τ ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Due to (39), we have  0 =  �  i∈E  µ∗  i ai +  �  i∈I  ν∗  i ∇ωi(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (41)  According to Assumption 2, there exists y ∈ X such that ⟨∇ωi(x∗), y−x∗⟩ < 0 for all i ∈ A(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Moreover, since A(xk) ⊆ A(x∗) when k is sufficiently large, we have i /∈ A(xk) if i /∈ A(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Therefore, we have νk  i = 0 for all i /∈ A(xk) as k → ∞, which implies ν∗  i = 0 for i /∈ A(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Then, taking inner products with y − x∗ on both sides of (41) yields  0 =  �  i∈A(x∗)  ν∗  i ⟨∇ωi(x∗), y − x∗⟩,  where the equality follows from ⟨ai, y − x∗⟩ = 0 for i ∈ E and ν∗  i = 0 for i /∈ A(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This,  together with ⟨∇ωi(x∗), y−x∗⟩ < 0 for all i ∈ A(x∗), gives ν∗  i = 0 for all i ∈ A(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Substituting  this and ν∗  i = 0 for i /∈ A(x∗) into (41), we have 0 = �  i∈E µ∗  i ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Noting that we assume that  {ai : i ∈ E} is linearly independent, we have µ∗  i = 0 for all i ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Therefore, ν∗  i = 0 for all i ∈ I  and µ∗  i = 0 for all i ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This contradicts (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (Case 2) Suppose that τ ∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We first consider the case of G(x∗) < H(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It follows from  the second line of (35) with k = kj, j → ∞, and (32) that limj→∞ λkj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This implies τ ∗ = 0,  which contradict (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We then must have G(x∗) = H(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This, together with the convexity  of G and (28) in Assumption 2, yields that there exists y ∈ X such that  ⟨¯sG − s∗  H, y − x∗⟩ ≤ G(y) − G(x∗) − ⟨s∗  H, y − x∗⟩ = G(y) − H(x∗) − ⟨s∗  H, y − x∗⟩ < 0, (42)  where ¯sG is an arbitrary subgradient of G at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' According to (39), there exists s∗  G ∈ ∂G(x∗)  such that  0 = τ ∗ (s∗  G − s∗  H) +  �  i∈E  µ∗  i ai +  �  i∈I  ν∗  i ∇ωi(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (43)  Taking inner products with y − x∗ on both sides yields  0 = τ ∗⟨s∗  G − s∗  H, y − x∗⟩ +  �  i∈A(x∗)  ν∗  i ⟨∇ωi(x∗), y − x∗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Note that ν∗  i ≥ 0 due to vk  i ≥ 0 for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This, together with (29) at x∗ and (42), implies  τ ∗ = 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We prove the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2  Convergence of the Entire Sequence to a KKT Point  In this subsection, we employ the analysis framework proposed in [2, 4] based on the K�L property  to study the sequential convergence of Algorithm 1 for β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Our first step is to show that the  sequence generated by Algorithm 1 satisfies sufficient decrease and relative error conditions with  respect to a potential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Motivated by the potential functions constructed in [46, 73], we  construct the following potential function  ϕ(x, y, z) := g(x) − ⟨x, y⟩ + h∗(y) + δ ¯F (·)≤0(x, z) + δX (x),  (44)  15  where  ¯F(x, z) := G(x) − ⟨x, z⟩ + H∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (45)  Then, we characterize the subdifferential of δ ¯F (·)≤0(x, z) using its structure and the convexity  of G and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We point out that this characterization holds for G and H being arbitrary proper  closed convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that Assumption 2 holds and (x, z) satisfies ¯F(x, z) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It holds that the  function δ ¯F (·) is regular and  ∂δ ¯F (·)≤0(x, z) =  ��  λ(s1 − z)  λ(−x + s2)  �  : s1 ∈ ∂G(x), s2 ∈ ∂H∗(z), λ ≥ 0, λ ¯F(x, z) = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (46)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To begin, let  S :=  �  (x, z) : ¯F(x, z) ≤ 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  In particular, we write S = ¯F −1(R−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Because G and H∗ are both convex functions, then ¯F is  locally Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It follows from [63, Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1] that ¯F : Rn×Rn → R is a strictly  continuous function since it is locally Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We obtain that y ∈ NR−( ¯F(x, z))  is equivalent to y ≥ 0, y ¯F(x, z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that the only vector y ∈ NR−( ¯F(x, z)) with  0 ∈ ∂(y ¯F)(x, z) is y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Moreover, R− is regular at ¯F(x, z) and ¯F is regular due to convexity  of G and H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' These, together with [63, Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='50], imply that S is regular, which further  implies δ ¯F(·) is regular by [63, Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='14], and (46) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  The rest of our proof is to show that the only vector y ∈ NR−( ¯F(x, z)) with 0 ∈ ∂(y ¯F)(x, z)  is y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, there exists y > 0 such that y ∈ NR−( ¯F(x, z))  with 0 ∈ ∂(y ¯F)(x, z), which implies  �  (x, z) :  ¯F(x, z) = 0, 0 ∈ ∂ ¯F(x, z)  �  ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (47)  Let (x, z) be such that ¯F(x, z) = 0 and 0 ∈ ∂ ¯F(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It follows from ¯F(x, z) = 0 and (45)  that  G(x) − ⟨x, z⟩ + H∗(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (48)  Using [3, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1] and 0 ∈ ∂ ¯F(x, z) with (45), we have  z ∈ ∂G(x), x ∈ ∂H∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  In particular, x ∈ ∂H∗(z) holds if and only if z ∈ ∂H(x) because H is a pointwise maximum of  continuous and convex functions, and thus closed and convex, which implies H∗(z) + H(x) =  ⟨x, z⟩ according to Young’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Using this and (48), we have G(x) = H(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It follows  from z ∈ ∂G(x) and the convexity of G that  G(y) ≥ G(x) + ⟨z, y − x⟩, for all y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Plugging z ∈ ∂H(x) and G(x) = H(x) into the above inequality yields that for z ∈ ∂H(x),  G(y) ≥ H(x) + ⟨z, y − x⟩, for all y ∈ Rn,  which contradicts (28) in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We prove the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  16  Now, we are ready to show that the sequence {(xk, sk  h, sk  H)} generated by Algorithm 1  satisfies the sufficient decrease and relative error conditions mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Let {(xk+1, sk  h, sk  H)} be the sequence  generated by Algorithm 1 with ρ + β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, the following statements hold:  (i) [Sufficient Decrease] The sequence {(xk+1, sk  h, sk  H)} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It holds for all k ≥ 1 that  ϕ(xk+1, sk  h, sk  H) − ϕ(xk, sk−1  h  , sk−1  H  ) ≤ −ρ + β  2  ∥xk+1 − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (ii) [Relative Error] There exists a constant κ > 0 such that for all k ≥ 0,  dist  �  0, ∂ϕ(xk+1, sk  h, sk  H)  �  ≤ κ∥xk+1 − xk∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' (i) It follows from (i) in Lemma 3 that {xk} ⊆ ¯  X is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This, together with the  fact that h and H are convex, implies that {(sk  h, sk  H)} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Therefore, the sequence  {(xk+1, sk  h, sk  H)} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' According to (45), we have for all k ≥ 0,  ¯F(xk+1, sk  H) = G(xk+1) − ⟨xk+1, sk  H⟩ + H∗(sk  H)  = G(xk+1) + H∗(sk  H) − ⟨xk, sk  H⟩ − ⟨xk+1 − xk, sk  H⟩  = G(xk+1) − H(xk) − ⟨xk+1 − xk, sk  H⟩ ≤ 0,  (49)  where the last equality follows from H(xk) + H∗(sk  H) = ⟨xk, sk  H⟩ due to Young’s inequality and  sk  H ∈ ∂H(xk), and the inequality is due to the constraint in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Moreover, it follows from  (22) and the ρ-strongly convexity of g that for all k ≥ 0,  g(xk+1) − ⟨sk  h, xk+1 − xk⟩ + ρ + β  2  ∥xk+1 − xk∥2 ≤ g(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (50)  This,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' together with (49) and xk ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' implies for all k ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  ϕ(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' sk  h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' sk  H) = g(xk+1) − ⟨xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' sk  h⟩ + h∗(sk  h)  ≤ g(xk) − ⟨sk  h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' xk⟩ − ρ + β  2  ∥xk+1 − xk∥2 + h∗(sk  h)  = g(xk) − h(xk) − ρ + β  2  ∥xk+1 − xk∥2  ≤ g(xk) − ⟨xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' sk−1  h  ⟩ + h∗(sk−1  h  ) − ρ + β  2  ∥xk+1 − xk∥2  = ϕ(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' sk−1  h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' sk−1  H  ) − ρ + β  2  ∥xk+1 − xk∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  where the first inequality uses (50),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' the second equality follows from h(xk) + h∗(sk  h) = ⟨xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' sk  h⟩  due to sk  h ∈ ∂h(xk) and Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' the second inequality follows from h(xk)+h∗(sk−1  h  ) ≥  ⟨xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' sk−1  h  ⟩ due to Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' and the last equality is due to xk ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' (44),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' and (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (ii) According to [63, Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='8(c)], we compute  ∂ϕ(x, y, z) = ∂  �  g(x) + h∗(y) + δ ¯F (·)≤0(x, z) + δX (x)  �  +  \uf8ee  \uf8ef\uf8f0  −y  −x  0  \uf8f9  \uf8fa\uf8fb  =  �  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  \uf8ee  \uf8ef\uf8f0  ∂g(x) − y + λ(∂G(x) − z) + NX(x)  −x + ∂h∗(y)  λ(−x + ∂H ∗ (z))  \uf8f9  \uf8fa\uf8fb : λ ≥ 0, λ ¯F(x, z) = 0  \uf8fc  \uf8f4  \uf8fd  \uf8f4  \uf8fe  ,  17  where the second equality uses [63, Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='9] and the fact that g, h∗ and δX are regular  due to the convexity and δ ¯F (·) is regular due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Therefore, we have  ∂ϕ(xk+1, sk  h, sk  H) =  �  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  \uf8ee  \uf8ef\uf8f0  ∂g(xk+1) − sk  h + λ(∂G(xk+1) − sk  H) + NX (xk+1)  −xk+1 + ∂h∗(sk  h)  λ(−xk+1 + ∂H∗(sk  H))  \uf8f9  \uf8fa\uf8fb :  λ ≥ 0, λ ¯F(xk+1, sk  H) = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (51)  It follows from Assumption 2 that the KKT system (35) holds for Problem (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then we have  λk ≥ 0 and  λk ¯F(xk+1, sk  H) = λk �  G(xk+1) − ⟨xk+1, sk  H⟩ + H∗(sk  H)  �  = λk �  G(xk+1) − H(xk) − ⟨xk+1 − xk, sk  H⟩  �  = 0,  (52)  where the first equality uses (45), the second equality follows from H(xk) + H∗(sk  H) = ⟨xk, sk  H⟩  due to sk  H ∈ ∂H(xk) and Young’s inequality, and the last equality follows from the second line  in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It follows from the last line in (35) that  β(xk − xk+1) ∈ ∂g(xk+1) − sk  h + λk �  ∂G(xk+1) − sk  H  �  + NX (xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This, together with (49), (51), (52) with λk ≥ 0, sk  h ∈ ∂h(xk), sk  H ∈ ∂H(xk), and the fact that  y ∈ ∂ψ(x) if and only if x ∈ ∂ψ∗(y) provided that ψ is a proper closed convex function, yields  that  \uf8ee  \uf8ef\uf8f0  β(xk − xk+1)  xk − xk+1  λk(xk − xk+1)  \uf8f9  \uf8fa\uf8fb ∈ ∂ϕ(xk+1, sk  h, sk  H)  This implies  dist  �  0, ∂ϕ(xk+1, sk  h, sk  H)  �  ≤ (β + 1 + λk)∥xk+1 − xk∥,  where λk ≥ 0 is bounded in (35) according to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, we complete the  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Since g, h, G, and H are continuous and convex functions and X is a closed and convex  set, we can verify that ϕ is a K�L function with exponent θ ∈ [0, 1) according to [9, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Using Lemma 5 and the analysis in [2, 3, 4, 9, 46, 73], we shall prove the sequential convergence  and the convergence rate of the sequence {xk} generated by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The proof is rather  standard and thus we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We refer the reader to [2, 46] for the detailed arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 2 hold, the function f is given in Problem (16),  and the level set  �  x ∈ X c : f(x) ≤ f(x0)  �  is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, the sequence {xk} generated by  Algorithm 1 with ρ + β > 0 converges to a KKT point x∗ of Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let θ ∈ [0, 1) denote  the K�L exponent of ϕ in (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' There exists an integer k∗ ≥ 1 such that the following statements  18  hold:  (i) If θ = 0, then {xk} converges finitely, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', xk = x∗ for all k ≥ k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (ii) If θ ∈ (0, 1/2], then {xk} converges linearly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', there exist c > 0 and q ∈ (0, 1) such that  for all k ≥ k∗,  ∥xk − x∗∥ ≤ cqk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (iii) If θ ∈ (1/2, 1), then {xk} converges sublinearly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', there exist c > 0 such that for all  k ≥ k∗,  ∥xk − x∗∥ ≤ ck− 1−θ  2θ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  It follows from Theorem 2 that the proximal DC algorithm achieves linear convergence when  the K�L exponent θ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Therefore, an interesting future direction is to investigate under what  conditions the K�L exponent of Problem (16) is 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [43, 33, 34, 45, 70, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='3  Iteration Complexity for Computing an Approximate KKT Point  In this subsection, we analyze the iteration complexity of Algorithm 1 for computing an ap-  proximate KKT point of Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Motivated by the analysis framework in [74] for DC  constrained DC programs with all functions being differentiable, we connect Algorithm 1 to a  variant of the Frank-Wolfe (FW) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  To simplify notation, let  w := (x, s, t),  q(w) := s − h(x),  Q(w) := t − H(x),  and  W := {w : x ∈ X, g(x) ≤ s, G(x) ≤ t} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  In particular, we should mention that q and Q are both concave functions and W is a convex  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We rewrite Problem (16) as follows by introducing auxiliary variables s, t ∈ R:  min  x∈X,s∈R,t∈Rs − h(x)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  g(x) ≤ s, G(x) ≤ t, t − H(x) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (53)  We further express Problem (53) as  min  w∈W q(w)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Q(w) ≤ 0,  (54)  Based on the above setup, we directly show the equivalence between the proximal DC iterations  in (22) and a variant of FW iterations applied to Problem (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The proximal DC iterations in (22) with  β ≥ 0 is equivalent to the following variant of FW iterations:  wk+1 ∈ argmin  w∈W  q(wk) + ⟨sk  q, w − wk⟩ + β  2 ∥w − wk∥2  T  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Q(wk) + ⟨sk  Q, w − wk⟩ ≤ 0,  (55)  19  where sk  q = (−sk  h, 1, 0), sk  h ∈ ∂h(xk), sk  Q = (−sk  H, 0, 1), sk  H ∈ ∂H(xk), and we define ∥z∥T =  ��n  i=1 z2  i for any z ∈ Rn+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The proof follows directly from the definitions of W, q(w), Q(w), and the fact that any  optimal solution of (55) must satisfy sk = g(xk) and s = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We next use the equivalent expression (54) to give an equivalent characterization of KKT  points (see Definition 2) of Problem (16) under the generalized MFCQ in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Given ¯w ∈ W, sq ∈ ∂q( ¯w) with g(¯x) ≤  ¯s, G(¯x) ≤ ¯t, and sQ ∈ ∂Q( ¯w), suppose that  ⟨sq, w − ¯w⟩ + β  2 ∥w − ¯w∥2  T ≥ 0  (56)  for all w ∈ W satisfying Q( ¯w) + ⟨sQ, w − ¯w⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, ¯x is a KKT point of Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' According to the statement of the lemma and [6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2], we obtain that  ¯w ∈ W is an optimal solution to the following convex problem:  min  w∈W  ⟨sq, w − ¯w⟩ + β  2 ∥w − ¯w∥2  T  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Q( ¯w) + ⟨sQ, w − ¯w⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Note that ⟨sq, w − ¯w⟩ = −⟨sh, x − ¯x⟩ + (s − ¯s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Moreover, the optimal solution of (55) must  satisfy s = g(x) and ¯s = g(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then we have  ⟨sq, w − ¯w⟩ + β  2 ∥w − ¯w∥2  T = g(x) − g(¯x) − ⟨sh, x − ¯x⟩ + β  2 ∥x − ¯x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (57)  Thus ¯x is an optimal solution to the following convex problem:  min  x∈X  g(x) − g(¯x) − ⟨sh, x − ¯x⟩ + β  2 ∥x − ¯x∥2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  G(x) − H(¯x) − ⟨sH, x − ¯x⟩ ≤ 0,  where sh ∈ ∂h(¯x) and sH ∈ ∂H(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This, together with the Slater’s condition due to Assump-  tion 2, implies that there exists λ ∈ R+ such that (¯x, λ) satisfies the KKT system in Definition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Consequently, studying the iteration complexity of Algorithm 1 for computing an approxi-  mate KKT point of Problem (16) is equivalent to that of the variant of the FW iterations (55)  for computing a point satisfying (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' However, we cannot expect to achieve a solution that  satisfies (56) in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Instead, we often obtain an approximate solution as shown in the next  theorem, which can be seen as an approximation of a KKT point of Problem (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The next  theorem gives the iteration complexity for achieving an approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  20  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let {xk} be the sequence generated by  Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, there exists ℓ ∈ [k] such that  ⟨sq, w − wℓ⟩ + β  2 ∥w − wℓ∥2  T ≥ −1  k  �  q(w0) − q∗�  ,  (58)  for all w ∈ W and Q(wl) + ⟨sl  Q, w − wl⟩ ≤ 0, where q∗ ∈ R is the optimal value of Problem  (54) and sl  Q ∈ ∂Q(wl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' According to Lemma 6, a sequence {wk} generated by iterations (55) satisfies wk =  (xk, sk, tk) for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Since q is a concave function and sk  q ∈ ∂q(wk), we have  ⟨sk  q, wk − wk+1⟩ ≤ q(wk) − q(wk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Averaging the above inequality over k yields  1  k  k  �  i=1  ⟨sk  q, wk − wk+1⟩ ≤ 1  k  �  q(w0) − q(wk+1)  �  ≤ 1  k  �  q(w0) − q∗�  ,  where the last inequality follows from the fact that q∗ ∈ R is the optimal value of Problem (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This implies that there exists an index ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , k} such that  ⟨sℓ  q, wℓ − wℓ+1⟩ ≤ 1  k  �  q(w0) − q∗�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (59)  Moreover, it follows from the optimality wk+1 to Problem (55) that for all w ∈ W satisfying  Q(wℓ) + ⟨sℓ  Q, w − wℓ⟩ ≤ 0,  ⟨sℓ  q, wℓ+1 − wℓ⟩ + β  2 ∥wℓ+1 − wℓ∥2  T ≤ ⟨sℓ  q, w − wℓ⟩ + β  2 ∥wℓ − w∥2  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This, together with (59), implies that it holds for all w ∈ W satisfying Q(wℓ)+⟨sℓ  Q, w−wℓ⟩ ≤ 0  that  ⟨sq, w − wℓ⟩ + β  2 ∥w − wℓ∥2  T ≥ ⟨sℓ  q, wℓ+1 − wℓ⟩ + β  2 ∥wℓ+1 − wℓ∥2  T ≥ −1  k  �  q(w0) − q∗�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We remark that in contrast to Theorems 1 and 2 that require ρ + β > 0, Theorem 3 can  be applied to analyze the case of ρ + β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It is worth noting that when β = 0, the standard  iteration complexity of the FW method for general nonconvex problems is O(1/  √  k) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=',  [38]), but the iteration complexity of our proposed FW method is improved to O(1/k) as we  construct a concave minimization surrogate using the DC structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  4  Extensions  In this section, we first discuss how to extend our approach to solve chance constrained problems  with the chance constraint estimated by general non-parametric estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We then extend  the proximal DC algorithm for solving Problem (16) with multiple DC constraints, which can  be used to solve chance constraint programs with multiple chance constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Finally, we show  that our technique can also be used to solve cardinality constrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1  Extension to L-Estimators of the Empirical Quantile  In statistics, an L-estimator is a linear combination of order statistics of a sample drawn from  the population distribution, which plays an important role in non-parametric estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The  main advantage of L-estimators is that they are easy to calculate and often resistant to outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Due to this, L-estimators have been widely used in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [17, 51, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This  naturally motivates us to apply the L-estimators to Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  To proceed, we formally introduce L-estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that a set of samples {Xi}N  i=1 is  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' according to some unknown distribution FX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In general, L-estimators of the empirical  quantile take the form �N  i=1 wiX[i], where w ∈ ∆ :=  �  u ∈ RN : 0 ≤ u ≤ 1, 1T u = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  In  statistics, there are many different L-estimators that outperform the empirical quantile in both  theory and practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [17, 21, 31, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, we consider some typical L-estimators of  the p empirical quantile for p ∈ (0, 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', X[M], where M = ⌈pN⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The first one is the weighted  average at X[M−1] (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [21, 31]) defined as  L1 = (1 − g)X[M−1] + gX[M],  where g = Np − M + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Another one is the kernel quantile estimator (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [44, 57]) defined  as  L2 =  N  �  i=1  �� i/N  (i−1)/N  1  hK  �x − p  h  �  dx  �  X[i],  where h > 0 is a constant and K(t) is a kernel function satisfying  � ∞  −∞ K(t)dt = 1, K(t) ≥ 0,  and K(−t) = K(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It is worth noting that this kernel quantile estimator can be viewed as a  smoothing version of the empirical quantile estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Now, we apply L-estimators to the SAA of the chance constrained program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Specifically,  replacing the the empirical quantile �C[M](x) in Problem (10) with its L-estimator yields the  following problem:  min  x∈X  �  f(x) :  N  �  i=1  wi �C[i](x) ≤ 0  �  ,  (60)  where the weight w ∈ ∆ is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' It is worth pointing out that Problem (2) is actually a special  case of Problem (60) by taking wM = 1 and wi = 0 for all i ̸= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, we reformulate this  problem into a DC constrained DC program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Before we proceed, let  ¯Z :=  �  x ∈ Rn :  N  �  i=1  wi �C[i](x) ≤ 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (61)  Similar to Lemma 1, we can also express the above constraint as a DC constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let  G(x) :=  N  �  i=1  wi  N  �  j=i  �C[j](x), H(x) :=  N−1  �  i=1  wi  N  �  j=i+1  �C[j](x),  (62)  22  where w ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Then, G and H are both continuous and convex functions, and the chance  constraint in ¯Z is equivalent to a DC constraint  G(x) − H(x) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Using the argument in Lemma 1, we can show that �N  j=i �C[j](x) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N are  convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Since each of G and H in (62) is a positive weighted sum of convex functions,  G and H are both convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' According to (15), we have for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , N − 1,  �C[i](x) =  N  �  j=i  �C[j](x) −  N  �  j=i+1  �C[j](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This yields that  N  �  i=1  wi �C[i](x) =  N−1  �  i=1  wi �C[i](x) + wN �C[N](x) =  N−1  �  i=1  wi  \uf8eb  \uf8ed  N  �  j=i  �C[j](x) −  N  �  j=i+1  �C[j](x)  \uf8f6  \uf8f8 + wN �C[N](x)  =  N  �  i=1  wi  N  �  j=i  �C[j](x) −  N−1  �  i=1  wi  N  �  j=i+1  �C[j](x) = G(x) − H(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We then obtain a DC constrained DC program for L-estimators of the empirical quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Consequently, we can still apply the proposed pDCA for solving the resulting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2  Extension to Multiple DC Constraints  In this subsection, we consider that Problem (16) has multiple DC constraints  Gi(x) − Hi(x) ≤ 0, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , K,  (63)  where Gi : Rn → R and Hi : Rn → R are continuous and convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' That is, we consider  the problem  min  x∈X  f(x) := g(x) − h(x)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Gi(x) − Hi(x) ≤ 0, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  (64)  We can still apply the proximal DC algorithm for solving this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Specifically, suppose  that an initial point x0 ∈ X satisfying Gi(x0) − Hi(x0) ≤ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , K is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' At the  k-th iteration, we choose sk  h ∈ ∂h(xk) and sk  Hi ∈ ∂Hi(xk) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , K, and generate the  next iterate xk+1 by solving the following convex subproblem  xk+1 ∈ argmin  x∈X  g(x) − h(xk) − ⟨sk  h, x − xk⟩ + β  2 ∥x − xk∥2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Gi(x) − Hi(xk) − ⟨sk  Hi, x − xk⟩ ≤ 0, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , K,  (65)  where β ≥ 0 is a penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In particular, we can also prove subsequential convergence  to a KKT point for the proximal DC algorithm by assuming the following generalized MFCQ:  23  Assumption 3 (Generalized MFCQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The generalized MFCQ holds for Problem (64), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=',  there exists y ∈ X such that for Gi(x) = Hi(x),  Gi(y) − Hi(x) − ⟨sHi, y − x⟩ < 0, for all sHi ∈ ∂Hi(x), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , K,  ⟨∇ωi(x), y − x⟩ < 0, for all i ∈ A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Using the similar argument in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1, we can obtain the following result:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 3 hold, the function f is given in Problem (64),  X is of the form of (26), and the level set  �  x ∈ X : f(x) ≤ f(x0), Gi(x) − Hi(x) ≤ 0, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , K  �  is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let {xk} be the sequence generated by (65) with ρ+β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, any accumulation  point of {xk} is a KKT point of Problem (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We remark that although it is pointed out in [56] that multiple DC constraints can be  combined into a single nondifferentiable DC constraint using the max-function, it makes the  constraint more complicated and thus a more difficult pDCA subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='3  Extension to Cardinality Constrained Optimization Problems  We consider the following cardinality constrained optimization problems:  min  x∈Rn {f(x) : ∥x∥0 ≤ K, x ∈ X} ,  (66)  where ∥x∥0 denotes the cardinality of the vector, and K is an interger satisfying 1 ≤ K ≤ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  This problem has found wide applications in diverse fields, such as quantitative finance [7, 22]  and signal processing [13], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' By introducing an auxiliary variable z ∈ Rn, the  cardinality constraint ∥x∥0 ≤ K is equivalent to z[N−K] ≤ 0, zi = |xi| for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This  implies that we can rewrite Problem (66) as  min  x∈X,z∈Rn  �  f(x) : z[N−K] ≤ 0, xi − zi ≤ 0, −xi − zi ≤ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  As in Lemma 1, we can further rewrite the constraint z[N−K] ≤ 0 into a DC constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then,  we can apply the proposed approach for solving the resulting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  5  Experimental Results  In this section, we conduct experiments to study the performance of our proposed method  on both synthetic and real data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' For ease of reference, we denote our proposed method  by pDCA (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  DCA) when β > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  β = 0) in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We also compare our  methods with some state-of-the-art methods, which are CVaR in [53], the bisection-based CVaR  method3 (Bi-CVaR) in [5, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1], mixed-integer program (MIP) in [1], an augmented  3The bisection based CVaR method is a heuristic approach that can improve the performance of CVaR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  24  Lagrangian decomposition method (ALDM) in [5], and a DC approximation-based successive  convex approximation method (SCA) in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Our codes are implemented in MATLAB 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  In particular, we use the optimization solver Gurobi (version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2) for solving linear, quadratic,  and mixed integer subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' All the experiments are conducted on a Linux server with  256GB RAM and 24-core AMD EPYC 7402 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='8GHz CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  For pDCA, we update the penalty parameter β in an adaptive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' That is, we set  βk+1 = βk/4 for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' For pDCA on each data set, we explore two different settings of  the regularization parameter β0, which will be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We respectively denote them by  pDCA-1 and pDCA-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We set the parameters of the remaining methods as those provided in  the corresponding papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' For the tested methods DCA, pDCA, Bi-CVaR, ALDM, and SCA,  we use the point returned by CVaR as their starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In each test, we terminate the  tested methods when |f k − f k+1|/ max{1, |f k+1|} ≤ 10−6, for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , or the running  time reaches 1800 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1  VaR-Constrained Portfolio Selection Problem  In this subsection, we study the VaR-constrained mean-variance portfolio selection problem,  which aims to minimize the risk while pursuing a targeted level of returns with probability at  least 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Let µ ∈ Rn and Σ ∈ Rn×n respectively denote expectation and covariance matrix  of the returns of n risky assets, and γ ∈ R+ denote the risk aversion factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' By letting x ∈ Rn  +  denote the allocation vector such that the weight of the i-th risky asset is xi for i ∈ [n], this  problem is formulated as follows:  min  x∈Rn γxT Σx − µT x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  P  �  ξT x ≥ R  �  ≥ 1 − α,  n  �  i=1  xi = 1, 0 ≤ xi ≤ u, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , n,  (67)  where R ∈ R+ is a prespecified level on the return and u ∈ R+ is an upper bound on the  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We use 2523 daily return data of 435 stocks included in Standard & Poor’s 500 Index between  March 2006 and March 2016, which can be downloaded from https://sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='tongji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='cn/semch_data/faculty_cv/xjz/ccop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Following [5], we generate the data input by choosing n = 100, 200, 300, 400, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' For  each n, we generate 5 instances from the daily return data set by randomly selecting n stocks  from the 435 stocks and N = 3n samples ˆξℓ for all ℓ ∈ [N] from the 2523 daily return data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Then, we compute the sample mean µ and sample covariance matrix Σ using these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We  set the remaining parameters as follows: R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='02%, γ = 2, and u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In the tests, we set  the initial regularization parameter β0 of pDCA-1 and pDCA-2 as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In  Table 1 and the other two tables below for the other two experiments, we use “fval” to denote  the averaged returned objective value for the test problems, “time” the averaged CPU time (in  seconds), and “prob” the empirical in-sample probability of the chance constraint, all of which  4Since we only check the running time at the end of each iteration, the actual finishing time of an algorithm  may be longer than this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  25  are averaged over 5 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We highlight the best values except those of MIP and CVaR  for items “fval” and “time” since MIP is not suitable for large-scale data sets and the solution  returned by CVaR is too conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Table 1: Comparison of different methods on the portfolio selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content='3827  time  1225  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='3467  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='093  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content='9493  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content='42  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content='9100  “*” indicates that the computed probability is lower than the targeted level in Problem (2), which implies the  returned solution is not feasible (the same for Table 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The magnitude of fval is 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We observe from Table 1 that although MIP achieves the lowest objective value, it is the  most time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In addition, we observe that pDCA is slightly better than DCA and  both pDCA and DCA generally outperform CVaR, Bi-CVaR, ALDM, and SCA in terms of  the objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Table 1 also demonstrates that CVaR is the fastest method, while DCA  and pDCA are comparable to the remaining ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Finally, we also observe that the in-sample  probabilities of DCA and pDCA are generally comparable to those of the other methods, except  that ALDM fails to satisfy the chance constraint for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='05 and sometimes is too conservative  for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  26  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2  Probabilistic Transportation Problem with Convex Objective  In this subsection, we consider a probabilistic version of the classical transportation problem,  which has been widely studied in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=', [5, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' This problem is to minimize  the transportation cost of delivering products from n suppliers to m customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The customer  demands are random and the j-th customer’s demand is represented by a random variable ξj  for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The i-th supplier has a limited production capacity θi ∈ R+ for each  i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The cost of shipping a unit of product from supplier i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , n} to customer  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m} is cij ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Suppose that the shipment quantities are required to be determined  before the customer demands are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' By letting xij denote the amount of shipment delivered  from supplier i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , n} to customer j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m}, this problem is formulated as  min  x∈Rn×m  n  �  i=1  m  �  j=1  cijxij  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content='  P  � n  �  i=1  xij ≥ ξj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m  �  ≥ 1 − α,  m  �  j=1  xij ≤ θi, xij ≥ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , n, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content='  27  In our experiments, we use the setting in Luedtke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' [50] to generate parameters (θ, c, ˆξ),  which is downloaded from http://homepages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content=' In particular, we  choose (n, m) = (40, 100) and N = 500, 1000, 1500, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We set β0 = 1, 10 for pDCA-1 and  pDCA-2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We report the experimental results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We observe that DCA  and pDCA in general can find significantly better solutions than CVaR and ALDM, and slightly  better solutions than Bi-CVaR and SCA in terms of objective values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Meanwhile, we see that  MIP returns either global optimal solutions or best objective values among all the algorithms  in the time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We also observe that the CPU time of the DCA is less than Bi-CVaR and  ALDM, much less than that of MIP and pDCA, and is slightly larger than that of CVaR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We should mention that pDCA is the most time-consuming among the tested methods, since  it solves a quadratic programming subproblem in each iteration, while other methods solve a  linear programming subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Table 2 also indicates that the in-sample probabilities of DCA  and pDCA are exactly the risk level 1 − α in all instances, while the in-sample probabilities of  ALDM and SCA may be either too loose or too conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='3  Probabilistic Transportation Problem with Non-Convex Objective  In this subsection, we consider a probabilistic version of the classical transportation problem  with a non-convex objective function, which has been studied in [5, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In particular, the setting  of this problem is exactly the same as that in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='2 except for the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Here,  we assume that the transportation cost from supplier i to customer j consists of the normal  cost cijxij and cost discount aijx2  ij (aij < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Consequently, this problem can be formulated as  min  x∈Rn×m  n  �  i=1  m  �  j=1  cijxij + aijx2  ij  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  P  � n  �  i=1  xij ≥ ξj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m  �  ≥ 1 − α,  m  �  j=1  xij ≤ θi, xij ≥ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , n, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' , m,  (69)  In our test, we set aij = −cij/ (2θi) for all i, j, and the remaining setting is the same as that in  the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  Since the objective function of this problem is non-convex, CVaR and Bi-CVaR cannot  handle this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Then, we only compare our proposed method with MIP, ALDM, and  SCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' To generate a feasible initial point, we apply CVaR to solve Problem (69) without cost  discount in the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We report the experimental results in Table 3, which are  similar to those of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  We further point out that although MIP achieves the lowest  objective value, it reaches the time limit for all the instances, which indicates the hardness of  the additional non-convex term in the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' In terms of objective values and running time,  we observe that DCA generally outperforms pDCA, ALDM, and SCA in most of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The  CPU time for DCA is similar to that of the convex case in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' The reason is similar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=',  the subproblems of DCA are all linear programs like the convex cases in (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We also observe  that the in-sample probabilities of DCA and pDCA are generally closer to the risk level 1 − α  than ALDM and SCA in all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  28  Table 3: Comparison of different methods on the probabilistic transportation problem with a  non-convex objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+page_content='  6  Conclusions  In this paper, we proposed a new DC reformulation based on the empirical quantile for solving  data-driven chance constrained programs and proposed a proximal DC algorithm to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We  proved the subsequential and sequential convergence to a KKT point of the proposed method  and derived the iteration complexity for computing an approximate KKT point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We point out  that our analysis holds for general DC constrained DC programs beyond those reformulated  from chance constrained programs and can be extended to DC programs with multiple DC  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' We also show possible extensions of our methods to L-estimators for quantile in  chance constrained programs and cardinality constrained programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content=' Finally, we demonstrated  the efficiency and efficacy of the proposed method via numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
+page_content='  29  Acknowledgements  We would like to thank Lai Tian (The Chinese University of Hong Kong) for the fruitful dis-  cussion of the theoretical analysis of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNAyT4oBgHgl3EQfj_gw/content/2301.00423v1.pdf'}
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+DESY-23-004
+The Tachyonic Higgs and the Inflationary Universe
+Bibhushan Shakya
+Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
+The Standard Model Higgs becomes tachyonic at high energy scales according to current mea-
+surements. This unstable regime of the Higgs potential can be realized in the early Universe during
+high scale inflation, potentially with catastrophic consequences.
+This letter highlights a crucial
+inherent feature of such configurations that has so far remained ignored: Higgs particle produc-
+tion out of vacuum induced by the rapidly evolving Higgs field, which gets exponentially enhanced
+due to the tachyonic instability. Such explosive particle production can rapidly drain energy away
+from the Higgs field, sustaining a significant density of Higgs particles even during inflation, and
+could initiate a qualitatively different form of preheating in parts of the post-inflationary Universe.
+Any study of the Higgs field in its tachyonic phase, either during or after inflation, must therefore
+take this substantial particle energy density into account, which could significantly affect the sub-
+sequent evolution of such systems. This could carry important implications for high scale inflation,
+post-inflationary preheating, observable signals in the cosmic microwave background, gravitational
+waves, and primordial black holes, as well as deeper concepts ranging from eternal inflation to the
+metastability of the electroweak vacuum.
+I.
+MOTIVATION
+Current
+measurements
+indicate
+that
+the
+Standard
+Model (SM) Higgs potential is unstable at high scales, and
+the electroweak (EW) vacuum that our Universe exists in is
+metastable, albeit with a decay lifetime significantly longer
+than the current age of the Universe. However, the Higgs
+could have briefly existed in this unstable regime in the
+early Universe due to quantum fluctuations during a pe-
+riod of high scale inflation. Such configurations have been
+extensively studied in the literature [1–14], and the con-
+sequences are believed to be catastrophic: the Higgs field
+rapidly evolves to regions of negative potential energy that
+can terminate inflation, resulting in crunching anti-de Sit-
+ter (AdS) space that grows to engulf all of spacetime, ren-
+dering the existence of a Universe such as ours impossible.
+This fate can be avoided with nonminimal modifications
+of the Higgs potential that stabilize it before reaching such
+regimes (see e.g. [8, 9, 15–26]). However, in the absence of
+such stabilizing corrections, the SM Higgs appears to be
+incompatible with inflation scales greater than the insta-
+bility scale of the Higgs potential.
+In this Letter, we study the effects of Higgs particle pro-
+duction in the tachyonic regime during inflation. It is well
+known that the tachyonic instability triggers an expon-
+tential growth of particle number [27–29]. Some previous
+papers [14, 30, 31] that considered particle production and
+tachyonic growth of inhomogeneities in this regime during
+inflation found such effects to be negligible; however, these
+papers only considered Hubble-induced effects, i.e. those
+sourced by the inflationary background.
+In this paper,
+we focus on particle production induced by the dynam-
+ics of the Higgs field itself. It is well known that a non-
+adiabatically changing background field can produce par-
+ticles out of vacuum; this phenomenon is encountered in
+many familiar contexts, such as the Schwinger mechanism,
+Hawking radiation from black holes, or gravitational par-
+ticle production. Although the energy density in the Higgs
+field is subdominant to the inflaton energy density in our
+regime of interest, which might have led previous studies
+to ignore this effect, we will see that particle production
+induced by the Higgs field evolution is an important effect,
+due to the fact that the Higgs field can reach significantly
+larger values than the Hubble scale during inflation.
+A substantial population of Higgs particles produced out
+of the Higgs field during inflation can have several impor-
+tant consequences. It can draw energy out of the Higgs
+field, slowing its evolution towards catastrophic values,
+as well as produce stabilizing thermal corrections to the
+Higgs potential.
+It can terminate inflation locally once
+its energy density becomes comparable to the inflaton en-
+ergy density, resulting in emergence out of inflation into a
+preheated state, much as in warm inflation scenarios [32],
+rather than into AdS. Even the collapse into AdS, cur-
+rently believed to be catastrophic, could become benign
+due to modified evolution due to the significant energy
+density in particles. Such considerations reopen the pos-
+sibility of restoring the Universe to the EW vacuum after
+reheating, thereby making high scale inflation compatible
+with the Higgs instability. The presence of a large den-
+sity of particles in some Hubble patches could also lead to
+various observables signals of such inhomogeneities, such
+as imprints in the cosmic microwave background (CMB),
+gravitational waves, and primordial black holes.
+The main purpose of this Letter is to demonstrate that
+excursions of the Higgs to large field values of its unstable
+potential is necessarily accompanied by a huge energy den-
+sity of Higgs particles, even during inflation, which can af-
+fect the subsequent evolution of the Higgs field. Section II
+describes the framework for the study. Section III presents
+the calculation of particle production from Higgs evolution
+and tachyonic instability during inflation.
+Backreaction
+effects of particle production are addressed in Section IV,
+followed by qualitative discussions of the post-inflationary
+evolution of such regions (Section V) and observable sig-
+nals of such configurations (Section VI). Section VII is de-
+voted to a discussion of open questions and broader impli-
+cations.
+arXiv:2301.08754v1  [hep-ph]  20 Jan 2023
+
+2
+II.
+FRAMEWORK: HIGGS EVOLUTION
+The Standard Model Higgs potential develops an insta-
+bility scale at ΛI ∼ 1011 GeV due to the Higgs quartic
+coupling running to negative values (see e.g. [33]). Above
+this scale, the Higgs potential can be written as
+V (h) ≈ −λ
+4 h4 .
+(1)
+We can approximate λ ≈ 0.01 for our purposes. The (field-
+dependent) Higgs mass in this regime is tachyonic:
+m2
+h(h) = Vhh = −3λh2 ≈ −(0.17 h)2 < 0 .
+(2)
+The inflaton potential is
+Vφ = 3
+8π H2M 2
+P ,
+(3)
+where MP ≈ 1.2 × 1019 GeV is the Planck scale, and H
+is the Hubble scale during inflation, which we take to be
+constant.
+The evolution of the Higgs field in this setup has been
+studied in detail in several previous works [2, 4, 5, 8–10,
+13, 14].
+For small (sub-Hubble) Higgs field values, the
+dynamics is dominated by quantum fluctuations of size
+∼
+H
+2π induced by inflation, resulting in random coherent
+“jumps” of the Higgs field within entire Hubble patches,
+which remains the dominant driving force until the Higgs
+reaches h ≈ (3/2πλ)1/3H ≈ 3.6H. Beyond this, classical
+evolution driven by the Higgs potential takes over, and the
+equation of motion of the Higgs field is
+¨h + 3H ˙h = dV
+dh .
+(4)
+We will solve for Higgs evolution in this classical regime,
+with initial conditions h = 3.6H and ˙h = 0, to obtain the
+Higgs field value as a function of time, h(t). In the early
+stages of this regime, Hubble friction causes the Higgs field
+to slow-roll for several e-folds of inflation, until it reaches
+h ∼ hsr ≡ (3/λ)1/2H ≈ 17.3H
+(end of slow-roll)
+(5)
+Beyond this point, Hubble friction becomes negligible, and
+the Higgs field diverges quickly to very large values in less
+than a single e-fold.
+Note that the inflaton energy density dominates over the
+Higgs potential energy until
+h ∼ hI ≡
+� 3
+2πλ
+�1/4�
+H MP
+(exit from inflation)
+(6)
+For this paper, we take the scale of inflation to be of the
+same order as the Higgs instability scale, H ∼ ΛI; then the
+inflaton energy density dominates until h ≳ hI ∼ 104H. If
+the Higgs field gets to such large values in a Hubble patch,
+this terminates inflation locally, and the region rapidly col-
+lapses into AdS. These collapsing regions grow to engulf
+the surrounding spacetime after inflation has ended glob-
+ally [10]; therefore, the existence of even a single Hubble
+patch where the Higgs field extends beyond the slow-roll
+regime is posited to be catastrophic for the existence of
+our Universe [9, 10, 34]. However, as we will see below, it
+is precisely in this window beyond slow-roll, hsr < h < hI,
+that particle production becomes important, and could af-
+fect the subsequent evolution.
+III.
+PARTICLE PRODUCTION AND
+TACHYONIC GROWTH
+We now consider particle production from the evolving
+Higgs field in the regime hsr < h < hI. It is well known
+that particle production during inflation requires non-
+adiabatic conditions. Beyond the Higgs slow-roll regime
+h > hsr, the Higgs mass indeed changes non-adiabatically,
+| ˙mh/m2
+h| ∼ 1, as can be verified numerically using Eq. 2
+and the numerical solution for Eq. 4. This non-adiabatic
+evolution of the Higgs mass can therefore excite Higgs par-
+ticles out of the vacuum. 1 The standard approach to cal-
+culate the number density of particles produced from a
+non-adiabatically changing background is via the compu-
+tation of Bogoliubov coefficients (see e.g. [35–37]). Here,
+we first present a semi-analytic estimate that is computa-
+tionally simpler and offers greater intuition before compar-
+ing with numerical solutions.
+When the Higgs mass evolves non-adiabatically, modes
+with momenta k ≲ |mh| get populated with occupation
+number nk = |βk|2 ∼ 1, where βk is the Bogoliubov coeffi-
+cient of a positive frequency mode, corresponding to parti-
+cle excitation [35–38]. When the mass is tachyonic, the co-
+efficient gets further enhanced exponentially via the tachy-
+onic instability as β ∼ e−iωt = e|ω|t, where ω2 = m2
+h + k2,
+for modes with ω2 < 0. The Higgs particle energy den-
+sity as a function of the Higgs field value can therefore be
+estimated as 2
+ρP (h) =
+� mh
+H
+d3k
+(2π)3 |ω(h, k)|nk =
+1
+2π2
+� mh
+H
+k2dk|ω(h, k)|nk
+(7)
+where ω2 (h, k) = m2
+h(h) + k2. The mode occupation num-
+ber is evaluated as
+nk = |βk|2 ≈ Exp
+�
+2
+�
+|ω(h, k)|dt
+�
+,
+(8)
+1 All other SM particles that obtain mass from the Higgs mechanism
+also have non-adiabatically varying masses and can get excited out
+of vacuum in this phase; however, their masses are not tachyonic,
+hence the effects of their production are negligible.
+2 Strictly speaking, the interpretation of nk as the number of parti-
+cles with energy |ωk| is robust only at a stable point of the theory,
+not in the unstable regime while the background is changing; nev-
+ertheless we will adopt this interpretation here, as is commonly
+done in the literature.
+
+3
+where the integral is taken over all tachyonic regimes,
+i.e. over all k and t where ω2 < 0 holds.
+For particle production during inflation, two additional
+considerations must be taken into account:
+• Inflation redshifts momenta and dilutes number den-
+sities exponentially fast: k → k/a ≈ ke−Ht.
+• The amplitudes of modes that become larger than
+the horizon size during inflation, ie k<H, get frozen,
+and do not grow further.
+Thus, exponential growth can only occur for modes in the
+window H < k < |mh|, which sets the limits of the inte-
+gral in Eq. 7. Since momenta are exponentially redshifted,
+it might appear that sub-horizon modes have no time to
+grow substantially before crossing the horizon. However,
+as mentioned earlier, the rapidly evolving Higgs field tran-
+sitions from hsr to hI within a single e-fold of inflation,
+hence the above effects are not substantial (but are never-
+theless incorporated in the calculation). The crucial point
+to note here is that even a fraction of an e-fold can realize
+a significant growth factor e|ω|t ∼ e|mh|t when |mh| ≫ H.
+The integral in the exponent in Eq. 8 at time t can be
+evaluated as
+� t
+ti
+|ω(h, k)|dtv =
+� t
+ti
+�
+Max[−m2
+h(tv) − (ke(t−tv))2, 0] dtv .
+(9)
+Here, the Max function restricts the integral to the tachy-
+onic regimes ω2 < 0, whereas the e(t−tv) factor accounts
+for the exponential redshifting of momenta due to infla-
+tion. We take the starting point ti as the time at which
+h ≈ hsr, as this is when the Higgs evolution becomes non-
+adiabatic; in practice, the integral is dominated by con-
+tributions from later times, when |mh| is larger, and is
+therefore fairly insensitive to the exact choice of ti.
+As a cross-check, the estimate in Eq. 8 can be compared
+with numerical solutions for the Bogoliubov coefficients.
+Ignoring the expansion of space, the mode function Xk for
+a scalar field evolves as
+X′′
+k + (k2 + m2
+h(t))Xk = 0 .
+(10)
+The field can be set to be in vacuum at time ti by choos-
+ing the initial conditions Xk(ti) = 1/�2ωk,0, X′
+k(ti) =
+−i
+�
+ωk,0/2, where ω2
+k,0 = k2 + m2
+h(ti). The mode occupa-
+tion number at subsequent times is then evaluated as
+nk(t) = |βk(t)|2 = |X′
+k|2 + |ωk Xk|2
+2|ωk|
+− 1
+2 .
+(11)
+Choosing ti as the time when h = hsr (again, the numerical
+results are also insensitive to the exact choice of ti), it can
+be verified that the estimate for |βk| in Eq. 8 agrees with
+the numerical results from the above prescription (solving
+Eq. 10, 11) within an O(1) factor for all values of (k, h) of
+interest. With this confirmation, we will henceforth use
+Eq. 8, which is computationally simpler, to calculate the
+occupation number of particles.
+50 100
+5001000
+5000104
+0.01
+0.10
+1
+10
+100
+1000
+h / H
+ρP / |V(h)|
+FIG. 1: Ratio of energy density in Higgs particles to the po-
+tential energy of the Higgs field, as defined in Eq. 12, without
+taking any backreaction effects into account.
+Next, consider the ratio of energy density in Higgs par-
+ticles to the Higgs potential energy at a given time t
+ρP (t)
+|V (h(t))| =
+2
+π2λ
+� |mh(t)|
+H
+k2dk |ω(h, k)| nk(t)
+h4(t)
+.
+(12)
+Fig. 1 plots this ratio in the regime hsr < h < hI. The
+particle energy density starts as a sub-percent level effect
+at hsr, but becomes comparable to the potential energy in
+the Higgs field for h ≳ 220 H, at which point backreaction
+effects of particle production need to be taken into account.
+Numerically, it is found that the integral is dominated by
+the lowest values of k, since these are the modes that have
+been growing exponentially for the longest duration.
+The growth of this ratio can be understood as follows.
+Approximating |ω(h, k)| ≈ |mh| =
+√
+3λh, we have
+ρP (t)
+|V (h(t))| = 2
+√
+3
+π2√
+λ
+� |mh(t)|
+H
+�k |βk|
+h
+�2 dk
+h .
+(13)
+The prefactor is O(1) and can be ignored. In the absence
+of exponential growth, βk ∼ 1, and the integral evaluates
+to (1/3)|m3
+h|, giving
+ρP (t)
+|V (h(t))| ∼ (1/3)(3λ)
+3
+2 ≈ 10−3, a sub-
+percent level effect (as also seen in Fig. 1 for h ∼ hsr).
+However, the expression in Eq. 13 suggests that if k|βk|
+grows faster than h with time, an increasingly larger frac-
+tion of the Higgs potential energy can get transferred to
+Higgs particles.
+An insightful way to understand that this indeed occurs
+is to note that the Bogoliubov coefficient |βk| scales as the
+Higgs velocity ˙h for small k.
+This can be seen clearly,
+as pointed out in [31], by taking the time derivative of
+the Higgs equation of motion Eq. 4, which reveals that ˙h
+also follows the simple harmonic oscillator equation with
+negative frequency −|mh|. Hence both ˙h and |βk| grow
+exponentially as ∼ e|mh|t. This observation enables us to
+write
+|βk(t)| ∼
+˙h(t)
+˙hh=hsr
+≈ h2
+h2sr
+(for k ≪ |mh|)
+(14)
+
+4
+For the final step, we have used the fact that when Hub-
+ble friction is negligible, energy conservation dictates that
+the potential energy lost by the Higgs field gets converted
+entirely into its kinetic energy,
+1
+2 ˙h2 ≈
+λ
+4 h4.
+With this,
+k|βk|/h ∼ kh/h2
+sr, which clearly grows rapidly with time.
+Note, in particular, that for the lowest modes k ∼ H,
+k|βk|/h ∼ Hh/h2
+sr becomes O(1) for h ∼ (hsr/H)2H ∼
+300 H, which is consistent with Fig. 1.
+Let us briefly discuss aspects that may disrupt the expo-
+nential enhancement of Higgs number density. The most
+worrisome is the decay of Higgs particles into other SM
+particles. Since the coherent Higgs field value h sets the
+background vacuum expectation value (vev) in the Hub-
+ble patch, all SM particles are correspondingly heavier in
+this dynamic phase. In particular, since the Higgs mass
+is |mh|, decays into SM states that interact most strongly
+with the Higgs, i.e. W and Z gauge bosons and the top
+quark, are kinematically blocked, and the dominant decay
+is into b¯b and WW ∗. The decay rate into b¯b, for instance, is
+Γh ∼ 3y2
+b/(8π)|mh| ∼ 10−5h, thus Γh < H in the regime of
+interest hsr < h < hI, and Higgs decays can be neglected.
+The rate for annihilation processes is parametrically simi-
+lar, hence annihilations are likewise negligible.
+IV.
+BACKREACTION EFFECTS
+The previous section has demonstrated that the rapidly
+evolving tachyonic Higgs field triggers an exponential
+growth of particles, whose energy density naively exceeds
+that available from the Higgs potential for h ≳ 220H.
+Hence it becomes necessary to take into account the back-
+reaction of the population of particles on the evolution of
+the Higgs field. The production of particles affects Higgs
+dynamics in two ways. First, particle production takes en-
+ergy out of the Higgs field, slowing its evolution towards
+higher field values. Second, it introduces thermal correc-
+tions to the Higgs potential.
+The former effect is more
+important, hence we address it first.
+The energy loss from the Higgs field due to particle pro-
+duction can be incorporated into the Higgs equation of mo-
+tion Eq. 4 by adding a new friction term to account for the
+energy transferred to particles. Its effect will be to slow
+the evolution of the Higgs, i.e. decrease ˙h, which will in
+turn suppress the production of particles (Eq. 14). Since
+the energy density of particles itself involves integrating
+over the Higgs evolution up to that point (Eq. 8, 12), this
+results in a nontrivial, coupled system that must be solved
+numerically. We will, instead, make use of analytical es-
+timates that are sufficient for a qualitative understanding
+of the evolution of the system.
+The key insight is to make use of the observation that
+the particle mode occupation number scales as the square
+of the Higgs velocity, nk ∼ ˙h2 (recall Eq. 14). This can be
+combined with Eq. 13 to obtain an approximate expression
+50
+100
+500 1000
+5000 104
+0.001
+0.010
+0.100
+1
+h/H
+fraction in Higgs kinetic energy
+FIG. 2: Fraction of Higgs potential energy converted to the
+kinetic energy of the rolling Higgs field,
+1
+2 ˙h2/|V (h)|. The re-
+mainder goes into Higgs particle production.
+for the energy density in Higgs particles
+ρP (h) ∼
+�λ
+3
+�2 � h
+H
+�2
+˙h2 .
+(15)
+This expression provides an excellent match to Fig. 1, as
+can be checked numerically.
+Beyond the slow-roll regime h > hsr, Hubble friction
+is negligible, hence energy conservation dictates that the
+sum of the Higgs kinetic energy and the energy density in
+particles must be equal to the energy released from the
+Higgs potential,
+1
+2
+˙h2 + ρP (h) ≈ |V (h)| .
+(16)
+Substituting ρP (h) from Eq. 15 enables us to solve for ˙h as
+a function of h,
+˙h ∼
+�
+λ/2
+�
+1 + 2
+� λ
+3
+�2 � h
+H
+�2 h2 .
+(17)
+The Higgs kinetic energy obtained from Eq. 17 is shown in
+Fig. 2. This shows that as the Higgs field rolls from hsr to
+hI, an increasingly greater fraction of the released poten-
+tial energy goes into particle production. At h ≳ 103H,
+the Higgs kinetic energy is only a percent level component,
+and almost all of the released energy is instead in particles.
+We can also estimate the thermal correction to the Higgs
+potential due to the presence of the particle bath.
+Al-
+though the particle ensemble is produced in coherent states
+of various momenta and has not had the time to thermal-
+ize, we may nevertheless use energy conservation to assign
+an effective temperature Teff,
+π2
+30T 4
+eff ≈ ρP ≈ |V (h)|
+⇒
+Teff ∼
+�15λ
+2π2
+�1/4
+h ∼ 0.3 h .
+(18)
+The temperature correction to the Higgs potential from a
+thermal bath of Higgs particles at temperature T is given
+
+5
+by (see e.g. [39, 40])
+∆VT = T 4
+2π2
+� ∞
+0
+dz z2 ln
+�
+�
+�1 − e
+−
+�
+z2+
+|m2
+h|
+T 2
+1 − e−z
+�
+�
+� .
+(19)
+With T = Teff above, it can be checked numerically that
+∆VT ∼ 0.03 |V (h)|, hence such thermal corrections do not
+appreciably modify the Higgs potential. 3
+Let us briefly discuss the nature of the growing popula-
+tion of Higgs particles. Calculations of particle production
+from vacuum assume that the produced particle distribu-
+tion is homogeneous and isotropic. However, it is also stan-
+dard to interpret a high occupation number of scalars in a
+coherent state as a classical scalar wave with spatial extent
+k−1 [27, 28] (however, see also [41, 42]), which therefore in-
+troduces sub-horizon sized spatial inhomogeneities in the
+otherwise homogeneous background field. The dynamics
+of such large inhomogeneities can form localized, pseudo-
+stable configurations known as oscillons, which remains
+a topic of active research [43–48].
+In the most extreme
+cases, the development of large inhomogeneities can cause
+the fragmentation of the Higgs field itself 4, at which point
+it makes little sense to talk about a coherently evolving
+background Higgs field. Detailed understanding of such
+aspects requires lattice studies and is beyond the scope of
+this work; here we simply emphasize that understanding
+the dynamics of such configurations is necessary to under-
+stand the eventual fate of a spatial region with large Higgs
+inhomogeneities.
+Independent of such details, it is clear that the pop-
+ulation of Higgs particles eventually becomes the domi-
+nant form of energy in the Universe as h approaches hI,
+and we expect this to terminate inflation locally (if infla-
+tion has not ended already due to inflaton dynamics) when
+ρP ∼ Vφ + V (h).
+V.
+POST-INFLATIONARY EVOLUTION
+We now discuss post-inflationary evolution of regions
+where the Higgs has evolved to large field values h > hsr,
+in particular whether such regions could be compatible
+with the existence of our Universe given the results in
+the previous sections. A definitive resolution of this ques-
+tion requires numerical simulations of the dynamics of the
+Higgs inhomogeneities as well as of spacetime itself, which
+is beyond the scope of this paper and will be addressed
+in future work [51]; here we only provide some qualitative
+discussions.
+3 This conclusion remains unchanged even if the Higgs population
+decays into a thermal bath of SM particles.
+4 Similar scenarios have recently been encountered, for instance, in
+the context of resonant particle production leading to axion frag-
+mentation [49, 50].
+Higgs instability during preheating has been studied in
+previous works (such as [20, 24, 26, 52–57]), but these focus
+on the Higgs in the EW minimum, studying the destabi-
+lizing effects of tachyonic instabilities, or resonant particle
+production through some inflaton-Higgs coupling. Here we
+are interested, instead, in configurations where the Higgs
+field is already in the unstable region, at h > hsr. Previous
+works that considered this regime painted a bleak picture,
+concluding that the existence of such regions in our past
+lightcone has catastrophic consequences [8–10, 34]. Such
+conclusions were primarily based on two observations (see
+in particular [10] for detailed discussions): (1) any region
+with h ≳ hsr rapidly diverges to h > hI within a single e-
+fold, descending into crunching AdS, and post-inflationary
+reheating effects likely cannot provide large enough ther-
+mal corrections to restore the Higgs to the EW vacuum
+in such a short timeframe; (2) while the interior of the
+AdS region collapses into a black hole, the boundary of
+the AdS region grows outwards, thereby engulfing other
+Hubble patches where the Higgs field might remain in the
+good (EW) vacuum. These results, however, were obtained
+within frameworks that only considered the interplay of
+the potential and kinetic energies of the Higgs and the in-
+flaton, and must be reassessed in the presence of a large
+energy density in particles.
+Particle production improves the former consideration
+(1) in two respects: the Higgs velocity is slower in this
+regime (as seen from Eq. 17 and Fig. 2), and the local Hub-
+ble patch emerges out of inflation in a partially preheated
+state. One might hope that the slower Higgs velocity could
+significantly delay its approach to hI; however, the time
+� hI
+hsr dh/˙h for the Higgs to roll from hsr to hI still evaluates
+(using Eq. 17) to roughly an e-fold. This is reasonable, as
+the friction from particle production becomes significant
+only towards the very late stages of this evolution. Nev-
+ertheless, the emergence out of inflation into a partially
+preheated state may aid with the preheating/reheating
+through the inflaton. In particular, if inflaton decay oc-
+curs via efficient resonant processes such as parametric res-
+onance or tachyonic preheating, the coherent Higgs popu-
+lation (or its annihilation/decay products) already present
+can seed the growth of such resonances, making these even
+more efficient. Such explosive particle production from the
+inflaton can stabilize the Higgs potential through thermal
+corrections, driving it back to the EW vacuum. However,
+it should be noted that such effects require some nonmini-
+mal coupling between the inflaton and the SM, which will
+in general also modify the Higgs potential.
+Such effects are unlikely to rescue Hubble patches where
+h ∼ hI ∼ 104 H, where |V (h)| ∼ Vφ, even with instanta-
+neous reheating. On the other hand, in regions with the
+Higgs at somewhat smaller field values, e.g. h ≲ 103H,
+we have |V (h)|/Vφ ≲ 10−4, hence releasing even a small
+fraction of the inflaton energy density into particles could
+provide large enough corrections to stabilize the Higgs po-
+tential.
+It is well known that preheating from resonant
+effects can draw a significant fraction of the inflaton en-
+
+6
+ergy density within one, or a few, inflaton oscillations [27–
+29]. Fig. 2 shows that the kinetic energy of the Higgs is a
+percent level of the total energy released from the Higgs
+potential in this regime, hence the Higgs spends over an or-
+der magnitude longer time in this regime as a consequence
+of particle production, improving the prospects of rescu-
+ing such patches. Understanding whether this can in fact
+be realized requires model-dependent studies with specific
+models of inflation, as well as a more careful treatment of
+the particle population (including the evolution of inho-
+mogeneities / oscillons discussed above), which is beyond
+the scope of this paper. Nevertheless, the above considera-
+tions at least make it more plausible that post-inflationary
+reheating can rescue regions with h > hsr from collapsing
+into AdS.
+Even the seemingly inevitable descent into AdS might
+not be catastrophic. The outward growth of these collaps-
+ing AdS regions, as found in [10] (see also [14, 34]), was
+based on simulations that only considered potential and
+kinetic energies. It is possible that the inclusion of parti-
+cles, which can become the dominant energy component in
+these patches, could change this outlook, and these AdS
+regions – more accurately, regions with negative poten-
+tial energy [58] but dominated by particle energy density
+– could evolve, likely collapsing into black holes, without
+destroying neighboring regions where the Higgs is in the
+EW vacuum. A complete resolution of this question re-
+quires numerical simulations of the spacetime dynamics.
+VI.
+OBSERVABLE SIGNALS
+Next, we briefly discuss some observable consequences
+of large excursions of the Higgs field to h > hsr in the
+early Universe. As demonstrated in the previous sections,
+such excursions are inevitably accompanied by a large en-
+ergy density of Higgs particles in these Hubble patches.
+Note that while such particle densities could be a signif-
+icant, even dominant, component of energy in the local
+patch, from a global viewpoint the Higgs field distribution
+is peaked around h ∼ 0 [1–14], and only a small frac-
+tion of inflating Hubble patches have the Higgs field at
+such large values. Therefore, at the end of inflation, this
+configuration results in very large inhomogeneities in an
+extremely small fraction of the Universe volume. Such in-
+homogeneities, if sufficiently numerous, can nevertheless
+leave several observable imprints in our Universe today.
+Again, the detailed nature of such signals can only be de-
+duced after the post-inflationary dynamics of the Higgs in-
+homogeneities and spacetime is better understood, hence
+we only make some brief qualitative statements here.
+Imprints in the CMB: Particle production during infla-
+tion is known to modify the primordial power spectrum
+[59], and produce CMB “hotspots” [60].
+Gravitational Waves:
+Fluctuations in the Higgs field
+generated during inflation can source stochastic gravita-
+tional waves [61–63], which can be observed with LISA,
+the Einstein Telescope, or Advanced-Ligo [61].
+Black Holes:
+The Higgs overdensities, if sufficiently
+large, can collapse into black holes, giving rise to a popula-
+tion of primordial black holes that could survive to present
+times [31, 63] (and, with the right parameters, could even
+account for dark matter [31]).
+Detailed studies of such signals in scenarios where the
+Higgs instability is accompanied by a large population of
+Higgs particles will be performed in future work [51].
+VII.
+DISCUSSION
+This Letter has highlighted the importance of particle
+production from Higgs dynamics in its tachyonic phase
+during and after high scale inflation. It is shown that non-
+adiabatic evolution of the Higgs field beyond its slow-roll
+regime, h > hsr ∼ 17H, excites Higgs particles out of
+the vacuum, and the tachyonic instability enhances these
+particle numbers exponentially, resulting in the released
+Higgs potential energy being converted almost entirely to
+particle energy density. Therefore, regions of space where
+the Standard Model Higgs field has fluctuated to such large
+values sustain a significant population of Higgs particles.
+Thus, any study of the Higgs field in this regime must take
+this sizeable particle density into account.
+This remarkable result has many important implica-
+tions.
+When such regions exit inflation, the emergent
+patch is already in a preheated state, which could facil-
+itate efficient dissipation of the inflaton energy density
+if the inflaton decays through resonant effects that can
+build on this pre-existing particle abundance. This opens
+the prospects of rapid thermal corrections that could re-
+store these regions to the electroweak vacuum instead of
+them evolving into anti-de Sitter space. Even if regions
+with large Higgs field values fall into AdS, it is not clear
+whether, in the presence of large particle densities, such oc-
+currences are necessarily catastrophic for the macroscopic
+Universe.
+Likewise, the presence of such large inhomo-
+geneities in parts of the post-inflationary Universe can also
+leave observable imprints in the CMB, gravitational waves,
+or primordial black holes, which would represent tantaliz-
+ing evidence of the realization of the tachyonic phase of
+the SM Higgs in our cosmological history.
+The results in this Letter could also hold relevance for
+questions of a more fundamental nature. The Higgs po-
+tential is metastable only for a very narrow range of pa-
+rameters (Higgs and top quark masses), hence its realiza-
+tion in nature is somewhat of a mystery. In this context,
+it is possible that the metastable vacuum exists because
+it serves some important purpose in the early Universe,
+which could be related to its tachyonic phase, and perhaps
+the explosive particle production that comes with it. The
+tachyonic regime also plays an intriguing role in the con-
+text of eternal inflation: any patch that inflates for too
+long will inevitably find itself in the tachyonic phase of
+the Higgs, which will trigger a rapid growth of inhomo-
+
+7
+geneities, terminating inflation locally; in this sense, the
+tachyonic Higgs could act as a regulator of eternal infla-
+tion. 5
+Several aspects and implications of Higgs particle pro-
+duction were only touched upon briefly and qualitatively
+here, and require further detailed study.
+Of paramount
+importance is a better understanding of the evolution of
+the large particle number densities or inhomogeneities,
+which requires lattice studies. It would also be insightful
+to study the post-inflationary evolution of patches with
+large Higgs number densities within specific models of in-
+flation and (p)reheating, to understand the extent to which
+large Higgs field values can be saved from collapsing into
+AdS. Numerical simulations of the negative potential en-
+ergy regimes with a significant particle energy density are
+needed to clarify whether such regions can be compati-
+ble with the existence of our Universe. Likewise, careful
+derivation of the nature of observable signals of the tachy-
+onic phase and accompanying particle production, in par-
+ticular in the CMB, gravitational waves, and primordial
+black holes, will be crucial in making more direct con-
+nections with ongoing and future experimental programs.
+These aspects will be addressed in greater detail in future
+work [51].
+Acknowledgments
+It is a pleasure to thank Gian Giudice for multiple in-
+sightful discussions at various stages of this project. The
+author is also grateful to Antonio Riotto, Geraldine Ser-
+vant, Alexander Westphal, and Kathryn Zurek for help-
+ful discussions and feedback, and thanks the Berkeley
+Center for Theoretical Physics and the Lawrence Berke-
+ley National Laboratory for hospitality during the com-
+pletion of this project.
+This work is supported by the
+Deutsche Forschungsgemeinschaft under Germany’s Excel-
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diff --git a/hNFAT4oBgHgl3EQf9B6c/content/tmp_files/load_file.txt b/hNFAT4oBgHgl3EQf9B6c/content/tmp_files/load_file.txt
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@@ -0,0 +1,720 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf,len=719
+page_content='DESY-23-004  The Tachyonic Higgs and the Inflationary Universe  Bibhushan Shakya  Deutsches Elektronen-Synchrotron DESY, Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 85, 22607 Hamburg, Germany  The Standard Model Higgs becomes tachyonic at high energy scales according to current mea-  surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' This unstable regime of the Higgs potential can be realized in the early Universe during  high scale inflation, potentially with catastrophic consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  This letter highlights a crucial  inherent feature of such configurations that has so far remained ignored: Higgs particle produc-  tion out of vacuum induced by the rapidly evolving Higgs field, which gets exponentially enhanced  due to the tachyonic instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Such explosive particle production can rapidly drain energy away  from the Higgs field, sustaining a significant density of Higgs particles even during inflation, and  could initiate a qualitatively different form of preheating in parts of the post-inflationary Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Any study of the Higgs field in its tachyonic phase, either during or after inflation, must therefore  take this substantial particle energy density into account, which could significantly affect the sub-  sequent evolution of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' This could carry important implications for high scale inflation,  post-inflationary preheating, observable signals in the cosmic microwave background, gravitational  waves, and primordial black holes, as well as deeper concepts ranging from eternal inflation to the  metastability of the electroweak vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  MOTIVATION  Current  measurements  indicate  that  the  Standard  Model (SM) Higgs potential is unstable at high scales, and  the electroweak (EW) vacuum that our Universe exists in is  metastable, albeit with a decay lifetime significantly longer  than the current age of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' However, the Higgs  could have briefly existed in this unstable regime in the  early Universe due to quantum fluctuations during a pe-  riod of high scale inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Such configurations have been  extensively studied in the literature [1–14], and the con-  sequences are believed to be catastrophic: the Higgs field  rapidly evolves to regions of negative potential energy that  can terminate inflation, resulting in crunching anti-de Sit-  ter (AdS) space that grows to engulf all of spacetime, ren-  dering the existence of a Universe such as ours impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  This fate can be avoided with nonminimal modifications  of the Higgs potential that stabilize it before reaching such  regimes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' [8, 9, 15–26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' However, in the absence of  such stabilizing corrections, the SM Higgs appears to be  incompatible with inflation scales greater than the insta-  bility scale of the Higgs potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  In this Letter, we study the effects of Higgs particle pro-  duction in the tachyonic regime during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' It is well  known that the tachyonic instability triggers an expon-  tential growth of particle number [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Some previous  papers [14, 30, 31] that considered particle production and  tachyonic growth of inhomogeneities in this regime during  inflation found such effects to be negligible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' however, these  papers only considered Hubble-induced effects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' those  sourced by the inflationary background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  In this paper,  we focus on particle production induced by the dynam-  ics of the Higgs field itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' It is well known that a non-  adiabatically changing background field can produce par-  ticles out of vacuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' this phenomenon is encountered in  many familiar contexts, such as the Schwinger mechanism,  Hawking radiation from black holes, or gravitational par-  ticle production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Although the energy density in the Higgs  field is subdominant to the inflaton energy density in our  regime of interest, which might have led previous studies  to ignore this effect, we will see that particle production  induced by the Higgs field evolution is an important effect,  due to the fact that the Higgs field can reach significantly  larger values than the Hubble scale during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  A substantial population of Higgs particles produced out  of the Higgs field during inflation can have several impor-  tant consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' It can draw energy out of the Higgs  field, slowing its evolution towards catastrophic values,  as well as produce stabilizing thermal corrections to the  Higgs potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  It can terminate inflation locally once  its energy density becomes comparable to the inflaton en-  ergy density, resulting in emergence out of inflation into a  preheated state, much as in warm inflation scenarios [32],  rather than into AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Even the collapse into AdS, cur-  rently believed to be catastrophic, could become benign  due to modified evolution due to the significant energy  density in particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Such considerations reopen the pos-  sibility of restoring the Universe to the EW vacuum after  reheating, thereby making high scale inflation compatible  with the Higgs instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The presence of a large den-  sity of particles in some Hubble patches could also lead to  various observables signals of such inhomogeneities, such  as imprints in the cosmic microwave background (CMB),  gravitational waves, and primordial black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The main purpose of this Letter is to demonstrate that  excursions of the Higgs to large field values of its unstable  potential is necessarily accompanied by a huge energy den-  sity of Higgs particles, even during inflation, which can af-  fect the subsequent evolution of the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Section II  describes the framework for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Section III presents  the calculation of particle production from Higgs evolution  and tachyonic instability during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Backreaction  effects of particle production are addressed in Section IV,  followed by qualitative discussions of the post-inflationary  evolution of such regions (Section V) and observable sig-  nals of such configurations (Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Section VII is de-  voted to a discussion of open questions and broader impli-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='08754v1  [hep-ph]  20 Jan 2023  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  FRAMEWORK: HIGGS EVOLUTION  The Standard Model Higgs potential develops an insta-  bility scale at ΛI ∼ 1011 GeV due to the Higgs quartic  coupling running to negative values (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Above  this scale, the Higgs potential can be written as  V (h) ≈ −λ  4 h4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (1)  We can approximate λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='01 for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The (field-  dependent) Higgs mass in this regime is tachyonic:  m2  h(h) = Vhh = −3λh2 ≈ −(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='17 h)2 < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (2)  The inflaton potential is  Vφ = 3  8π H2M 2  P ,  (3)  where MP ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='2 × 1019 GeV is the Planck scale, and H  is the Hubble scale during inflation, which we take to be  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The evolution of the Higgs field in this setup has been  studied in detail in several previous works [2, 4, 5, 8–10,  13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  For small (sub-Hubble) Higgs field values, the  dynamics is dominated by quantum fluctuations of size  ∼  H  2π induced by inflation, resulting in random coherent  “jumps” of the Higgs field within entire Hubble patches,  which remains the dominant driving force until the Higgs  reaches h ≈ (3/2πλ)1/3H ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='6H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Beyond this, classical  evolution driven by the Higgs potential takes over, and the  equation of motion of the Higgs field is  ¨h + 3H ˙h = dV  dh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (4)  We will solve for Higgs evolution in this classical regime,  with initial conditions h = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='6H and ˙h = 0, to obtain the  Higgs field value as a function of time, h(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' In the early  stages of this regime, Hubble friction causes the Higgs field  to slow-roll for several e-folds of inflation, until it reaches  h ∼ hsr ≡ (3/λ)1/2H ≈ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='3H  (end of slow-roll)  (5)  Beyond this point, Hubble friction becomes negligible, and  the Higgs field diverges quickly to very large values in less  than a single e-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Note that the inflaton energy density dominates over the  Higgs potential energy until  h ∼ hI ≡  � 3  2πλ  �1/4�  H MP  (exit from inflation)  (6)  For this paper, we take the scale of inflation to be of the  same order as the Higgs instability scale, H ∼ ΛI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' then the  inflaton energy density dominates until h ≳ hI ∼ 104H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' If  the Higgs field gets to such large values in a Hubble patch,  this terminates inflation locally, and the region rapidly col-  lapses into AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' These collapsing regions grow to engulf  the surrounding spacetime after inflation has ended glob-  ally [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' therefore, the existence of even a single Hubble  patch where the Higgs field extends beyond the slow-roll  regime is posited to be catastrophic for the existence of  our Universe [9, 10, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' However, as we will see below, it  is precisely in this window beyond slow-roll, hsr < h < hI,  that particle production becomes important, and could af-  fect the subsequent evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  PARTICLE PRODUCTION AND  TACHYONIC GROWTH  We now consider particle production from the evolving  Higgs field in the regime hsr < h < hI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' It is well known  that particle production during inflation requires non-  adiabatic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Beyond the Higgs slow-roll regime  h > hsr, the Higgs mass indeed changes non-adiabatically,  | ˙mh/m2  h| ∼ 1, as can be verified numerically using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 2  and the numerical solution for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' This non-adiabatic  evolution of the Higgs mass can therefore excite Higgs par-  ticles out of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 1 The standard approach to cal-  culate the number density of particles produced from a  non-adiabatically changing background is via the compu-  tation of Bogoliubov coefficients (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' [35–37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Here,  we first present a semi-analytic estimate that is computa-  tionally simpler and offers greater intuition before compar-  ing with numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  When the Higgs mass evolves non-adiabatically, modes  with momenta k ≲ |mh| get populated with occupation  number nk = |βk|2 ∼ 1, where βk is the Bogoliubov coeffi-  cient of a positive frequency mode, corresponding to parti-  cle excitation [35–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' When the mass is tachyonic, the co-  efficient gets further enhanced exponentially via the tachy-  onic instability as β ∼ e−iωt = e|ω|t, where ω2 = m2  h + k2,  for modes with ω2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The Higgs particle energy den-  sity as a function of the Higgs field value can therefore be  estimated as 2  ρP (h) =  � mh  H  d3k  (2π)3 |ω(h, k)|nk =  1  2π2  � mh  H  k2dk|ω(h, k)|nk  (7)  where ω2 (h, k) = m2  h(h) + k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The mode occupation num-  ber is evaluated as  nk = |βk|2 ≈ Exp  �  2  �  |ω(h, k)|dt  �  ,  (8)  1 All other SM particles that obtain mass from the Higgs mechanism  also have non-adiabatically varying masses and can get excited out  of vacuum in this phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' however, their masses are not tachyonic,  hence the effects of their production are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  2 Strictly speaking, the interpretation of nk as the number of parti-  cles with energy |ωk| is robust only at a stable point of the theory,  not in the unstable regime while the background is changing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' nev-  ertheless we will adopt this interpretation here, as is commonly  done in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  3  where the integral is taken over all tachyonic regimes,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' over all k and t where ω2 < 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  For particle production during inflation, two additional  considerations must be taken into account:  Inflation redshifts momenta and dilutes number den-  sities exponentially fast: k → k/a ≈ ke−Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The amplitudes of modes that become larger than  the horizon size during inflation, ie k<H, get frozen,  and do not grow further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Thus, exponential growth can only occur for modes in the  window H < k < |mh|, which sets the limits of the inte-  gral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Since momenta are exponentially redshifted,  it might appear that sub-horizon modes have no time to  grow substantially before crossing the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' However,  as mentioned earlier, the rapidly evolving Higgs field tran-  sitions from hsr to hI within a single e-fold of inflation,  hence the above effects are not substantial (but are never-  theless incorporated in the calculation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The crucial point  to note here is that even a fraction of an e-fold can realize  a significant growth factor e|ω|t ∼ e|mh|t when |mh| ≫ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The integral in the exponent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 8 at time t can be  evaluated as  � t  ti  |ω(h, k)|dtv =  � t  ti  �  Max[−m2  h(tv) − (ke(t−tv))2, 0] dtv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (9)  Here, the Max function restricts the integral to the tachy-  onic regimes ω2 < 0, whereas the e(t−tv) factor accounts  for the exponential redshifting of momenta due to infla-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' We take the starting point ti as the time at which  h ≈ hsr, as this is when the Higgs evolution becomes non-  adiabatic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' in practice, the integral is dominated by con-  tributions from later times, when |mh| is larger, and is  therefore fairly insensitive to the exact choice of ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  As a cross-check, the estimate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 8 can be compared  with numerical solutions for the Bogoliubov coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Ignoring the expansion of space, the mode function Xk for  a scalar field evolves as  X′′  k + (k2 + m2  h(t))Xk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (10)  The field can be set to be in vacuum at time ti by choos-  ing the initial conditions Xk(ti) = 1/�2ωk,0, X′  k(ti) =  −i  �  ωk,0/2, where ω2  k,0 = k2 + m2  h(ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The mode occupa-  tion number at subsequent times is then evaluated as  nk(t) = |βk(t)|2 = |X′  k|2 + |ωk Xk|2  2|ωk|  − 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (11)  Choosing ti as the time when h = hsr (again, the numerical  results are also insensitive to the exact choice of ti), it can  be verified that the estimate for |βk| in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 8 agrees with  the numerical results from the above prescription (solving  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 10, 11) within an O(1) factor for all values of (k, h) of  interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' With this confirmation, we will henceforth use  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 8, which is computationally simpler, to calculate the  occupation number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  50 100  5001000  5000104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='10  1  10  100  1000  h / H  ρP / |V(h)|  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 1: Ratio of energy density in Higgs particles to the po-  tential energy of the Higgs field, as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 12, without  taking any backreaction effects into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Next, consider the ratio of energy density in Higgs par-  ticles to the Higgs potential energy at a given time t  ρP (t)  |V (h(t))| =  2  π2λ  � |mh(t)|  H  k2dk |ω(h, k)| nk(t)  h4(t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (12)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 1 plots this ratio in the regime hsr < h < hI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The  particle energy density starts as a sub-percent level effect  at hsr, but becomes comparable to the potential energy in  the Higgs field for h ≳ 220 H, at which point backreaction  effects of particle production need to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Numerically, it is found that the integral is dominated by  the lowest values of k, since these are the modes that have  been growing exponentially for the longest duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The growth of this ratio can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Approximating |ω(h, k)| ≈ |mh| =  √  3λh, we have  ρP (t)  |V (h(t))| = 2  √  3  π2√  λ  � |mh(t)|  H  �k |βk|  h  �2 dk  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (13)  The prefactor is O(1) and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' In the absence  of exponential growth, βk ∼ 1, and the integral evaluates  to (1/3)|m3  h|, giving  ρP (t)  |V (h(t))| ∼ (1/3)(3λ)  3  2 ≈ 10−3, a sub-  percent level effect (as also seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 1 for h ∼ hsr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  However, the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 13 suggests that if k|βk|  grows faster than h with time, an increasingly larger frac-  tion of the Higgs potential energy can get transferred to  Higgs particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  An insightful way to understand that this indeed occurs  is to note that the Bogoliubov coefficient |βk| scales as the  Higgs velocity ˙h for small k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  This can be seen clearly,  as pointed out in [31], by taking the time derivative of  the Higgs equation of motion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 4, which reveals that ˙h  also follows the simple harmonic oscillator equation with  negative frequency −|mh|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Hence both ˙h and |βk| grow  exponentially as ∼ e|mh|t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' This observation enables us to  write  |βk(t)| ∼  ˙h(t)  ˙hh=hsr  ≈ h2  h2sr  (for k ≪ |mh|)  (14)  4  For the final step, we have used the fact that when Hub-  ble friction is negligible, energy conservation dictates that  the potential energy lost by the Higgs field gets converted  entirely into its kinetic energy,  1  2 ˙h2 ≈  λ  4 h4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  With this,  k|βk|/h ∼ kh/h2  sr, which clearly grows rapidly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Note, in particular, that for the lowest modes k ∼ H,  k|βk|/h ∼ Hh/h2  sr becomes O(1) for h ∼ (hsr/H)2H ∼  300 H, which is consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Let us briefly discuss aspects that may disrupt the expo-  nential enhancement of Higgs number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The most  worrisome is the decay of Higgs particles into other SM  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Since the coherent Higgs field value h sets the  background vacuum expectation value (vev) in the Hub-  ble patch, all SM particles are correspondingly heavier in  this dynamic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' In particular, since the Higgs mass  is |mh|, decays into SM states that interact most strongly  with the Higgs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' W and Z gauge bosons and the top  quark, are kinematically blocked, and the dominant decay  is into b¯b and WW ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The decay rate into b¯b, for instance, is  Γh ∼ 3y2  b/(8π)|mh| ∼ 10−5h, thus Γh < H in the regime of  interest hsr < h < hI, and Higgs decays can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The rate for annihilation processes is parametrically simi-  lar, hence annihilations are likewise negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  BACKREACTION EFFECTS  The previous section has demonstrated that the rapidly  evolving tachyonic Higgs field triggers an exponential  growth of particles, whose energy density naively exceeds  that available from the Higgs potential for h ≳ 220H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Hence it becomes necessary to take into account the back-  reaction of the population of particles on the evolution of  the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The production of particles affects Higgs  dynamics in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' First, particle production takes en-  ergy out of the Higgs field, slowing its evolution towards  higher field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Second, it introduces thermal correc-  tions to the Higgs potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The former effect is more  important, hence we address it first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The energy loss from the Higgs field due to particle pro-  duction can be incorporated into the Higgs equation of mo-  tion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 4 by adding a new friction term to account for the  energy transferred to particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Its effect will be to slow  the evolution of the Higgs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' decrease ˙h, which will in  turn suppress the production of particles (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Since  the energy density of particles itself involves integrating  over the Higgs evolution up to that point (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 8, 12), this  results in a nontrivial, coupled system that must be solved  numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' We will, instead, make use of analytical es-  timates that are sufficient for a qualitative understanding  of the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The key insight is to make use of the observation that  the particle mode occupation number scales as the square  of the Higgs velocity, nk ∼ ˙h2 (recall Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' This can be  combined with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 13 to obtain an approximate expression  50  100  500 1000  5000 104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='100  1  h/H  fraction in Higgs kinetic energy  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 2: Fraction of Higgs potential energy converted to the  kinetic energy of the rolling Higgs field,  1  2 ˙h2/|V (h)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The re-  mainder goes into Higgs particle production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  for the energy density in Higgs particles  ρP (h) ∼  �λ  3  �2 � h  H  �2  ˙h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (15)  This expression provides an excellent match to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 1, as  can be checked numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Beyond the slow-roll regime h > hsr, Hubble friction  is negligible, hence energy conservation dictates that the  sum of the Higgs kinetic energy and the energy density in  particles must be equal to the energy released from the  Higgs potential,  1  2  ˙h2 + ρP (h) ≈ |V (h)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (16)  Substituting ρP (h) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 15 enables us to solve for ˙h as  a function of h,  ˙h ∼  �  λ/2  �  1 + 2  � λ  3  �2 � h  H  �2 h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (17)  The Higgs kinetic energy obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 17 is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' This shows that as the Higgs field rolls from hsr to  hI, an increasingly greater fraction of the released poten-  tial energy goes into particle production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' At h ≳ 103H,  the Higgs kinetic energy is only a percent level component,  and almost all of the released energy is instead in particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  We can also estimate the thermal correction to the Higgs  potential due to the presence of the particle bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Al-  though the particle ensemble is produced in coherent states  of various momenta and has not had the time to thermal-  ize, we may nevertheless use energy conservation to assign  an effective temperature Teff,  π2  30T 4  eff ≈ ρP ≈ |V (h)|  ⇒  Teff ∼  �15λ  2π2  �1/4  h ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='3 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (18)  The temperature correction to the Higgs potential from a  thermal bath of Higgs particles at temperature T is given  5  by (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' [39, 40])  ∆VT = T 4  2π2  � ∞  0  dz z2 ln  �  �  �1 − e  −  �  z2+  |m2  h|  T 2  1 − e−z  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  (19)  With T = Teff above, it can be checked numerically that  ∆VT ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='03 |V (h)|, hence such thermal corrections do not  appreciably modify the Higgs potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 3  Let us briefly discuss the nature of the growing popula-  tion of Higgs particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Calculations of particle production  from vacuum assume that the produced particle distribu-  tion is homogeneous and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' However, it is also stan-  dard to interpret a high occupation number of scalars in a  coherent state as a classical scalar wave with spatial extent  k−1 [27, 28] (however, see also [41, 42]), which therefore in-  troduces sub-horizon sized spatial inhomogeneities in the  otherwise homogeneous background field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The dynamics  of such large inhomogeneities can form localized, pseudo-  stable configurations known as oscillons, which remains  a topic of active research [43–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  In the most extreme  cases, the development of large inhomogeneities can cause  the fragmentation of the Higgs field itself 4, at which point  it makes little sense to talk about a coherently evolving  background Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Detailed understanding of such  aspects requires lattice studies and is beyond the scope of  this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' here we simply emphasize that understanding  the dynamics of such configurations is necessary to under-  stand the eventual fate of a spatial region with large Higgs  inhomogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Independent of such details, it is clear that the pop-  ulation of Higgs particles eventually becomes the domi-  nant form of energy in the Universe as h approaches hI,  and we expect this to terminate inflation locally (if infla-  tion has not ended already due to inflaton dynamics) when  ρP ∼ Vφ + V (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  POST-INFLATIONARY EVOLUTION  We now discuss post-inflationary evolution of regions  where the Higgs has evolved to large field values h > hsr,  in particular whether such regions could be compatible  with the existence of our Universe given the results in  the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' A definitive resolution of this ques-  tion requires numerical simulations of the dynamics of the  Higgs inhomogeneities as well as of spacetime itself, which  is beyond the scope of this paper and will be addressed  in future work [51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' here we only provide some qualitative  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  3 This conclusion remains unchanged even if the Higgs population  decays into a thermal bath of SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  4 Similar scenarios have recently been encountered, for instance, in  the context of resonant particle production leading to axion frag-  mentation [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Higgs instability during preheating has been studied in  previous works (such as [20, 24, 26, 52–57]), but these focus  on the Higgs in the EW minimum, studying the destabi-  lizing effects of tachyonic instabilities, or resonant particle  production through some inflaton-Higgs coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Here we  are interested, instead, in configurations where the Higgs  field is already in the unstable region, at h > hsr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Previous  works that considered this regime painted a bleak picture,  concluding that the existence of such regions in our past  lightcone has catastrophic consequences [8–10, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Such  conclusions were primarily based on two observations (see  in particular [10] for detailed discussions): (1) any region  with h ≳ hsr rapidly diverges to h > hI within a single e-  fold, descending into crunching AdS, and post-inflationary  reheating effects likely cannot provide large enough ther-  mal corrections to restore the Higgs to the EW vacuum  in such a short timeframe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' (2) while the interior of the  AdS region collapses into a black hole, the boundary of  the AdS region grows outwards, thereby engulfing other  Hubble patches where the Higgs field might remain in the  good (EW) vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' These results, however, were obtained  within frameworks that only considered the interplay of  the potential and kinetic energies of the Higgs and the in-  flaton, and must be reassessed in the presence of a large  energy density in particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Particle production improves the former consideration  (1) in two respects: the Higgs velocity is slower in this  regime (as seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 17 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 2), and the local Hub-  ble patch emerges out of inflation in a partially preheated  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' One might hope that the slower Higgs velocity could  significantly delay its approach to hI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' however, the time  � hI  hsr dh/˙h for the Higgs to roll from hsr to hI still evaluates  (using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 17) to roughly an e-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' This is reasonable, as  the friction from particle production becomes significant  only towards the very late stages of this evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Nev-  ertheless, the emergence out of inflation into a partially  preheated state may aid with the preheating/reheating  through the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' In particular, if inflaton decay oc-  curs via efficient resonant processes such as parametric res-  onance or tachyonic preheating, the coherent Higgs popu-  lation (or its annihilation/decay products) already present  can seed the growth of such resonances, making these even  more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Such explosive particle production from the  inflaton can stabilize the Higgs potential through thermal  corrections, driving it back to the EW vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' However,  it should be noted that such effects require some nonmini-  mal coupling between the inflaton and the SM, which will  in general also modify the Higgs potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Such effects are unlikely to rescue Hubble patches where  h ∼ hI ∼ 104 H, where |V (h)| ∼ Vφ, even with instanta-  neous reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' On the other hand, in regions with the  Higgs at somewhat smaller field values, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' h ≲ 103H,  we have |V (h)|/Vφ ≲ 10−4, hence releasing even a small  fraction of the inflaton energy density into particles could  provide large enough corrections to stabilize the Higgs po-  tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  It is well known that preheating from resonant  effects can draw a significant fraction of the inflaton en-  6  ergy density within one, or a few, inflaton oscillations [27–  29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 2 shows that the kinetic energy of the Higgs is a  percent level of the total energy released from the Higgs  potential in this regime, hence the Higgs spends over an or-  der magnitude longer time in this regime as a consequence  of particle production, improving the prospects of rescu-  ing such patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Understanding whether this can in fact  be realized requires model-dependent studies with specific  models of inflation, as well as a more careful treatment of  the particle population (including the evolution of inho-  mogeneities / oscillons discussed above), which is beyond  the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Nevertheless, the above considera-  tions at least make it more plausible that post-inflationary  reheating can rescue regions with h > hsr from collapsing  into AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Even the seemingly inevitable descent into AdS might  not be catastrophic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The outward growth of these collaps-  ing AdS regions, as found in [10] (see also [14, 34]), was  based on simulations that only considered potential and  kinetic energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' It is possible that the inclusion of parti-  cles, which can become the dominant energy component in  these patches, could change this outlook, and these AdS  regions – more accurately, regions with negative poten-  tial energy [58] but dominated by particle energy density  – could evolve, likely collapsing into black holes, without  destroying neighboring regions where the Higgs is in the  EW vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' A complete resolution of this question re-  quires numerical simulations of the spacetime dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  OBSERVABLE SIGNALS  Next, we briefly discuss some observable consequences  of large excursions of the Higgs field to h > hsr in the  early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' As demonstrated in the previous sections,  such excursions are inevitably accompanied by a large en-  ergy density of Higgs particles in these Hubble patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Note that while such particle densities could be a signif-  icant, even dominant, component of energy in the local  patch, from a global viewpoint the Higgs field distribution  is peaked around h ∼ 0 [1–14], and only a small frac-  tion of inflating Hubble patches have the Higgs field at  such large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Therefore, at the end of inflation, this  configuration results in very large inhomogeneities in an  extremely small fraction of the Universe volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Such in-  homogeneities, if sufficiently numerous, can nevertheless  leave several observable imprints in our Universe today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Again, the detailed nature of such signals can only be de-  duced after the post-inflationary dynamics of the Higgs in-  homogeneities and spacetime is better understood, hence  we only make some brief qualitative statements here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Imprints in the CMB: Particle production during infla-  tion is known to modify the primordial power spectrum  [59], and produce CMB “hotspots” [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Gravitational Waves:  Fluctuations in the Higgs field  generated during inflation can source stochastic gravita-  tional waves [61–63], which can be observed with LISA,  the Einstein Telescope, or Advanced-Ligo [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Black Holes:  The Higgs overdensities, if sufficiently  large, can collapse into black holes, giving rise to a popula-  tion of primordial black holes that could survive to present  times [31, 63] (and, with the right parameters, could even  account for dark matter [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Detailed studies of such signals in scenarios where the  Higgs instability is accompanied by a large population of  Higgs particles will be performed in future work [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  DISCUSSION  This Letter has highlighted the importance of particle  production from Higgs dynamics in its tachyonic phase  during and after high scale inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' It is shown that non-  adiabatic evolution of the Higgs field beyond its slow-roll  regime, h > hsr ∼ 17H, excites Higgs particles out of  the vacuum, and the tachyonic instability enhances these  particle numbers exponentially, resulting in the released  Higgs potential energy being converted almost entirely to  particle energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Therefore, regions of space where  the Standard Model Higgs field has fluctuated to such large  values sustain a significant population of Higgs particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Thus, any study of the Higgs field in this regime must take  this sizeable particle density into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  This remarkable result has many important implica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  When such regions exit inflation, the emergent  patch is already in a preheated state, which could facil-  itate efficient dissipation of the inflaton energy density  if the inflaton decays through resonant effects that can  build on this pre-existing particle abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' This opens  the prospects of rapid thermal corrections that could re-  store these regions to the electroweak vacuum instead of  them evolving into anti-de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Even if regions  with large Higgs field values fall into AdS, it is not clear  whether, in the presence of large particle densities, such oc-  currences are necessarily catastrophic for the macroscopic  Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Likewise, the presence of such large inhomo-  geneities in parts of the post-inflationary Universe can also  leave observable imprints in the CMB, gravitational waves,  or primordial black holes, which would represent tantaliz-  ing evidence of the realization of the tachyonic phase of  the SM Higgs in our cosmological history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  The results in this Letter could also hold relevance for  questions of a more fundamental nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The Higgs po-  tential is metastable only for a very narrow range of pa-  rameters (Higgs and top quark masses), hence its realiza-  tion in nature is somewhat of a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' In this context,  it is possible that the metastable vacuum exists because  it serves some important purpose in the early Universe,  which could be related to its tachyonic phase, and perhaps  the explosive particle production that comes with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The  tachyonic regime also plays an intriguing role in the con-  text of eternal inflation: any patch that inflates for too  long will inevitably find itself in the tachyonic phase of  the Higgs, which will trigger a rapid growth of inhomo-  7  geneities, terminating inflation locally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' in this sense, the  tachyonic Higgs could act as a regulator of eternal infla-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' 5  Several aspects and implications of Higgs particle pro-  duction were only touched upon briefly and qualitatively  here, and require further detailed study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Of paramount  importance is a better understanding of the evolution of  the large particle number densities or inhomogeneities,  which requires lattice studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' It would also be insightful  to study the post-inflationary evolution of patches with  large Higgs number densities within specific models of in-  flation and (p)reheating, to understand the extent to which  large Higgs field values can be saved from collapsing into  AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Numerical simulations of the negative potential en-  ergy regimes with a significant particle energy density are  needed to clarify whether such regions can be compati-  ble with the existence of our Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' Likewise, careful  derivation of the nature of observable signals of the tachy-  onic phase and accompanying particle production, in par-  ticular in the CMB, gravitational waves, and primordial  black holes, will be crucial in making more direct con-  nections with ongoing and future experimental programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  These aspects will be addressed in greater detail in future  work [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  Acknowledgments  It is a pleasure to thank Gian Giudice for multiple in-  sightful discussions at various stages of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content=' The  author is also grateful to Antonio Riotto, Geraldine Ser-  vant, Alexander Westphal, and Kathryn Zurek for help-  ful discussions and feedback, and thanks the Berkeley  Center for Theoretical Physics and the Lawrence Berke-  ley National Laboratory for hospitality during the com-  pletion of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
+page_content='  This work is supported by the  Deutsche Forschungsgemeinschaft under Germany’s Excel-  lence Strategy - EXC 2121 Quantum Universe - 390833306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFAT4oBgHgl3EQf9B6c/content/2301.08754v1.pdf'}
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+Tree-structured Policy Planning with Learned Behavior Models
+Yuxiao Chen1, Peter Karkus1, Boris Ivanovic1, Xinshuo Weng1, and Marco Pavone1,2
+Abstract— Autonomous vehicles (AVs) need to reason about
+the multimodal behavior of neighboring agents while planning
+their own motion. Many existing trajectory planners seek a
+single trajectory that performs well under all plausible futures
+simultaneously, ignoring bi-directional interactions and thus
+leading to overly conservative plans. Policy planning, whereby
+the ego agent plans a policy that reacts to the environment’s
+multimodal behavior, is a promising direction as it can account
+for the action-reaction interactions between the AV and the
+environment. However, most existing policy planners do not
+scale to the complexity of real autonomous vehicle applications:
+they are either not compatible with modern deep learning
+prediction models, not interpretable, or not able to generate
+high quality trajectories. To fill this gap, we propose Tree
+Policy Planning (TPP), a policy planner that is compatible
+with state-of-the-art deep learning prediction models, generates
+multistage motion plans, and accounts for the influence of ego
+agent on the environment behavior. The key idea of TPP is to
+reduce the continuous optimization problem into a tractable
+discrete MDP through the construction of two tree structures:
+an ego trajectory tree for ego trajectory options, and a scenario
+tree for multi-modal ego-conditioned environment predictions.
+We demonstrate the efficacy of TPP in closed-loop simulations
+based on real-world nuScenes dataset and results show that TPP
+scales to realistic AV scenarios and significantly outperforms
+non-policy baselines.
+I. INTRODUCTION
+A key challenge of motion planning for autonomous
+vehicles (AV)s is reasoning about the interaction between the
+ego vehicle and neighboring agents. The task is commonly
+divided into two subproblems: trajectory prediction for other
+agents, and ego motion planning with prediction. Trajectory
+prediction has seen substantial progress in recent years,
+coming from simple kineamtic models [1] to powerful deep
+learning models [2]–[6] capable of generating high-quality,
+multi-modal predictions. However, motion planning with
+such high-capacity prediction models remains a challenge.
+Typical AV planners simplify the ego behavior planning
+problem into trajectory planning, where a single trajectory
+is sought that minimizes an expected cost over all predicted
+futures within the planning horizon [7]. Such trajectory
+planning leads to overly conservative plans because it ignores
+the new information to be acquired about other agents in
+subsequent time steps, and the effects of ego actions on the
+behavior of other agents.
+In fact, in decision-making literature, people have long
+been pursuing a closed-loop policy instead of an open-loop
+1The
+authors
+are
+with
+NVIDIA
+Research,
+NVIDIA
+Cor-
+poration,
+2788
+San
+Tomas
+Expressway,
+Santa
+Clara,
+CA.
+{yuxiaoc,pkarkus,bivanovic,xweng}@nvidia.com
+2Marco
+Pavone
+is
+with
+the
+Department
+of
+Aeronautics
+and
+Astronautics,
+Stanford
+University,
+and
+with
+NVIDIA
+Research.
+{pavone@stanford.edu, mpavone@nvidia.com}
+plan as the former is shown to be optimal for problems such
+as Markov decision processes (MDP) [8].
+To highlight the difference between a motion policy and a
+single trajectory, consider the scenario depicted on the right
+end of Fig. 1. The ego vehicle (blue) needs to move pass the
+adjacent vehicle (red). Within the planning horizon, the red
+vehicle either maintains its current lane or cuts in front of the
+ego vehicle. A traditional planner seeks a single trajectory
+that performs well in both situations and thus will not choose
+to pass since lane change is a plausible choice for the red
+vehicle at current time. Alternatively, a motion policy may
+choose to nudge forward for the first part of the horizon,
+observe (by leveraging a forecasting model) the behavior of
+the adjacent vehicle, and then decide the subsequent motion.
+If the red vehicle already started a lane change, the ego
+vehicle brakes to avoid collision; if the red vehicle stays
+in its own lane, since the ego vehicle is already next to the
+red vehicle, a sudden lane change is very unlikely, the ego
+vehicle can then move past the red vehicle.
+In the field of robotics, policy planning often takes the
+form of contingency planning [10], [11], with the key idea
+of preparing multiple plans for different likely futures. To
+differentiate the two concepts, we view contingency planning
+as a simplified policy planning variant with only one stage of
+reasoning. Unfortunately, continuous-space policy planning
+with high-capacity prediction models is computationally pro-
+hibitive, and despite various simplifications [13], [14], [19],
+practical policy planning for AVs remains an open problem.
+In this paper we introduce Tree Policy Planning (TPP), a
+practical policy planner that meets the requirements of real-
+world autonomous driving, specifically, TPP is designed with
+the following desiderata:
+1) compatible with state-of-the-art deep-learned predic-
+tive models for trajectory forecasting
+2) plans with closed-loop ego-conditioning, i.e., factors in
+the ego vehicle’s influence on other agents
+DL
+EC
+Interp
+MS
+Scalability
+Multi-mode planning [9]–[12]
+×
+×
+✓
+×
+poor
+Branch MPC [13]
+×
+✓
+✓
+✓
+poor
+POMDP [14]–[16]
+×
+✓
+✓
+✓
+poor
+Game theoretic [17], [18]
+×
+✓
+✓
+×
+very poor
+Lookout [19]
+✓
+×
+✓
+×
+good
+CfO [20]
+✓
+✓
+×
+×
+medium
+TPP (ours)
+✓
+✓
+✓
+✓
+good
+TABLE I: Comparison of existing policy planning methods
+and TPP where “DL” is short for compatibility with deep-
+learned prediction models, “EC” stands for ego-conditioning,
+“Interp” is short for interpretable, “MS” stands for mul-
+tistage, and “Scalability” means the capability to scale to
+complicated scenes with large number of agents.
+arXiv:2301.11902v1  [cs.RO]  27 Jan 2023
+
+Fig. 1: Structure of TPP: the ego motion sampler generates the trajectory tree, which is fed to the prediction model for
+ego-conditioned prediction to generate the scenario tree. Then we solve for the ego motion policy via dynamic programming.
+3) allows for multistage reasoning that leverages the
+agents’ future reactive behavior
+4) interpretable and easily tunable
+5) scales to scenarios with a large number of agents
+We compare existing policy planning works with TPP in
+Table I. To the best of our knowledge, TPP is the first
+algorithm that fulfils our desiderata.
+The key idea of TPP is to convert the continuous space
+motion planning problem into a tractable finite-horizon MDP
+by constructing two trees: an ego trajectory tree carrying ego
+motion candidates, and a scenario tree that contains multiple
+behavior modes for neighboring agents. The algorithm is
+illustrated in Fig. 1. It consists of three components: an
+ego motion sampler that generates the ego trajectory tree, a
+deep-learned multi-stage prediction model that generates the
+scenario tree, and a dynamic program module that solves for
+the optimal policy with respect to the constructed MDP.
+We evaluate TPP in closed-loop simulation based on a
+recently proposed realistic data-driven traffic model [21]. Our
+results show that TPP achieves good closed-loop driving per-
+formance in real-world scenarios from the nuScenes dataset
+[22], and significantly outperforms non-policy planning base-
+lines. The algorithm is highly parallelizable, and can run in
+real time.
+II. RELATED WORKS
+Various contingency planning algorithms have been pro-
+posed and they can be viewed as different approxima-
+tions of policy planning. A complete policy planning algo-
+rithm should be capable of multi-stage reasoning with bi-
+directional interactions between the agent and the environ-
+ment. Since the ego is always reacting to the environment,
+the concept of bi-directional interaction is also referred to as
+ego-conditioning, emphasizing on the environment reacting
+to the ego agent. The application of autonomous driving
+also asks for high quality trajectories, accurate prediction,
+interpretability, and scalability. All these features combined
+makes it difficult to realize a non-compromising policy
+planning algorithm for autonomous vehicles. As a result,
+different simplification methods have been proposed. Limited
+to one stage, [10] proposed a trajectory optimization method
+with one trajectory for each scene evolution mode, similar
+ideas are also seen in [11], [12] where a predefined set of
+modes are evaluated online. [19] uses a deep-learned predic-
+tion model to generate the modes. [13] performs multistage
+reasoning via branch MPC and leveraged ego-conditioning,
+yet requires a simple differentiable prediction model.
+Partially Observed Markov Decision Process (POMDP)
+[14]–[16] is a classic policy planning method, yet cannot
+generate high quality trajectories or use powerful deep-
+learned prediction models. Game theoretic approaches are
+well-known for reasoning about bi-directional interactions,
+yet assume some simple fixed behavior model of the en-
+vironment, e.g. rational [18] or noisy Boltzmann [17]. [23]
+performs policy planning by training a neural network to
+react to the environment behavior, yet gives up on multistage
+reasoning, and is not interpretable and hard to tune.
+Table I summarizes the various existing methods and their
+compromises, and we would like to show in the remainder
+of the paper that TPP closes the gap to a practical policy
+planning method for AV.
+III. TREE POLICY PLANNING
+TPP generates the ego motion policy in three steps as
+shown in Fig. 1. First, the ego motion sampler generates
+an ego trajectory tree, r, given the current ego state and
+a lane map (when available). The ego trajectory tree con-
+sists of connected, dynamically feasible trajectory segments
+that together represent candidate ego-vehicle trajectories the
+planner can choose from. Second, the prediction module is
+a deep learned model that generates multi-modal predictions
+of the environment in the form of a scenario tree, e. Finally,
+the dynamic programming module converts r and e into an
+MDP and finds an optimal policy via dynamic programming.
+We use the following notation. Let sr denote the state of
+the ego vehicle and se denote the state of the environment,
+including the state of nearby agents and the location of static
+obstacles. For the trajectory tree, ri
+j denotes the j-th node of
+stage i and and the root node is simply r0. The nodes in
+the scenario tree are labeled the same way. We enforce that
+all nodes of the same stage contain trajectories of the same
+duration with ti
+0 and ti
+f being the starting and ending time
+of the i-th stage. The trajectory of a any node other than the
+root needs to follow the trajectory of its parent node. For both
+trees, Par(·) denotes the parent of a node and Ch(·) denote
+the set of children of a node, e.g., Par(r1
+1) = r0, Ch(e1
+0) =
+{e2
+0,e2
+1} for the two trees in Fig. 1.
+Next, we discuss the three components in detail.
+
+Ego motion sampler
+323.
+Dynamic
+program
+Prediction modelA. Ego-motion sampler
+The ego motion sampler generates a set of dynamically
+feasible trajectory options for planning to consider, and
+organizes them into a trajectory tree.
+Terminal point sampling. We first sample a set of terminal
+points and then find a feasible trajectory to the terminal point.
+Terminal points are sampled from a fixed set of acceleration-
+steering values, as well as along (and near to) lane centerlines
+when lane information is available.
+Spline fitting.
+We then fit a spline from the current ego
+state to the terminal point following [24]. Given the current
+ego state sr[0] = [X0,Y0,v0,ψ0]⊺ and a terminal condition
+sr[T] = [XT,YT,vT,ψT]⊺, we use two separate polynomials
+X[t] and Y[t] to parameterize the X,Y coordinates such that
+the resulting spline connects the two points and the boundary
+conditions are met:
+X[0] = X0
+X[T] = XT
+Y[0] = Y0
+Y[T] = YT
+˙X[0] = v0 cosψ0
+˙X[T] = vT cosψT
+˙X[0] = v0 sinψ0
+˙X[T] = vT sinψT.
+(1)
+With four boundary conditions on each polynomial, two 3rd
+order (cubic) polynomial can be uniquely determined, which
+allows us to directly compute the coefficient of X[t] and Y[t]
+in closed-form and calculate the trajectory from X[t] and Y[t].
+Finally we drop trajectories that violate dynamic constraints.
+Tree construction.
+In practice, trajectory trees are grown
+stage by stage, starting from the current state as root r0.
+We set an upper bound on the number of children a node
+can have, and randomly drop children nodes if the limit
+is exceeded. In each stage we grow children branches in
+parallel, thus the time complexity is linear in stage number.
+B. Multi-stage prediction model
+A critical part of TPP is the prediction model that gener-
+ates a scenario tree. For policy planning, we need a scene-
+centric, ego-conditioned, and multi-stage prediction model.
+We will first review these requirements and then show ex-
+amples of deep-learning models satisfying our requirements.
+Requirements. First, the prediction model needs to predict
+multiple modes of the trajectory distribution for each agent.
+Second, predictions must be scene-centric, i.e., outputting
+modes of the joint trajectory distribution of all relevant agents
+in the scene. Branches of our scenario tree corresponds to
+modes of the joint distribution so that our planner will be
+able to evaluate ego trajectory candidates against joint futures
+effectively. Examples for multi-modal scene-scentric pre-
+dictions include [6], [25]. Third, we need ego-conditioning
+(EC), i.e., prediction results should be conditioned on the
+ego vehicle’s hypothetical future motion. Ego conditioning
+allows the planner to leverage the reactive behaviors of
+the environment towards the ego vehicle and is a known
+technique in prediction literature [3], [6]. However, training
+a EC prediction model remains challenging as the ground
+truth under counterfactual ego future motion is not known.
+Lastly, we need to generate modes of the predicted trajec-
+tory distribution in multiple stages. That is, the generated sce-
+nario tree contains multiple stages with each node branching
+into multiple children after each stage. Multistage prediction
+allows the ego agent to depend its future motion on the
+future multimodal observation of the environment (similar
+to a decision tree) and the stage number can be freely set
+as a design choice. Details on converting existing models to
+enable multistage prediction can be found in the full version.
+Model architectures.
+To evaluate TPP’s flexibility w.r.t.
+different prediction models, we tested it with three vastly
+distinct prediction models. The first model is a rasterized
+model with CNN backbone and a CVAE; the second model
+is the PredictionNet [2] model with a Unet backbone, and
+the third model is the Agentformer model [25]. All mod-
+els are modified to enable multistage prediction and ego-
+conditioning. Details can be found in the full version.
+C. Dynamic programming
+The dynamic programming module takes the ego trajec-
+tory tree r and the scenario tree e, along with a handcrafted
+cost function l and finds an optimal ego motion policy.
+Intuitively, our algorithm constructs a discrete MDP over
+r and e and finds an optimal policy through dynamic pro-
+gramming. More specifically, states of the MDP are nodes
+of the scenario/trajectory trees that represent the joint ego
+and environment state; transitions are given by the tree
+parent/children edges in the two trees; and rewards are
+defined by the integral of the cost over the time segment. We
+then solve for the optimal finite-horizon policy for the dis-
+crete MDP using dynamic programming, i.e., calculating the
+value function backwards in stage via the Bellman equation.
+With the optimal policy, the continuous-time trajectory is
+recovered from the trajectory tree by executing the trajectory
+segments according to the policy. In practice we use a cost
+function l with standard terms for collision avoidance, lane
+keeping, goal reaching, and ride comfort; and a dynamically
+extended unicycle model [26] for the vehicle dynamics.
+Next we provide a formal discussion of the TPP algorithm.
+IV. FORMAL DISCUSSION
+A. Analysis without ego-conditioning
+In our motion planning problem we are interested in
+minimizing the expected cumulative cost over a horizon:
+J =
+� T
+0 l(s(t),se(t))dt,
+where l is the running cost function that depends on the
+ego state s and the environment state se, T is the planning
+horizon. Our planning algorithm operates on the two trees
+r and e with the same number of stages, so the cumulative
+cost can be rewritten as
+J =
+N
+∑
+i=0
+Li(ri,ei),
+(2)
+where ri and ei are the nodes the ego vehicle and the environ-
+ment take at stage i, and Li(ri,ei) =
+� ti
+f
+ti
+0 l(sri(t),sei
+e (t))dt. Due
+
+to the stochasticity of the environment, we aim to minimize
+Ee0:N[∑N
+i=0 Li(ri,ei)], where the expectation is on ei, the actual
+branch the environment takes and the distribution is given
+as part of the prediction. For simplicity, we shall present the
+result assuming e is not ego-conditioned, i.e., there is a single
+e, and later show that the result can be easily extended to
+the case with ego-conditioning.
+The end result is a policy that chooses which node to exe-
+cute for the next stage given the current ego and environment
+node, i.e., ri+1 = π⋆(ri,ei), where π⋆ is the optimal policy.
+The form of the policy is determined by the assumption about
+the information flow, i.e., during stage i, the ego vehicle can
+observe the environment’s response ei, and choose the ego
+motion among the children nodes of the current node ri. To
+obtain the optimal policy, a value function is needed:
+V(ri,ei) = Eei:N[
+N
+∑
+k=i
+Lk(rk,ek)],
+(3)
+which is the expected cost-to-go. The dynamic program goes
+backwards in stage and is trivial for the last stage:
+V(rN,eN) = LN(rN,eN).
+For stage i where i < N, we have
+V(ri,ei) = Li(ri,ei)+
+min
+ri+1∈Ch(ri)Eei+1[V(ri+1,ei+1)|ei].
+Since ei+1 is enumerable, the expectation is calculated as
+Eei+1[V(ri+1,ei+1)|ei] =
+∑
+ei+1
+j
+∈Ch(ei)
+P[ei+1
+j
+|ei]V(ri+1,ei+1
+j
+),
+where the branching probability of the scenario tree is given
+by the prediction model. Further define
+Q(ri+1,ei) := Li(ri,ei)+Eei+1[V(ri+1,ei+1)|ei]
+= Li(ri,ei)+
+∑
+ei+1
+j
+∈Ch(ei)
+P[ei+1
+j
+|ei]V(ri+1,ei+1
+j
+).
+where ri = Par(ri+1). We name this function Q function as
+it is analogous to the Q function in reinforcement learning
+where here bk+1 is the action we need to choose.
+The value function is then obtained backwards in stage:
+V(ri,ei) =
+�
+Li(ri,ei),
+i = N
+minri+1∈Ch(ri) Q(ri+1,ei),
+i < N
+(4)
+The optimal policy is then
+π⋆(ri,ei) = argmin
+ri+1∈Ch(ri)
+Q(ri+1,ei).
+(5)
+Theorem 1 (Optimality): The policy in (5) is the optimal
+policy with respect to the cost in (2).
+Proof: The proof is by induction. For stage N, the cost
+is a constant, π⋆ is trivially optimal, and V(rN,eN) is the cost-
+to-go. For 0 ≤ i < N, assume V(ri+1,ei+1) is the expected
+cost-to-go function at stage i + 1, then by the principle of
+optimality, the optimal node to take at stage i+1 is
+ri+1,⋆ = argmin
+ri+1∈Ch(ri)
+Li(ri,ei)+Eei+1[V(ri+1,ei+1)|ei],
+Fig. 2: Flattened trajectory tree as ego-conditioning modes
+which is exactly π⋆(ri,ei). This in turn shows that
+V(ri,ei) = Li(ri,ei)+Eei+1[V(ri+1,⋆,ei+1)|ei]
+=
+min
+ri+1∈Ch(ri)Q(ri+1,ei).
+This is true for all nodes in r and e, then by induction,
+π⋆(si,ei) is the optimal policy for all i ∈ {0,1,...,N}.
+B. Extension to ego-conditioned prediction.
+In the case where the scenario tree is ego-conditioned,
+the algorithm structure stays roughly the same. To generate
+the ego-conditioned trajectory prediction, we first run the
+ego motion sampler to generate the trajectory tree, then the
+trajectory tree is ”flattened ” to create the ego-conditioning
+(EC) modes. E.g., for trees in Fig. 1, the flattened EC modes
+are shown in Fig. 2. Under each EC mode, the prediction
+model generates a distinct scenario tree conditioned on the
+ego vehicle’s motion. To keep the planning problem under
+ego-conditioning well-defined, the ensemble of EC scenario
+trees needs to satisfy a condition called causal consistency.
+Definition 1 (Causal consistency): The ensemble of EC
+scenario tree is causally consistent (CC) if for any stage
+i ≤ N, any two scenario trees under different EC modes,
+if the conditioned ego trajectories are identical up to stage i,
+the two scenario trees are identical up to stage i.
+This condition guarantees the causality of the policy
+planning problem as for any particular stage i within the
+prediction horizon, no future information beyond stage i
+affects the prediction up to stage i. While causal consistency
+seems trivial, it may be violated by a multi-stage prediction
+model without special care. For example, if the EC trajectory
+is encoded as a whole for all stages, the scenario trees
+generated are not causally consistent.
+Proposition 1 (Sufficient condition for CC): Assume that
+a multi-stage ego-conditioned prediction model generates the
+trajectory prediction stage by stage and the EC prediction is
+deterministic, then the generated ensemble of EC scenario
+trees is causally consistent.
+Remark 1: For stochastic prediction models, the generated
+scenario trees are causally consistent if the random seeds are
+shared among all EC modes with the same EC trajectory.
+The proof is omitted here. It is straightforward to show
+that the 3 prediction models in Section III-B are CC.
+Given causally consistent EC predictions, only two terms
+change in the policy planning algorithm. First, all nodes
+of the scenario tree now depend on one of the nodes of
+trajectory tree of the same stage, and are denoted as ei(ri) to
+emphasize the dependence. Note that the dependence is on ri
+instead of the whole ego trajectory due to causal consistency.
+Second, the branching probability now depends on the ego
+
+Stage 2, branch 1
+ Stage 0.
+Stage i, branch i...
+Stage 2, branch 2
+Stage 1, branch 2
+Stage 2, branch 3Rasterized
+PredictionNet
+Agentformer
+Mean ADE
+0.67
+0.25
+0.33
+Mean FDE
+1.3
+0.85
+1.01
+TABLE II: Prediction metrics average displacement error
+(ADE) and final displacement error (FDE) of the three
+models. The mean is taken over all modes.
+node, and is denoted as P[ei+1
+j
+|ei(ri),ri]. Interestingly, the
+value function can be defined exactly the same way as
+V(ri,ei(ri)) = Eei:N[
+N
+∑
+k=i
+Lk(rk,ek(rk))].
+(6)
+Similarly, the Q function is now
+Q(ri+1,ei(ri)) := Li(ri,ei(ri))+Eei+1[V(ri+1,ei+1(ri+1))|ei(ri)]
+= Li(ri,ei(ri))+
+∑
+ei+1
+j
+∈Ch(ei(ri),ri+1)
+P[ei+1
+j
+|ei(ri),ri]V(ri+1,ei+1
+j
+),
+where Ch(ei(ri),ri+1) denotes the set of children nodes
+following ei(ri) on the tree associated with ri+1. The optimal
+policy is then defined as
+π⋆(ri,ei(ri)) = argmin
+ri+1∈Ch(ri)
+Q(ri+1,ei(ri)).
+(7)
+Theorem 2: Given a causally consistent ensemble of EC
+scenario trees, the policy in (7) is the optimal policy with
+respect to the cost function J = ∑N
+i=0 Li(ri,ei(ri)).
+Proof:
+The proof essentially follows the same steps
+as the proof of Theorem 1. Starting from stage N, the cost
+is constant, thus π⋆ is trivially optimal and V(rN,eN(rN))
+is the cost-to-go. For all 0 ≤ i < N, assume for all
+ri+1, V(ri+1,ei+1(ri+1)) is the expected cost-to-go func-
+tion at stage i + 1. Then for any ri, the cost-to-go to
+take any children node ri+1 ∈ Ch(ri) is Li(ri,ei(ri)) +
+E[V(ri+1,ei+1(ri+1))|ei(ri)], which is exactly Q(ri+1,ei(ri)).
+It follows that π⋆(ri,ei(ri)) gives the optimal bi+1 within
+Ch(ri) and the cost-to-go function of stage i is computed as
+V(ri,ei(ri)) =
+min
+ri+1∈Ch(ri)Q(ri+1,ei(ri)).
+The optimality of π⋆ is then proved by induction.
+V. EXPERIMENTS
+A. Training of prediction model
+We configure TPP with three different prediction models.
+Apart from the model architecture, the experimental setup is
+the same. Prediction models are trained with the nuScenes
+dataset [27], comprised of 1000 scenes lasting for 20 seconds
+collected in urban areas of Boston and Singapore. For sim-
+plicity, only vehicles are considered. All prediction models
+use 2 stages and the branching factor is 4.
+Loss function. Our training loss is a weighted sum of
+the following terms: prediction loss that penalizes the error
+between the predicted trajectories and the ground truth; EC
+collision loss that penalizes collisions between the predicted
+trajectories and the conditioned ego trajectory; collision
+loss that penalizes collisions between predicted trajectories
+of different agents in the scene; and additional standard
+regularization losses specific to the model architecture.
+Fig. 3: Example scene from simulation: the ego vehicle (blue)
+cuts in front of the red vehicle and then swerve around the
+green static vehicle in front. The ego trajectory tree is shown
+in magenta, the colored line shows agents’ actual trajectory.
+Ego conditioning. One natural choice of the ego trajectory
+for ego-conditioning is the ground truth future trajectory. In
+addition, to expose the model to more diverse ego trajecto-
+ries, we perturb the ground truth ego trajectory with noise
+generated via the Ornstein–Uhlenbeck process as alternative
+ego trajectories. However, since the ground truth for sur-
+rounding agents under the counterfactual ego motion is not
+available, for the lack of a better choice, we still use the origi-
+nal ground truth as the target for the EC trajectory prediction.
+We penalize collision between the predicted trajectories of
+surrounding agents and the conditioned ego trajectory so that
+the model learns to avoid the conditioned ego trajectory.
+Training results. We report standard ADE/FDE prediction
+metrics for the three prediction models in Table II. The re-
+sults show that all three models learn reasonable predictions.
+B. Experimental setup
+Closed-loop simulation.
+We evaluate TPP in BITS, a
+closed-loop AV simulation environment proposed in [21].
+Each simulation scene is initialized with a scene from the
+nuScenes dataset and then evolves forward with a policy
+for each agent. One agent is chosen as the ego vehicle
+and controlled by TPP or its alternatives. All other agents
+runs the learned BITS policy that generates realistic and
+diverse behavior of road vehicles. Additionally, to incite
+more interactions between agents, we ”spawn” new agents
+around the ego vehicle to challenge the planner throughout
+the simulation duration.
+Metrics.
+We use three key metrics: collision rate, offroad
+rate, and area coverage. The first two are calculated as the
+percentage of time steps where the ego vehicle is in collision
+(offroad). The coverage metric is calculated with a kernel
+density estimation (KDE) procedure following [21], and it
+captures the effectiveness of the ego plan in terms of liveness
+and distance travelled. The evaluation is done on 100 scenes
+from the evaluation split of nuScenes (not used for training).
+Baselines.
+We consider three baselines, the first two are
+non-contingent counterpart of TPP. For fair comparison, the
+two baselines share the same ego-motion sampler and ego-
+conditioning prediction model as TPP, the first baseline,
+named non-contingency robust (NCR), considers all possible
+trajectories predicted by the prediction model (similar to
+
+(t=0)
+(t=1.5s)
+(t=3s)
+(t=4.5s)Planning
+Prediction
+Crash Offroad Coverage ↑
+algorithm
+model
+rate ↓
+rate ↓
+TPP (ours)
+Rasterized
+0.46% 0.64%
+164
+Non-Cont. Robust
+Rasterized
+0.80%
+0.61%
+163
+Non-Cont. Greedy
+Rasterized
+1.23%
+1.00%
+168
+TPP (ours)
+PredictionNet
+1.02% 0.10%
+174
+Non-Cont. Robust
+PredictionNet
+1.34%
+2.07%
+166
+Non-Cont. Greedy
+PredictionNet
+1.60%
+0.45%
+170
+TPP (ours)
+Agentformer
+0.97% 0.20%
+174
+Non-Cont. Robust
+Agentformer
+1.08%
+0.63%
+158
+Non-Cont. Greedy
+Agentformer
+2.52%
+0.98%
+163
+TPP (ours)
+Agentformer (Gaussian) 3.92%
+5.53%
+150
+Non-Cont. Robust Agentformer (Gaussian) 1.40%
+0.40%
+144
+Non-Cont. Greedy Agentformer (Gaussian) 1.04%
+0.25%
+149
+TPP (ours)
+Kinematic model
+1.75%
+0.51%
+160
+TABLE III: Closed loop simulation results with TPP and
+the baselines, the default Agentformer model uses a discrete
+latent space and Agentformer (Gaussian) is its variant with
+a Gaussian latent space. TPP significantly outperform the
+baselines with the same ego trajectory tree and prediction
+model (bold means best).
+[28]); while the second baseline, named non-policy greedy
+(NCG), only avoids the most likely prediction mode (similar
+to [7]). The third baseline uses a kinematic prediction model
+with 2 modes: maintaining speed and braking, and share the
+same multi-stage tree structure.
+C. Results
+Fig. 3 shows snapshots of the simulation with TPP where
+the ego vehicle (blue) cuts in front of the red vehicle and
+then change lane again to avoid the parked green vehicle.
+The magenta lines show the ego trajectory tree, and the
+colorful line shows the rollout trajectories of the agents. The
+quantitative results are shown in Tables III and IV, and we
+make the following observations. Simulation videos can be
+found here.
+TPP outperforms the baselines.
+As shown in Table III,
+TPP significantly outperforms the two non-policy baselines
+on crash and offroad rate under all 3 prediction models while
+achieving similar coverage. Even under a simple kinematic
+model, the performance is better than non-policy planners
+with sophisticated prediction models. TPP with deep-learned
+prediction models performs better than with a simple kine-
+matic model, indicating the deep-learned prediction model
+benefits the closed-loop performance.
+Better predictions do not mean better plan. Interestingly,
+from Table III, the ”better” prediction model measured
+by ADE/FDE does not necessarily lead to better closed-
+loop performance. For example, the rasterized model is the
+simplest and ”worst” model among the three, yet TPP with
+it outperforms the other two in terms of crash rate.
+Ego-conditioning is useful. Table IV compares TPP with
+or without ego-conditioning. Ego-conditioning improves the
+closed-loop performance considerably in most cases, espe-
+cially the coverage metric, hinting that the ego vehicle is
+more comfortable moving through traffic when it expects
+reaction from other agents.
+Temporal consistency is important. Additionally, we tested
+the Agentformer model with Gaussian latent space, where the
+prediction requires sampling of the latent variable at every
+Crash rate↓
+Offroad rate↓
+Coverage ↑
+Rasterized
+0.46% / 0.53%
+0.60% / 1.05%
+164 / 163
+PredictionNet
+1.00% / 0.63%
+0.10% / 0.71%
+174 / 151
+Agentformer
+0.97% / 1.30%
+0.20% / 0.53%
+174 / 163
+TABLE IV: Ablation study of ego-conditioning, the first and
+second number are the metrics with and without ego condi-
+tioning. Most metrics see degradation when ego-conditioning
+is disabled (bold means better).
+Rasterized
+PredictionNet
+Agentformer
+Prediction w/ EC
+150ms
+5.00s
+2.60s
+Prediction w/o EC
+37ms
+75ms
+150ms
+Ego-motion sampling
+30ms
+Dynamic programming
+11ms
+TABLE V: Runtime of the three key components
+inference call, compared to using discrete latent space where
+sampling is not needed. While the prediction ADE/FDE is
+similar, the closed-loop performance with the Gaussian latent
+is significantly worse. We suspect this is due to the poor
+temporal consistency between predictions of consecutive
+time steps. While both TPP and the two baselines saw
+performance degradation under the Gaussian latent, the pol-
+icy planner relies on more sophisticated prediction/planning
+interaction, thus suffers more than the two baselines.
+Runtime analysis.
+Table V shows the runtime of the
+three components of TPP on an Nvidia 3090 GPU (most
+computation is done on GPU). It should be noted that all
+runtime is recorded on unoptimized Python code and there
+is significant space of improvement. For example, the long
+runtime for PredictionNet is mainly due to rasterization,
+and we know its runtime can be as low as 5ms with
+the proper engineering [2], e.g., efficient CUDA code for
+rasterization. While the runtime for prediction models is not
+very informative and is not the focus of this paper, it is quite
+clear that ego motion sampling and dynamic programming
+can be done very efficiently. With proper optimization, we
+believe that TPP can be easily implemented in real time. We
+plan to release the code upon publication.
+VI. CONCLUSION
+We present TPP, a policy planner capable of generating
+multistage motion policies that react to the environment. It
+is centered around an ego trajectory tree and a scenario
+tree that predicts the environment behavior, both containing
+multiple stages. Ego-conditioning is applied to leverage the
+reactive behavior of the environment under the ego vehicle’s
+presence. The closed-loop simulation result shows that TPP
+significantly outperform two non-policy benchmarks and the
+runtime test suggest that such sophisticated policy planner
+can be run in real time. Our planned future works include
+(1) study more flexible topology of the trees and enable
+event-triggered branching (2) improve the efficiency of com-
+putation, especially under ego-conditioning (3) adaptive and
+personalized policy planning.
+ACKNOWLEDGEMENT
+We would like to thank Bilal Kartal, Siva Hari, and
+Edward Schmerling for the insightful discussion.
+
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+
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@@ -0,0 +1,586 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf,len=585
+page_content='Tree-structured Policy Planning with Learned Behavior Models  Yuxiao Chen1, Peter Karkus1, Boris Ivanovic1, Xinshuo Weng1, and Marco Pavone1,2  Abstract— Autonomous vehicles (AVs) need to reason about  the multimodal behavior of neighboring agents while planning  their own motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Many existing trajectory planners seek a  single trajectory that performs well under all plausible futures  simultaneously, ignoring bi-directional interactions and thus  leading to overly conservative plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Policy planning, whereby  the ego agent plans a policy that reacts to the environment’s  multimodal behavior, is a promising direction as it can account  for the action-reaction interactions between the AV and the  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' However, most existing policy planners do not  scale to the complexity of real autonomous vehicle applications:  they are either not compatible with modern deep learning  prediction models, not interpretable, or not able to generate  high quality trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' To fill this gap, we propose Tree  Policy Planning (TPP), a policy planner that is compatible  with state-of-the-art deep learning prediction models, generates  multistage motion plans, and accounts for the influence of ego  agent on the environment behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The key idea of TPP is to  reduce the continuous optimization problem into a tractable  discrete MDP through the construction of two tree structures:  an ego trajectory tree for ego trajectory options, and a scenario  tree for multi-modal ego-conditioned environment predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We demonstrate the efficacy of TPP in closed-loop simulations  based on real-world nuScenes dataset and results show that TPP  scales to realistic AV scenarios and significantly outperforms  non-policy baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' INTRODUCTION  A key challenge of motion planning for autonomous  vehicles (AV)s is reasoning about the interaction between the  ego vehicle and neighboring agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The task is commonly  divided into two subproblems: trajectory prediction for other  agents, and ego motion planning with prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Trajectory  prediction has seen substantial progress in recent years,  coming from simple kineamtic models [1] to powerful deep  learning models [2]–[6] capable of generating high-quality,  multi-modal predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' However, motion planning with  such high-capacity prediction models remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Typical AV planners simplify the ego behavior planning  problem into trajectory planning, where a single trajectory  is sought that minimizes an expected cost over all predicted  futures within the planning horizon [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Such trajectory  planning leads to overly conservative plans because it ignores  the new information to be acquired about other agents in  subsequent time steps, and the effects of ego actions on the  behavior of other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  In fact, in decision-making literature, people have long  been pursuing a closed-loop policy instead of an open-loop  1The  authors  are  with  NVIDIA  Research,  NVIDIA  Cor-  poration,  2788  San  Tomas  Expressway,  Santa  Clara,  CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  {yuxiaoc,pkarkus,bivanovic,xweng}@nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='com  2Marco  Pavone  is  with  the  Department  of  Aeronautics  and  Astronautics,  Stanford  University,  and  with  NVIDIA  Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  {pavone@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='edu, mpavone@nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='com}  plan as the former is shown to be optimal for problems such  as Markov decision processes (MDP) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  To highlight the difference between a motion policy and a  single trajectory, consider the scenario depicted on the right  end of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The ego vehicle (blue) needs to move pass the  adjacent vehicle (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Within the planning horizon, the red  vehicle either maintains its current lane or cuts in front of the  ego vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' A traditional planner seeks a single trajectory  that performs well in both situations and thus will not choose  to pass since lane change is a plausible choice for the red  vehicle at current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Alternatively, a motion policy may  choose to nudge forward for the first part of the horizon,  observe (by leveraging a forecasting model) the behavior of  the adjacent vehicle, and then decide the subsequent motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  If the red vehicle already started a lane change, the ego  vehicle brakes to avoid collision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' if the red vehicle stays  in its own lane, since the ego vehicle is already next to the  red vehicle, a sudden lane change is very unlikely, the ego  vehicle can then move past the red vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  In the field of robotics, policy planning often takes the  form of contingency planning [10], [11], with the key idea  of preparing multiple plans for different likely futures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' To  differentiate the two concepts, we view contingency planning  as a simplified policy planning variant with only one stage of  reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Unfortunately, continuous-space policy planning  with high-capacity prediction models is computationally pro-  hibitive, and despite various simplifications [13], [14], [19],  practical policy planning for AVs remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  In this paper we introduce Tree Policy Planning (TPP), a  practical policy planner that meets the requirements of real-  world autonomous driving, specifically, TPP is designed with  the following desiderata:  1) compatible with state-of-the-art deep-learned predic-  tive models for trajectory forecasting  2) plans with closed-loop ego-conditioning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' factors in  the ego vehicle’s influence on other agents  DL  EC  Interp  MS  Scalability  Multi-mode planning [9]–[12]  ×  ×  ✓  ×  poor  Branch MPC [13]  ×  ✓  ✓  ✓  poor  POMDP [14]–[16]  ×  ✓  ✓  ✓  poor  Game theoretic [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' [18]  ×  ✓  ✓  ×  very poor  Lookout [19]  ✓  ×  ✓  ×  good  CfO [20]  ✓  ✓  ×  ×  medium  TPP (ours)  ✓  ✓  ✓  ✓  good  TABLE I: Comparison of existing policy planning methods  and TPP where “DL” is short for compatibility with deep-  learned prediction models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' “EC” stands for ego-conditioning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  “Interp” is short for interpretable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' “MS” stands for mul-  tistage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' and “Scalability” means the capability to scale to  complicated scenes with large number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='11902v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='RO]  27 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 1: Structure of TPP: the ego motion sampler generates the trajectory tree, which is fed to the prediction model for  ego-conditioned prediction to generate the scenario tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Then we solve for the ego motion policy via dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  3) allows for multistage reasoning that leverages the  agents’ future reactive behavior  4) interpretable and easily tunable  5) scales to scenarios with a large number of agents  We compare existing policy planning works with TPP in  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' To the best of our knowledge, TPP is the first  algorithm that fulfils our desiderata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  The key idea of TPP is to convert the continuous space  motion planning problem into a tractable finite-horizon MDP  by constructing two trees: an ego trajectory tree carrying ego  motion candidates, and a scenario tree that contains multiple  behavior modes for neighboring agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The algorithm is  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' It consists of three components: an  ego motion sampler that generates the ego trajectory tree, a  deep-learned multi-stage prediction model that generates the  scenario tree, and a dynamic program module that solves for  the optimal policy with respect to the constructed MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We evaluate TPP in closed-loop simulation based on a  recently proposed realistic data-driven traffic model [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Our  results show that TPP achieves good closed-loop driving per-  formance in real-world scenarios from the nuScenes dataset  [22], and significantly outperforms non-policy planning base-  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The algorithm is highly parallelizable, and can run in  real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' RELATED WORKS  Various contingency planning algorithms have been pro-  posed and they can be viewed as different approxima-  tions of policy planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' A complete policy planning algo-  rithm should be capable of multi-stage reasoning with bi-  directional interactions between the agent and the environ-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Since the ego is always reacting to the environment,  the concept of bi-directional interaction is also referred to as  ego-conditioning, emphasizing on the environment reacting  to the ego agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The application of autonomous driving  also asks for high quality trajectories, accurate prediction,  interpretability, and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' All these features combined  makes it difficult to realize a non-compromising policy  planning algorithm for autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' As a result,  different simplification methods have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Limited  to one stage, [10] proposed a trajectory optimization method  with one trajectory for each scene evolution mode, similar  ideas are also seen in [11], [12] where a predefined set of  modes are evaluated online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' [19] uses a deep-learned predic-  tion model to generate the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' [13] performs multistage  reasoning via branch MPC and leveraged ego-conditioning,  yet requires a simple differentiable prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Partially Observed Markov Decision Process (POMDP)  [14]–[16] is a classic policy planning method, yet cannot  generate high quality trajectories or use powerful deep-  learned prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Game theoretic approaches are  well-known for reasoning about bi-directional interactions,  yet assume some simple fixed behavior model of the en-  vironment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' rational [18] or noisy Boltzmann [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' [23]  performs policy planning by training a neural network to  react to the environment behavior, yet gives up on multistage  reasoning, and is not interpretable and hard to tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Table I summarizes the various existing methods and their  compromises, and we would like to show in the remainder  of the paper that TPP closes the gap to a practical policy  planning method for AV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' TREE POLICY PLANNING  TPP generates the ego motion policy in three steps as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' First, the ego motion sampler generates  an ego trajectory tree, r, given the current ego state and  a lane map (when available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The ego trajectory tree con-  sists of connected, dynamically feasible trajectory segments  that together represent candidate ego-vehicle trajectories the  planner can choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Second, the prediction module is  a deep learned model that generates multi-modal predictions  of the environment in the form of a scenario tree, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Finally,  the dynamic programming module converts r and e into an  MDP and finds an optimal policy via dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Let sr denote the state of  the ego vehicle and se denote the state of the environment,  including the state of nearby agents and the location of static  obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For the trajectory tree, ri  j denotes the j-th node of  stage i and and the root node is simply r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The nodes in  the scenario tree are labeled the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' We enforce that  all nodes of the same stage contain trajectories of the same  duration with ti  0 and ti  f being the starting and ending time  of the i-th stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The trajectory of a any node other than the  root needs to follow the trajectory of its parent node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For both  trees, Par(·) denotes the parent of a node and Ch(·) denote  the set of children of a node, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=', Par(r1  1) = r0, Ch(e1  0) =  {e2  0,e2  1} for the two trees in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Next, we discuss the three components in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Ego motion sampler  323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Dynamic  program  Prediction modelA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Ego-motion sampler  The ego motion sampler generates a set of dynamically  feasible trajectory options for planning to consider, and  organizes them into a trajectory tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Terminal point sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' We first sample a set of terminal  points and then find a feasible trajectory to the terminal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Terminal points are sampled from a fixed set of acceleration-  steering values, as well as along (and near to) lane centerlines  when lane information is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Spline fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We then fit a spline from the current ego  state to the terminal point following [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Given the current  ego state sr[0] = [X0,Y0,v0,ψ0]⊺ and a terminal condition  sr[T] = [XT,YT,vT,ψT]⊺, we use two separate polynomials  X[t] and Y[t] to parameterize the X,Y coordinates such that  the resulting spline connects the two points and the boundary  conditions are met:  X[0] = X0  X[T] = XT  Y[0] = Y0  Y[T] = YT  ˙X[0] = v0 cosψ0  ˙X[T] = vT cosψT  ˙X[0] = v0 sinψ0  ˙X[T] = vT sinψT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  (1)  With four boundary conditions on each polynomial, two 3rd  order (cubic) polynomial can be uniquely determined, which  allows us to directly compute the coefficient of X[t] and Y[t]  in closed-form and calculate the trajectory from X[t] and Y[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Finally we drop trajectories that violate dynamic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Tree construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  In practice, trajectory trees are grown  stage by stage, starting from the current state as root r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We set an upper bound on the number of children a node  can have, and randomly drop children nodes if the limit  is exceeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' In each stage we grow children branches in  parallel, thus the time complexity is linear in stage number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Multi-stage prediction model  A critical part of TPP is the prediction model that gener-  ates a scenario tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For policy planning, we need a scene-  centric, ego-conditioned, and multi-stage prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We will first review these requirements and then show ex-  amples of deep-learning models satisfying our requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' First, the prediction model needs to predict  multiple modes of the trajectory distribution for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Second, predictions must be scene-centric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=', outputting  modes of the joint trajectory distribution of all relevant agents  in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Branches of our scenario tree corresponds to  modes of the joint distribution so that our planner will be  able to evaluate ego trajectory candidates against joint futures  effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Examples for multi-modal scene-scentric pre-  dictions include [6], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Third, we need ego-conditioning  (EC), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=', prediction results should be conditioned on the  ego vehicle’s hypothetical future motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Ego conditioning  allows the planner to leverage the reactive behaviors of  the environment towards the ego vehicle and is a known  technique in prediction literature [3], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' However, training  a EC prediction model remains challenging as the ground  truth under counterfactual ego future motion is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Lastly, we need to generate modes of the predicted trajec-  tory distribution in multiple stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' That is, the generated sce-  nario tree contains multiple stages with each node branching  into multiple children after each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Multistage prediction  allows the ego agent to depend its future motion on the  future multimodal observation of the environment (similar  to a decision tree) and the stage number can be freely set  as a design choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Details on converting existing models to  enable multistage prediction can be found in the full version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  To evaluate TPP’s flexibility w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  different prediction models, we tested it with three vastly  distinct prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The first model is a rasterized  model with CNN backbone and a CVAE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' the second model  is the PredictionNet [2] model with a Unet backbone, and  the third model is the Agentformer model [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' All mod-  els are modified to enable multistage prediction and ego-  conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Details can be found in the full version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Dynamic programming  The dynamic programming module takes the ego trajec-  tory tree r and the scenario tree e, along with a handcrafted  cost function l and finds an optimal ego motion policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Intuitively, our algorithm constructs a discrete MDP over  r and e and finds an optimal policy through dynamic pro-  gramming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' More specifically, states of the MDP are nodes  of the scenario/trajectory trees that represent the joint ego  and environment state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' transitions are given by the tree  parent/children edges in the two trees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' and rewards are  defined by the integral of the cost over the time segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' We  then solve for the optimal finite-horizon policy for the dis-  crete MDP using dynamic programming, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=', calculating the  value function backwards in stage via the Bellman equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  With the optimal policy, the continuous-time trajectory is  recovered from the trajectory tree by executing the trajectory  segments according to the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' In practice we use a cost  function l with standard terms for collision avoidance, lane  keeping, goal reaching, and ride comfort;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' and a dynamically  extended unicycle model [26] for the vehicle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Next we provide a formal discussion of the TPP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' FORMAL DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Analysis without ego-conditioning  In our motion planning problem we are interested in  minimizing the expected cumulative cost over a horizon:  J =  � T  0 l(s(t),se(t))dt,  where l is the running cost function that depends on the  ego state s and the environment state se, T is the planning  horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Our planning algorithm operates on the two trees  r and e with the same number of stages, so the cumulative  cost can be rewritten as  J =  N  ∑  i=0  Li(ri,ei),  (2)  where ri and ei are the nodes the ego vehicle and the environ-  ment take at stage i, and Li(ri,ei) =  � ti  f  ti  0 l(sri(t),sei  e (t))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Due  to the stochasticity of the environment, we aim to minimize  Ee0:N[∑N  i=0 Li(ri,ei)], where the expectation is on ei, the actual  branch the environment takes and the distribution is given  as part of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For simplicity, we shall present the  result assuming e is not ego-conditioned, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=', there is a single  e, and later show that the result can be easily extended to  the case with ego-conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  The end result is a policy that chooses which node to exe-  cute for the next stage given the current ego and environment  node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=', ri+1 = π⋆(ri,ei), where π⋆ is the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  The form of the policy is determined by the assumption about  the information flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=', during stage i, the ego vehicle can  observe the environment’s response ei, and choose the ego  motion among the children nodes of the current node ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' To  obtain the optimal policy, a value function is needed:  V(ri,ei) = Eei:N[  N  ∑  k=i  Lk(rk,ek)],  (3)  which is the expected cost-to-go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The dynamic program goes  backwards in stage and is trivial for the last stage:  V(rN,eN) = LN(rN,eN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  For stage i where i < N, we have  V(ri,ei) = Li(ri,ei)+  min  ri+1∈Ch(ri)Eei+1[V(ri+1,ei+1)|ei].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Since ei+1 is enumerable, the expectation is calculated as  Eei+1[V(ri+1,ei+1)|ei] =  ∑  ei+1  j  ∈Ch(ei)  P[ei+1  j  |ei]V(ri+1,ei+1  j  ),  where the branching probability of the scenario tree is given  by the prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Further define  Q(ri+1,ei) := Li(ri,ei)+Eei+1[V(ri+1,ei+1)|ei]  = Li(ri,ei)+  ∑  ei+1  j  ∈Ch(ei)  P[ei+1  j  |ei]V(ri+1,ei+1  j  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  where ri = Par(ri+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' We name this function Q function as  it is analogous to the Q function in reinforcement learning  where here bk+1 is the action we need to choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  The value function is then obtained backwards in stage:  V(ri,ei) =  �  Li(ri,ei),  i = N  minri+1∈Ch(ri) Q(ri+1,ei),  i < N  (4)  The optimal policy is then  π⋆(ri,ei) = argmin  ri+1∈Ch(ri)  Q(ri+1,ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  (5)  Theorem 1 (Optimality): The policy in (5) is the optimal  policy with respect to the cost in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Proof: The proof is by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For stage N, the cost  is a constant, π⋆ is trivially optimal, and V(rN,eN) is the cost-  to-go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For 0 ≤ i < N, assume V(ri+1,ei+1) is the expected  cost-to-go function at stage i + 1, then by the principle of  optimality, the optimal node to take at stage i+1 is  ri+1,⋆ = argmin  ri+1∈Ch(ri)  Li(ri,ei)+Eei+1[V(ri+1,ei+1)|ei],  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 2: Flattened trajectory tree as ego-conditioning modes  which is exactly π⋆(ri,ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' This in turn shows that  V(ri,ei) = Li(ri,ei)+Eei+1[V(ri+1,⋆,ei+1)|ei]  =  min  ri+1∈Ch(ri)Q(ri+1,ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  This is true for all nodes in r and e, then by induction,  π⋆(si,ei) is the optimal policy for all i ∈ {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=',N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Extension to ego-conditioned prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  In the case where the scenario tree is ego-conditioned,  the algorithm structure stays roughly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' To generate  the ego-conditioned trajectory prediction, we first run the  ego motion sampler to generate the trajectory tree, then the  trajectory tree is ”flattened ” to create the ego-conditioning  (EC) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=', for trees in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 1, the flattened EC modes  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Under each EC mode, the prediction  model generates a distinct scenario tree conditioned on the  ego vehicle’s motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' To keep the planning problem under  ego-conditioning well-defined, the ensemble of EC scenario  trees needs to satisfy a condition called causal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Definition 1 (Causal consistency): The ensemble of EC  scenario tree is causally consistent (CC) if for any stage  i ≤ N, any two scenario trees under different EC modes,  if the conditioned ego trajectories are identical up to stage i,  the two scenario trees are identical up to stage i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  This condition guarantees the causality of the policy  planning problem as for any particular stage i within the  prediction horizon, no future information beyond stage i  affects the prediction up to stage i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' While causal consistency  seems trivial, it may be violated by a multi-stage prediction  model without special care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For example, if the EC trajectory  is encoded as a whole for all stages, the scenario trees  generated are not causally consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Proposition 1 (Sufficient condition for CC): Assume that  a multi-stage ego-conditioned prediction model generates the  trajectory prediction stage by stage and the EC prediction is  deterministic, then the generated ensemble of EC scenario  trees is causally consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Remark 1: For stochastic prediction models, the generated  scenario trees are causally consistent if the random seeds are  shared among all EC modes with the same EC trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  The proof is omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' It is straightforward to show  that the 3 prediction models in Section III-B are CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Given causally consistent EC predictions, only two terms  change in the policy planning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' First, all nodes  of the scenario tree now depend on one of the nodes of  trajectory tree of the same stage, and are denoted as ei(ri) to  emphasize the dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Note that the dependence is on ri  instead of the whole ego trajectory due to causal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Second, the branching probability now depends on the ego  Stage 2, branch 1  Stage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Stage i, branch i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Stage 2, branch 2  Stage 1, branch 2  Stage 2, branch 3Rasterized  PredictionNet  Agentformer  Mean ADE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='33  Mean FDE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='01  TABLE II: Prediction metrics average displacement error  (ADE) and final displacement error (FDE) of the three  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The mean is taken over all modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  node, and is denoted as P[ei+1  j  |ei(ri),ri].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Interestingly, the  value function can be defined exactly the same way as  V(ri,ei(ri)) = Eei:N[  N  ∑  k=i  Lk(rk,ek(rk))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  (6)  Similarly, the Q function is now  Q(ri+1,ei(ri)) := Li(ri,ei(ri))+Eei+1[V(ri+1,ei+1(ri+1))|ei(ri)]  = Li(ri,ei(ri))+  ∑  ei+1  j  ∈Ch(ei(ri),ri+1)  P[ei+1  j  |ei(ri),ri]V(ri+1,ei+1  j  ),  where Ch(ei(ri),ri+1) denotes the set of children nodes  following ei(ri) on the tree associated with ri+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The optimal  policy is then defined as  π⋆(ri,ei(ri)) = argmin  ri+1∈Ch(ri)  Q(ri+1,ei(ri)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  (7)  Theorem 2: Given a causally consistent ensemble of EC  scenario trees, the policy in (7) is the optimal policy with  respect to the cost function J = ∑N  i=0 Li(ri,ei(ri)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Proof:  The proof essentially follows the same steps  as the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Starting from stage N, the cost  is constant, thus π⋆ is trivially optimal and V(rN,eN(rN))  is the cost-to-go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For all 0 ≤ i < N, assume for all  ri+1, V(ri+1,ei+1(ri+1)) is the expected cost-to-go func-  tion at stage i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Then for any ri, the cost-to-go to  take any children node ri+1 ∈ Ch(ri) is Li(ri,ei(ri)) +  E[V(ri+1,ei+1(ri+1))|ei(ri)], which is exactly Q(ri+1,ei(ri)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  It follows that π⋆(ri,ei(ri)) gives the optimal bi+1 within  Ch(ri) and the cost-to-go function of stage i is computed as  V(ri,ei(ri)) =  min  ri+1∈Ch(ri)Q(ri+1,ei(ri)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  The optimality of π⋆ is then proved by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' EXPERIMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Training of prediction model  We configure TPP with three different prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Apart from the model architecture, the experimental setup is  the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Prediction models are trained with the nuScenes  dataset [27], comprised of 1000 scenes lasting for 20 seconds  collected in urban areas of Boston and Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For sim-  plicity, only vehicles are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' All prediction models  use 2 stages and the branching factor is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Our training loss is a weighted sum of  the following terms: prediction loss that penalizes the error  between the predicted trajectories and the ground truth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' EC  collision loss that penalizes collisions between the predicted  trajectories and the conditioned ego trajectory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' collision  loss that penalizes collisions between predicted trajectories  of different agents in the scene;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' and additional standard  regularization losses specific to the model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 3: Example scene from simulation: the ego vehicle (blue)  cuts in front of the red vehicle and then swerve around the  green static vehicle in front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The ego trajectory tree is shown  in magenta, the colored line shows agents’ actual trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Ego conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' One natural choice of the ego trajectory  for ego-conditioning is the ground truth future trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' In  addition, to expose the model to more diverse ego trajecto-  ries, we perturb the ground truth ego trajectory with noise  generated via the Ornstein–Uhlenbeck process as alternative  ego trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' However, since the ground truth for sur-  rounding agents under the counterfactual ego motion is not  available, for the lack of a better choice, we still use the origi-  nal ground truth as the target for the EC trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We penalize collision between the predicted trajectories of  surrounding agents and the conditioned ego trajectory so that  the model learns to avoid the conditioned ego trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Training results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' We report standard ADE/FDE prediction  metrics for the three prediction models in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The re-  sults show that all three models learn reasonable predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Experimental setup  Closed-loop simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We evaluate TPP in BITS, a  closed-loop AV simulation environment proposed in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Each simulation scene is initialized with a scene from the  nuScenes dataset and then evolves forward with a policy  for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' One agent is chosen as the ego vehicle  and controlled by TPP or its alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' All other agents  runs the learned BITS policy that generates realistic and  diverse behavior of road vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Additionally, to incite  more interactions between agents, we ”spawn” new agents  around the ego vehicle to challenge the planner throughout  the simulation duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We use three key metrics: collision rate, offroad  rate, and area coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The first two are calculated as the  percentage of time steps where the ego vehicle is in collision  (offroad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The coverage metric is calculated with a kernel  density estimation (KDE) procedure following [21], and it  captures the effectiveness of the ego plan in terms of liveness  and distance travelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The evaluation is done on 100 scenes  from the evaluation split of nuScenes (not used for training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  We consider three baselines, the first two are  non-contingent counterpart of TPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For fair comparison, the  two baselines share the same ego-motion sampler and ego-  conditioning prediction model as TPP, the first baseline,  named non-contingency robust (NCR), considers all possible  trajectories predicted by the prediction model (similar to  (t=0)  (t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='5s)  (t=3s)  (t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='5s)Planning  Prediction  Crash Offroad Coverage ↑  algorithm  model  rate ↓  rate ↓  TPP (ours)  Rasterized  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='46% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='64%  164  Non-Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Robust  Rasterized  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='80%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='61%  163  Non-Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Greedy  Rasterized  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='23%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='00%  168  TPP (ours)  PredictionNet  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='02% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='10%  174  Non-Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Robust  PredictionNet  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='34%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='07%  166  Non-Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Greedy  PredictionNet  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='60%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='45%  170  TPP (ours)  Agentformer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='97% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='20%  174  Non-Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Robust  Agentformer  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='08%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='63%  158  Non-Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Greedy  Agentformer  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='52%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='98%  163  TPP (ours)  Agentformer (Gaussian) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='92%  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='53%  150  Non-Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Robust Agentformer (Gaussian) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='40%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='40%  144  Non-Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Greedy Agentformer (Gaussian) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='04%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='25%  149  TPP (ours)  Kinematic model  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='75%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='51%  160  TABLE III: Closed loop simulation results with TPP and  the baselines, the default Agentformer model uses a discrete  latent space and Agentformer (Gaussian) is its variant with  a Gaussian latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' TPP significantly outperform the  baselines with the same ego trajectory tree and prediction  model (bold means best).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  [28]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' while the second baseline, named non-policy greedy  (NCG), only avoids the most likely prediction mode (similar  to [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The third baseline uses a kinematic prediction model  with 2 modes: maintaining speed and braking, and share the  same multi-stage tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Results  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' 3 shows snapshots of the simulation with TPP where  the ego vehicle (blue) cuts in front of the red vehicle and  then change lane again to avoid the parked green vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  The magenta lines show the ego trajectory tree, and the  colorful line shows the rollout trajectories of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The  quantitative results are shown in Tables III and IV, and we  make the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Simulation videos can be  found here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  TPP outperforms the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  As shown in Table III,  TPP significantly outperforms the two non-policy baselines  on crash and offroad rate under all 3 prediction models while  achieving similar coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Even under a simple kinematic  model, the performance is better than non-policy planners  with sophisticated prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' TPP with deep-learned  prediction models performs better than with a simple kine-  matic model, indicating the deep-learned prediction model  benefits the closed-loop performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Better predictions do not mean better plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Interestingly,  from Table III, the ”better” prediction model measured  by ADE/FDE does not necessarily lead to better closed-  loop performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For example, the rasterized model is the  simplest and ”worst” model among the three, yet TPP with  it outperforms the other two in terms of crash rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Ego-conditioning is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Table IV compares TPP with  or without ego-conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Ego-conditioning improves the  closed-loop performance considerably in most cases, espe-  cially the coverage metric, hinting that the ego vehicle is  more comfortable moving through traffic when it expects  reaction from other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Temporal consistency is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Additionally, we tested  the Agentformer model with Gaussian latent space, where the  prediction requires sampling of the latent variable at every  Crash rate↓  Offroad rate↓  Coverage ↑  Rasterized  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='46% / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='53%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='60% / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='05%  164 / 163  PredictionNet  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='00% / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='63%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='10% / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='71%  174 / 151  Agentformer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='97% / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='30%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='20% / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='53%  174 / 163  TABLE IV: Ablation study of ego-conditioning, the first and  second number are the metrics with and without ego condi-  tioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Most metrics see degradation when ego-conditioning  is disabled (bold means better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Rasterized  PredictionNet  Agentformer  Prediction w/ EC  150ms  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='00s  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='60s  Prediction w/o EC  37ms  75ms  150ms  Ego-motion sampling  30ms  Dynamic programming  11ms  TABLE V: Runtime of the three key components  inference call, compared to using discrete latent space where  sampling is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' While the prediction ADE/FDE is  similar, the closed-loop performance with the Gaussian latent  is significantly worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' We suspect this is due to the poor  temporal consistency between predictions of consecutive  time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' While both TPP and the two baselines saw  performance degradation under the Gaussian latent, the pol-  icy planner relies on more sophisticated prediction/planning  interaction, thus suffers more than the two baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Runtime analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  Table V shows the runtime of the  three components of TPP on an Nvidia 3090 GPU (most  computation is done on GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' It should be noted that all  runtime is recorded on unoptimized Python code and there  is significant space of improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' For example, the long  runtime for PredictionNet is mainly due to rasterization,  and we know its runtime can be as low as 5ms with  the proper engineering [2], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=', efficient CUDA code for  rasterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' While the runtime for prediction models is not  very informative and is not the focus of this paper, it is quite  clear that ego motion sampling and dynamic programming  can be done very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' With proper optimization, we  believe that TPP can be easily implemented in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' We  plan to release the code upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' CONCLUSION  We present TPP, a policy planner capable of generating  multistage motion policies that react to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' It  is centered around an ego trajectory tree and a scenario  tree that predicts the environment behavior, both containing  multiple stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Ego-conditioning is applied to leverage the  reactive behavior of the environment under the ego vehicle’s  presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' The closed-loop simulation result shows that TPP  significantly outperform two non-policy benchmarks and the  runtime test suggest that such sophisticated policy planner  can be run in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content=' Our planned future works include  (1) study more flexible topology of the trees and enable  event-triggered branching (2) improve the efficiency of com-  putation, especially under ego-conditioning (3) adaptive and  personalized policy planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  We would like to thank Bilal Kartal, Siva Hari, and  Edward Schmerling for the insightful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFKT4oBgHgl3EQfxS4R/content/2301.11902v1.pdf'}
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+1st Place Solution for ECCV 2022 OOD-CV
+Challenge Object Detection Track
+Wei Zhao1,⋆, Binbin Chen1,⋆, Weijie Chen1,2,†, Shicai Yang1, Di Xie1,
+Shiliang Pu1, and Yueting Zhuang2
+1 Hikvision Research Institute, Hangzhou China
+2 Zhejiang University, Hangzhou, China
+{zhaowei29, chenbinbin8, chenweijie5, yangshicai, xiedi,
+pushiliang.hri}@hikvision.com, {chenweijie, yzhuang}@zju.edu.cn
+Abstract. OOD-CV challenge‡ is an out-of-distribution generalization
+task. To solve this problem in object detection track, we propose a sim-
+ple yet effective Generalize-then-Adapt (G&A) framework, which
+is composed of a two-stage domain generalization part and a one-stage
+domain adaptation part. The domain generalization part is implemented
+by a Supervised Model Pretraining stage using source data for
+model warm-up and a Weakly Semi-Supervised Model Pretrain-
+ing stage using both source data with box-level label and auxiliary data
+(ImageNet-1K) with image-level label for performance boosting. The
+domain adaptation part is implemented as a Source-Free Domain
+Adaptation paradigm, which only uses the pre-trained model and the
+unlabeled target data to further optimize in a self-supervised training
+manner. The proposed G&A framework help us achieve the first place
+on the object detection leaderboard of the OOD-CV challenge. Code will
+be released in https://github.com/hikvision-research/OOD-CV.
+Keywords: Weakly Semi-Supervised Object Detection, Out-of-Distribution
+Generalization, Test-Time Domain Adaptive Object Detection
+1
+Method
+To improve the model robustness in the unknown target domains, we propose a
+simple yet effective Generalize-then-Adapt (G&A) framework to solve the
+model degradation problem for object detection under domain shift. Specifi-
+cally, this framework is composed of a two-stage domain generalization part and
+a one-stage domain adaptation part: 1) Supervised Model Pre-training.
+A strong baseline is built against domain shift, which exploits labeled source
+data with various strong data augmentation strategies so as to simulate po-
+tential out-of-distribution data. 2) Weakly Semi-Supervised Model Pre-
+training. Previous work demonstrates that using extra auxiliary training can
+⋆ Equal contributions
+† Corresponding author
+‡ https://www.ood-cv.org/
+arXiv:2301.04796v1  [cs.CV]  12 Jan 2023
+
+2
+W. Zhao et al.
+S
+…
+…
+…
+…
+aeroplane
+…
+…
+car
+car
+aeroplane
+multi-aug source data
+weakly labeled data
+Generalize: enrich 
+source domain
+…
+…
+…
+Adapt: shift to 
+target domain
+target data
+S: source domain 
+T: target domain
+T
+S
+only model 
+transmission
+Fig. 1. The proposed Generalize-then-Adapt framework, which firstly enriches the
+source domain as much as possible via strong data augmentation and weakly semi-
+supervised object detection by exploiting the image-level labeled ImageNet-1K as an
+auxiliary training data, and then shifts the model performance from the enriched source
+domains to the given target domain via test-time domain adaptation.
+further enhance out-of-distribution generalization ability [9]. ImageNet-1K [13]
+can be viewed as an auxiliary training data with only image-level label. In
+this way, the pre-trained object detector in the first stage can be further op-
+timized on the labeled source data (Robin training set [18] with box-level la-
+bel) and the weakly-labeled source data (ImageNet-1K with image-level label),
+termed Weakly Semi-Supervised Object Detection, which is implemented via
+a Class-Specific Pseudo-Labeling method in this report. 3) Source-Free Do-
+main Adaptation is utilized for Test-Time Training, which adapts the model
+to the target domain by merely exploiting the source pre-trained object de-
+tector and the unlabeled target data without accessing source data [1, 8]. In
+this challenge, it is simply implemented as a Mean-Teacher based Self-Training
+mechanism. The overview of the proposed method is shown in Fig. 2. After inte-
+grating Test-Time Augmentation and Model Ensemble strategies, our solution
+ranks the 1st place on the object detection leaderboard of the OOD-CV challenge
+https://codalab.lisn.upsaclay.fr/competitions/6784#results.
+Our thinking behind this framework: Compared with conventional do-
+main adaptation methods, which train the source data and the target data
+jointly (although this paradigm is forbidden in this challenge), the proposed
+G&A framework is more feasible in real-world scenarios, which decouples the
+joint-training paradigm into a domain generalization stage by merely exploiting
+source data and a test-time domain adaptation stage by merely exploiting target
+data as shown in Fig.1. G&A only allows pre-trained model transmission with-
+
+ECCV 2022 OOD-CV Challenge Object Detection Track
+3
+Labeled Data
+(Robin)
+Labeled Data
+(Robin)
+Unlabeled Data
+(from ImageNet)
+Class-Specific 
+Pseudo-Labeling 
+Weak-Aug
+Strong-Aug
+Strong-Aug
+Strong-Aug
+Supervision
+Supervision
+EMA
+Stage1: Strong Baseline
+Unlabeled Data
+(Robin)
+EMA
+Pseudo-Labeling 
+Weak-Aug
+Strong-Aug
+Supervision
+Stage2: Weakly Semi-Supervised Object Detection
+Stage3: Test Time Training
+Mix-Up
+Color-Jitter
+Foggy or Snow
+Flip
+Resize
+Mask Paste
+Strong Augmentation
+……
+Fig. 2. An overview of the proposed Generalize-then-Adapt framework, which is com-
+posed of three stages: 1) Strong Baseline (using Robin training set with box-level label),
+2) Weakly Semi-Supervised Object Detection (using Robin training set with box-level
+label and ImageNet-1K with image-level label), 3) Test-Time Training (using unlabeled
+Robin testing data).
+out source data exchange for the sake of avoiding data expansive transmission
+and data privacy leakage. The generalization step is usually carried out on the
+server side, while the adaptation step is usually conducted on the client side for
+model self-evolution. Actually, these two steps can be viewed as an upstream
+operation and a downstream operation. However, the existing works in OOD
+community usually focus on the domain generalization step or the test-time do-
+main adaptation step, without taking these two steps into an integration. We
+hope our solution with superior performance on the leaderboard of this challenge
+can inspire the community to focus on how to integrate these two steps so as to
+further resist model degradation under domain shifts.
+2
+Implementation Details
+2.1
+Dependencies
+– Backbone: ConvNext-Large [11]
+– Detection Neck: FPN [10], DyHead [3]
+– Detection Head: DDOD [2], TOOD [4], VFNet [16], Auto-Assign [20],
+PAA [7]
+– Ensemble: WBF [14]
+– Dataset: ROBIN [18], ImageNet-1K [13]
+– Weakly-Supervised Semantic Segmentation: MCTformer [15]
+
+4
+W. Zhao et al.
+2.2
+Training Description
+AUG
+sub-AUG
+probability
+random resize
+/
+1.0
+mask-level copy-paste
+/
+0.3
+random-choice
+mixup
+0.3
+blur,noise
+0.3
+color jitter
+0.3
+random-erase
+0.3
+foggy, snow
+0.3
+random horizontal flipping
+/
+0.5
+Table 1. Strong data augmentation.
+Strong baseline. A ConvNext-Large based DDOD detection model is trained
+using the ROBIN training dataset with strong data augmentation. In this stage,
+the source data is enriched from the original source domain to a wider domain
+that includes both the original source domain as well as some novel domains to
+resist domain shifts.
+Weakly semi-supervised object detection. To further generalize the object
+detector, a subset of ImageNet-1K (with the same labels as the 10 categories in
+ROBIN task) is used as the additional data to enrich the source data with a
+weakly semi-supervised object detection method based on class-specific pseudo-
+labeling. Firstly, the pseudo-boxes of ImageNet-1K are generated by forwarding
+the weak augmented ImageNet-1K dataset to the EMA (Exponential Moving
+Average) model. Due to the image-level class supervision prior, we only keep the
+pseudo-boxes belonging to the corresponding image-level labels. Then, the pre-
+trained model obtained in the first stage is further optimized using the ImageNet-
+1K dataset and the ROBIN dataset [18] with strong data augmentation. The
+detail about stage2 is shown in Fig. 2-stage2.
+Test-Time Training. A source-free domain adaptation based test-time train-
+ing strategy is used to adapt the model to the testing domain. A simple mean-
+teacher based self-training mechanism is used in this stage. The weak augmented
+test data is fed to the EMA model to generate the pseudo label. And then these
+labels and the corresponding data are used to train the detector with strong
+data augmentation. The final EMA model is selected for testing.
+Technique details. We build the detector with the backbone of ConvNext-
+Large and the detection head of DDOD with FPN [10]. For data augmentation,
+we apply mixup [17], color jitter, foggy [6], snow [6], resize, horizontal flipping,
+
+ECCV 2022 OOD-CV Challenge Object Detection Track
+5
+rotation, blur, noise, random erase [19], and mask-level copy-paste. To obtain
+more vivid occlusion effects, we introduce MCTformer [15] to generate pseudo
+masks of the salient objects in the subset of ImageNet-1K. Then, these masks
+are used in the mask-level copy-paste strategy [5], which are pasted onto the
+training images to simulate object occlusion. The data augmentation setting
+is shown in Table 1. Note that both mixup and mask-level copy-paste use the
+ImageNet-1K dataset, and these augmentations are not used in the test-time
+domain adaptation part. The optimizer for the three training stages described
+above is AdamW [12]. The learning rate is 0.0001. The batch size is 2 per GPU. A
+multi-scale training mechanism is used, and the image height is rescaled between
+[480, 800] randomly.
+2.3
+Testing Description
+In the normal test stage, we only use a single scale (width 1333, height 800). For
+test-time augmentation (TTA), we use multi-scale and horizontal flipping tricks.
+The multi-scale trick includes 11 scales, and the image height is evenly ranging
+from 480 to 800 .
+3
+Experimental Results
+3.1
+Ablation Study
+Method S1* S1 S2 S3 TTA OOD-AP50
+(Phase-1)
+OOD-AP50
+(Phase-2)
+DDOD
+✓
+74.43
+/
+✓
+83.24
+/
+✓ ✓
+85.34
+68.26
+✓ ✓ ✓
+85.94
+71.34
+✓ ✓ ✓
+✓
+86.21
+/
+Table 2. Ablation study. S1, S2 and S3 are short for stage-1, stage-2 and stage-3. S1*
+only uses weak augmentation in stage-1. TTA is short for test-time augmentation.
+Table 2 shows the ablation study of the proposed method on the phase-1
+dataset. The DDOD based ConvNext is the basic object detection framework in
+this ablation study. The results of S1 and S2 show that the generalization ability
+of the detection model is gradually enhanced as the training data is enriched.
+The result of S3 shows that the test-time domain adaptation can effectively shift
+the model performance from the enriched source domain to the target domain.
+Besides, the TTA can further improve the model performance.
+
+6
+W. Zhao et al.
+Model
+OOD-AP50
+(Phase-1)
+OOD-AP50
+(Phase-2)
+DDOD
+85.94
+71.34
++TTA
+86.21
+/
+DDOD+Dyhead1
+85.42
+/
++TTA
+85.92
+/
+DDOD+Dyhead2
+85.70
+/
++TTA
+86.08
+/
+TOOD
+84.93
+/
++TTA
+85.92
+/
+Auto-Assign
+83.96
+/
++TTA
+84.69
+/
+VFNet
+83.69
+/
++TTA
+84.66
+/
+DDOD†
+85.04
+/
++TTA
+85.76
+/
+PAA
+85.39
+/
+Ensemble
+86.67
+73.89
+Table 3. Model ensemble performance. † uses random rotation and random vertical
+flipping except for the strong data augmentations in Table 2. We use WBF [14] to
+fuse the vanilla test results and TTA results from the models of DDOD [2], DDOD
+[2]+DyHead [3], TOOD [4], Auto-Assign [20], VFNet [16], DDOD† [2] and PAA [7].
+“Dyhead1” and “Dyhead2” represent different number of Dyhead blocks.
+Environment
+Descriptions
+system and hardware Ubuntu20.04+cuda11.3+cudnn8+Tesla V100(32GB)×8
+python and pytorch
+python=3.8.5+pytorch=1.10
+python library
+mmcv-full=1.6.1+mmdet=2.25.1+mmcls=0.23.2
+Table 4. Language and implementation details.
+3.2
+Ensembles And Fusion Strategies
+Experimental evidence shows that the ensemble method can get further perfor-
+mance improvement. We ensemble the predicted boxes from the 15 predictions
+(8 different models and 7 of them are further performed TTA) by weighted box
+fusion (WBF). ConvNext-Large [11] is the backbone for all models, and the train-
+ing settings are exactly the same. Detailed results are shown in Table 3. After
+ensemble, the OOD accuracy (mAP50) is improved from 71.34% to 73.89%.
+
+ECCV 2022 OOD-CV Challenge Object Detection Track
+7
+Stage
+Time
+Descriptions
+Stage-1 (training) 13h45min13s fp16, test-interval=4epoch, total-interval=36epoch
+Stage-2 (training) 9h44min26s fp16, test-interval=1epoch, total-interval=12epoch
+Stage-3 (training) 5h36min03s
+fp16, test-interval=1epoch, total-interval=6epoch
+Testing
+26.3 fps
+fp32
+Table 5. Training/testing time (single model, batch-size=2×8). The test-time augmen-
+tation and model ensemble counterparts can be roughly determined from the single-
+model results, which is omitted here for simplicity.
+ConvNext+DDOD
+Input height 800
+Flops
+834.61 GFLOPs
+Params
+204.71 M
+ConvNext+DDOD (Multi-Scale)
+Input height 480:800:32
+Flops
+14687.86 GFLOPs
+Params
+204.71 M
+Table 6. Method complexity and model parameters (single model)
+3.3
+Model Complexity Analysis
+The environment we use is given in Table 4. The training and testing time
+for DDOD model are shown in Table 5. The proposed solution requires three
+training stages as shown in Table 5. Therefore, the training time is longer than
+the base detection framework. However, the test efficiency is the same as the basis
+detection framework. Please refer to Table 6 for model complexity analysis.
+4
+Conclusion
+We thank the organizing committee for providing data [18] that allowed us to
+study the robustness of object detectors under domain shifts.
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+
diff --git a/n9E3T4oBgHgl3EQf7QsQ/content/tmp_files/load_file.txt b/n9E3T4oBgHgl3EQf7QsQ/content/tmp_files/load_file.txt
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@@ -0,0 +1,395 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf,len=394
+page_content='1st Place Solution for ECCV 2022 OOD-CV  Challenge Object Detection Track  Wei Zhao1,⋆, Binbin Chen1,⋆, Weijie Chen1,2,†, Shicai Yang1, Di Xie1,  Shiliang Pu1, and Yueting Zhuang2  1 Hikvision Research Institute, Hangzhou China  2 Zhejiang University, Hangzhou, China  {zhaowei29, chenbinbin8, chenweijie5, yangshicai, xiedi,  pushiliang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='hri}@hikvision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='com, {chenweijie, yzhuang}@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='cn  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' OOD-CV challenge‡ is an out-of-distribution generalization  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' To solve this problem in object detection track, we propose a sim-  ple yet effective Generalize-then-Adapt (G&A) framework, which  is composed of a two-stage domain generalization part and a one-stage  domain adaptation part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The domain generalization part is implemented  by a Supervised Model Pretraining stage using source data for  model warm-up and a Weakly Semi-Supervised Model Pretrain-  ing stage using both source data with box-level label and auxiliary data  (ImageNet-1K) with image-level label for performance boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The  domain adaptation part is implemented as a Source-Free Domain  Adaptation paradigm, which only uses the pre-trained model and the  unlabeled target data to further optimize in a self-supervised training  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The proposed G&A framework help us achieve the first place  on the object detection leaderboard of the OOD-CV challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Code will  be released in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='com/hikvision-research/OOD-CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Keywords: Weakly Semi-Supervised Object Detection, Out-of-Distribution  Generalization, Test-Time Domain Adaptive Object Detection  1  Method  To improve the model robustness in the unknown target domains, we propose a  simple yet effective Generalize-then-Adapt (G&A) framework to solve the  model degradation problem for object detection under domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Specifi-  cally, this framework is composed of a two-stage domain generalization part and  a one-stage domain adaptation part: 1) Supervised Model Pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  A strong baseline is built against domain shift, which exploits labeled source  data with various strong data augmentation strategies so as to simulate po-  tential out-of-distribution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' 2) Weakly Semi-Supervised Model Pre-  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Previous work demonstrates that using extra auxiliary training can  ⋆ Equal contributions  † Corresponding author  ‡ https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='ood-cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='org/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='04796v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='CV]  12 Jan 2023  2  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  S  …  …  …  …  aeroplane  …  …  car  car  aeroplane  multi-aug source data  weakly labeled data  Generalize: enrich  source domain  …  …  …  Adapt: shift to  target domain  target data  S: source domain  T: target domain  T  S  only model  transmission  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The proposed Generalize-then-Adapt framework, which firstly enriches the  source domain as much as possible via strong data augmentation and weakly semi-  supervised object detection by exploiting the image-level labeled ImageNet-1K as an  auxiliary training data, and then shifts the model performance from the enriched source  domains to the given target domain via test-time domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  further enhance out-of-distribution generalization ability [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' ImageNet-1K [13]  can be viewed as an auxiliary training data with only image-level label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' In  this way, the pre-trained object detector in the first stage can be further op-  timized on the labeled source data (Robin training set [18] with box-level la-  bel) and the weakly-labeled source data (ImageNet-1K with image-level label),  termed Weakly Semi-Supervised Object Detection, which is implemented via  a Class-Specific Pseudo-Labeling method in this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' 3) Source-Free Do-  main Adaptation is utilized for Test-Time Training, which adapts the model  to the target domain by merely exploiting the source pre-trained object de-  tector and the unlabeled target data without accessing source data [1, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' In  this challenge, it is simply implemented as a Mean-Teacher based Self-Training  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The overview of the proposed method is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' After inte-  grating Test-Time Augmentation and Model Ensemble strategies, our solution  ranks the 1st place on the object detection leaderboard of the OOD-CV challenge  https://codalab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='lisn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='upsaclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='fr/competitions/6784#results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Our thinking behind this framework: Compared with conventional do-  main adaptation methods, which train the source data and the target data  jointly (although this paradigm is forbidden in this challenge), the proposed  G&A framework is more feasible in real-world scenarios, which decouples the  joint-training paradigm into a domain generalization stage by merely exploiting  source data and a test-time domain adaptation stage by merely exploiting target  data as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' G&A only allows pre-trained model transmission with-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='ECCV 2022 OOD-CV Challenge Object Detection Track  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Labeled Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='(Robin)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Labeled Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='(Robin)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Unlabeled Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='(from ImageNet)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Class-Specific  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Pseudo-Labeling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Weak-Aug  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Strong-Aug  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Strong-Aug  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Strong-Aug  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Supervision  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Supervision  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='EMA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Stage1: Strong Baseline  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Unlabeled Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='(Robin)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='EMA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Pseudo-Labeling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Weak-Aug  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Strong-Aug  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Supervision  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Stage2: Weakly Semi-Supervised Object Detection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Stage3: Test Time Training  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Mix-Up  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Color-Jitter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Foggy or Snow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Flip  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Resize  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Mask Paste  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='Strong Augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' An overview of the proposed Generalize-then-Adapt framework, which is com-  posed of three stages: 1) Strong Baseline (using Robin training set with box-level label),  2) Weakly Semi-Supervised Object Detection (using Robin training set with box-level  label and ImageNet-1K with image-level label), 3) Test-Time Training (using unlabeled  Robin testing data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  out source data exchange for the sake of avoiding data expansive transmission  and data privacy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The generalization step is usually carried out on the  server side, while the adaptation step is usually conducted on the client side for  model self-evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Actually, these two steps can be viewed as an upstream  operation and a downstream operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' However, the existing works in OOD  community usually focus on the domain generalization step or the test-time do-  main adaptation step, without taking these two steps into an integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' We  hope our solution with superior performance on the leaderboard of this challenge  can inspire the community to focus on how to integrate these two steps so as to  further resist model degradation under domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  2  Implementation Details  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='1  Dependencies  – Backbone: ConvNext-Large [11]  – Detection Neck: FPN [10], DyHead [3]  – Detection Head: DDOD [2], TOOD [4], VFNet [16], Auto-Assign [20],  PAA [7]  – Ensemble: WBF [14]  – Dataset: ROBIN [18], ImageNet-1K [13]  – Weakly-Supervised Semantic Segmentation: MCTformer [15]  4  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='2  Training Description  AUG  sub-AUG  probability  random resize  /  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='0  mask-level copy-paste  /  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3  random-choice  mixup  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3  blur,noise  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3  color jitter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3  random-erase  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3  foggy, snow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3  random horizontal flipping  /  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='5  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Strong data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Strong baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' A ConvNext-Large based DDOD detection model is trained  using the ROBIN training dataset with strong data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' In this stage,  the source data is enriched from the original source domain to a wider domain  that includes both the original source domain as well as some novel domains to  resist domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Weakly semi-supervised object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' To further generalize the object  detector, a subset of ImageNet-1K (with the same labels as the 10 categories in  ROBIN task) is used as the additional data to enrich the source data with a  weakly semi-supervised object detection method based on class-specific pseudo-  labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Firstly, the pseudo-boxes of ImageNet-1K are generated by forwarding  the weak augmented ImageNet-1K dataset to the EMA (Exponential Moving  Average) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Due to the image-level class supervision prior, we only keep the  pseudo-boxes belonging to the corresponding image-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Then, the pre-  trained model obtained in the first stage is further optimized using the ImageNet-  1K dataset and the ROBIN dataset [18] with strong data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The  detail about stage2 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' 2-stage2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Test-Time Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' A source-free domain adaptation based test-time train-  ing strategy is used to adapt the model to the testing domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' A simple mean-  teacher based self-training mechanism is used in this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The weak augmented  test data is fed to the EMA model to generate the pseudo label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' And then these  labels and the corresponding data are used to train the detector with strong  data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The final EMA model is selected for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Technique details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' We build the detector with the backbone of ConvNext-  Large and the detection head of DDOD with FPN [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' For data augmentation,  we apply mixup [17], color jitter, foggy [6], snow [6], resize, horizontal flipping,  ECCV 2022 OOD-CV Challenge Object Detection Track  5  rotation, blur, noise, random erase [19], and mask-level copy-paste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' To obtain  more vivid occlusion effects, we introduce MCTformer [15] to generate pseudo  masks of the salient objects in the subset of ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Then, these masks  are used in the mask-level copy-paste strategy [5], which are pasted onto the  training images to simulate object occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The data augmentation setting  is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Note that both mixup and mask-level copy-paste use the  ImageNet-1K dataset, and these augmentations are not used in the test-time  domain adaptation part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The optimizer for the three training stages described  above is AdamW [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The batch size is 2 per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' A  multi-scale training mechanism is used, and the image height is rescaled between  [480, 800] randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3  Testing Description  In the normal test stage, we only use a single scale (width 1333, height 800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' For  test-time augmentation (TTA), we use multi-scale and horizontal flipping tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  The multi-scale trick includes 11 scales, and the image height is evenly ranging  from 480 to 800 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  3  Experimental Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='1  Ablation Study  Method S1* S1 S2 S3 TTA OOD-AP50  (Phase-1)  OOD-AP50  (Phase-2)  DDOD  ✓  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='43  /  ✓  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='24  /  ✓ ✓  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='34  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='26  ✓ ✓ ✓  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='94  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='34  ✓ ✓ ✓  ✓  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='21  /  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' S1, S2 and S3 are short for stage-1, stage-2 and stage-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' S1*  only uses weak augmentation in stage-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' TTA is short for test-time augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Table 2 shows the ablation study of the proposed method on the phase-1  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The DDOD based ConvNext is the basic object detection framework in  this ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The results of S1 and S2 show that the generalization ability  of the detection model is gradually enhanced as the training data is enriched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  The result of S3 shows that the test-time domain adaptation can effectively shift  the model performance from the enriched source domain to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Besides, the TTA can further improve the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  6  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Model  OOD-AP50  (Phase-1)  OOD-AP50  (Phase-2)  DDOD  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='94  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='34  +TTA  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='21  /  DDOD+Dyhead1  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='42  /  +TTA  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='92  /  DDOD+Dyhead2  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='70  /  +TTA  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='08  /  TOOD  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='93  /  +TTA  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='92  /  Auto-Assign  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='96  /  +TTA  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='69  /  VFNet  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='69  /  +TTA  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='66  /  DDOD†  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='04  /  +TTA  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='76  /  PAA  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='39  /  Ensemble  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='67  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='89  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Model ensemble performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' † uses random rotation and random vertical  flipping except for the strong data augmentations in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' We use WBF [14] to  fuse the vanilla test results and TTA results from the models of DDOD [2], DDOD  [2]+DyHead [3], TOOD [4], Auto-Assign [20], VFNet [16], DDOD† [2] and PAA [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  “Dyhead1” and “Dyhead2” represent different number of Dyhead blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  Environment  Descriptions  system and hardware Ubuntu20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='04+cuda11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3+cudnn8+Tesla V100(32GB)×8  python and pytorch  python=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='5+pytorch=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='10  python library  mmcv-full=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='1+mmdet=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='1+mmcls=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='2  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Language and implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='2  Ensembles And Fusion Strategies  Experimental evidence shows that the ensemble method can get further perfor-  mance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' We ensemble the predicted boxes from the 15 predictions  (8 different models and 7 of them are further performed TTA) by weighted box  fusion (WBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' ConvNext-Large [11] is the backbone for all models, and the train-  ing settings are exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Detailed results are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' After  ensemble, the OOD accuracy (mAP50) is improved from 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='34% to 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='89%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  ECCV 2022 OOD-CV Challenge Object Detection Track  7  Stage  Time  Descriptions  Stage-1 (training) 13h45min13s fp16, test-interval=4epoch, total-interval=36epoch  Stage-2 (training) 9h44min26s fp16, test-interval=1epoch, total-interval=12epoch  Stage-3 (training) 5h36min03s  fp16, test-interval=1epoch, total-interval=6epoch  Testing  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3 fps  fp32  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Training/testing time (single model, batch-size=2×8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The test-time augmen-  tation and model ensemble counterparts can be roughly determined from the single-  model results, which is omitted here for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  ConvNext+DDOD  Input height 800  Flops  834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='61 GFLOPs  Params  204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='71 M  ConvNext+DDOD (Multi-Scale)  Input height 480:800:32  Flops  14687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='86 GFLOPs  Params  204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='71 M  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Method complexity and model parameters (single model)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='3  Model Complexity Analysis  The environment we use is given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The training and testing time  for DDOD model are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' The proposed solution requires three  training stages as shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Therefore, the training time is longer than  the base detection framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' However, the test efficiency is the same as the basis  detection framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content=' Please refer to Table 6 for model complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
+page_content='  4  Conclusion  We thank the organizing committee for providing data [18] that allowed us to  study the robustness of object detectors under domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQf7QsQ/content/2301.04796v1.pdf'}
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+arXiv:2301.04139v1  [gr-qc]  10 Jan 2023
+Electromagnetic Effects on the
+Complexity of Static Cylindrical
+Object in f(G, T) Gravity
+M. Sharif1 ∗and K. Hassan2 †
+1 Department of Mathematics and Statistics, The University of Lahore,
+1-KM Defence Road Lahore, Pakistan.
+2 Department of Mathematics, University of the Punjab,
+Quaid-e-Azam Campus, Lahore-54590, Pakistan.
+Abstract
+In this paper, we investigate complexity of anisotropic cylindrical
+object under the influence of electromagnetic field in f(G, T) the-
+ory, where G and T indicate the Gauss-Bonnet term and trace of
+the stress-energy tensor, respectively. For this purpose, we calculate
+the modified field equations, non-conservation equation and mass dis-
+tributions that assist in comprehending the structure of astrophysical
+objects. The Riemann tensor is divided into different structure scalars,
+among which one is called the complexity factor. This factor is used
+to measure complexity of the system due to the involvement of in-
+homogeneous energy density, anisotropic pressure and charge.
+The
+vanishing of the complexity factor is employed as a constraint to for-
+mulate charged static solutions for the Gokhroo-Mehra model and
+polytropic equation of state. We conclude that the presence of charge
+reduces the complexity of the anisotropic system.
+Keywords:
+Anisotropic fluid; Self-gravitating systems; f(G, T) gravity;
+Complexity factor.
+PACS: 04.40.-b; 04.50.Kd; 04.40.Dg
+∗msharif.math@pu.edu.pk
+†komalhassan3@gmail.com
+1
+
+1
+Introduction
+General theory of relativity (GR) improved the fundamental ideas about our
+universe and the notion of gravity. Einstein assumed that the universe was
+neither contracting nor expanding, so he considered it as a static epoch.
+Later, Hubble confirmed the expansion of the cosmos by relating the reces-
+sion velocity of galaxies and distance. The inexplicable force known as dark
+energy is thought to be responsible for the accelerated expansion of the uni-
+verse. Supernovae and microwave background radiations are among different
+astrophysical events that unveil the fast cosmic expansion [1]. The ΛCDM
+model is the best model to explain the current accelerated expansion of the
+universe, however, it has two major challenges: fine-tuning and cosmic co-
+incidence. Thus, some modifications in GR (so called modified theories of
+gravity) were introduced to explain the expanding universe.
+In this regard, Nojiri and Odintsov [2] revised the Einstein-Hilbert action
+by adding the function of Gauss-Bonnet (GB) invariant, which resulted in
+f(G) gravity. This also helped in explaining how the cosmos transitioned
+from a decelerating to an accelerating epoch. The GB invariant is a com-
+bination of the Riemann tensor (Rψχβα), Ricci tensor (Rψχ) and curvature
+scalar (R) as
+G = −4RψχRψχ + RψχβαRψχβα + R2.
+Bamba et al. [3] studied the early-time bounce as well as late-time acceler-
+ation by considering the scale factor in the form of an exponential function
+by reconstructing f(G) model. Abbas et al. [4] examined viable and sta-
+ble aspects of stellar objects by utilizing the Krori-Barua metric together
+with a power-law model in f(G) theory. Sharif and Ikram [5] addressed the
+inflationary characteristics of homogeneous and isotropic universe using fea-
+sible f(G) model. For a plane symmetric structure, Shamir and Saeed [6]
+investigated the exponential and power-law solutions in this theory. Sharif
+and Ramzan [7] applied the embedding class-1 method to examine physical
+features of anisotropic self-gravitating objects.
+Sharif and Ikram [8] introduced trace of the energy-momentum tensor in
+f(G) action and proposed a new modified theory named as f(G, T) theory.
+They discussed energy conditions using FRW metric in this theory. The same
+authors [9] used the perturbation approach to check the stable behavior of
+the Einstein universe. For a certain range of the parameters, Hossienkhani et
+al. [10] demonstrated that weak and null energy conditions show consistency
+2
+
+with anisotropic f(G, T) universe. Sharif and Naeem [11] developed a new
+solution to analyze viability and stability of different stellar configurations.
+Shamir [12] investigated the oscillating universe for two models, such as linear
+and logarithmic trace terms of f(G, T) gravity.
+In the evolution and stable configuration of the celestial entities, electro-
+magnetic field plays a crucial role as it suppresses the gravitational force.
+A significant quantity of charge is required to counteract gravitational at-
+traction and maintain the stabilizing behavior of self-gravitating entities. In
+astrophysical realm, the first static charged solution of the Einstein-Maxwell
+field equations was presented by Reissner and Nordstrom. Bondi [13] im-
+posed the known cylindrical wave solution of the free space on the static field
+and observed the propagation of energy from such waves. Ivanov [14] studied
+general and regular solutions of the field equations in three different classes
+for the charged cylindrical object. Esculpi and Aloma [15] investigated the
+influence of anisotropy on charged compact objects using linear equation
+of state that relates radial and tangential pressures.
+Sharif and Naz [16]
+used dynamical field equations to investigate the collapsing phenomenon of a
+cylindrically charged star and evaluated the relationship between GB terms,
+physical variables and Weyl tensor in f(G) gravity.
+We have worked on
+Tolman IV and Krori-Barua spacetimes to illustrate physical features of the
+compact object in uncharged/charged configurations, respectively, in f(G, T)
+theory [17].
+The complexity of self-gravitating objects is greatly influenced by feasible
+characteristics of astronomical entities, such as temperature, heat, pressure,
+energy density, etc. Therefore, a mathematical formula that includes all sig-
+nificant components must be used to determine complexity of cosmic entities.
+The notion of complexity in terms of entropy and information was introduced
+by L´opez-Ruiz et al. [18]. Initially, this concept was implemented on two
+physical objects, i.e., perfect crystal and ideal gas. The molecules of perfect
+crystal are grouped in an ordered manner (due to the symmetric nature),
+thus it has zero entropy. In contrast, as the molecules of ideal gas are scat-
+tered randomly, so it shows maximum entropy. The probability distribution
+around the symmetric component of a perfect crystal adds little information
+because of its symmetry, and a small portion is enough to characterize all of
+its properties. However, maximum amount of information is available while
+studying a tiny region of ideal gas. Both these structures exhibit different
+behavior, yet they are allocated zero complexity. Another appropriate con-
+cept of complexity was established on the basis of how different probabilistic
+3
+
+states depart from the equiprobable distribution of the structure [19]. Using
+this concept, both ideal gas and perfect crystal are considered to give zero
+complexity. The energy density was employed in the revised methodology
+(in place of probability distribution) to evaluate the complexity of celestial
+objects [20]. However, other state variables like heat, pressure, temperature,
+etc. were not included in this benchmark.
+Recently, Herrera [21] redefined the notion of complexity in terms of in-
+homogeneous energy density, pressure anisotropy and Tolman mass for the
+case of static anisotropic fluid source. In this approach, different structure
+scalars are obtained through the orthogonal breakdown of the Riemann ten-
+sor. A scalar that encompasses all the above-mentioned parameters is called
+complexity factor. Sharif and Butt [22] studied how the complexity of spher-
+ical configuration was affected in the presence of an electromagnetic field.
+Herrera and his collaborators [23] extended the concept of complexity for
+dynamical dissipative configuration and addressed two modes of evolution.
+The same authors [24] also considered an axially symmetric structure to
+calculate its complexity. Contreras et al. [25] utilized the temporal metric
+component of Durgapal IV and V solutions along with vanishing complexity
+condition for both charged/uncharged cases to discuss feasibility of the con-
+structed models. Contreras and Stuchlik [26] formulated a simple protocol
+to develop the anisotropic interior solutions via vanishing condition. Abbas
+and Nazar [27] calculated the complexity of various structures in the realm
+of non-minimally coupled f(R) gravity. Sharif et al. [28] applied Herrera’s
+approach to the spherically symmetric matter source in f(G) gravity and
+found an increment in complexity. A large body of literature is found on
+the application of complexity to different geometries in the background of
+various modified theories [29]. Yousaf et al. [30] inspected the complexity
+producing factor for the charged/uncharged spheres in f(G, T) gravity.
+When a strong gravitational field is present, the departure from spherical
+to non-spherical geometries becomes crucial for assessing the solution. Levi
+Civita developed vacuum cylindrically symmetric spacetime that inspired as-
+tronomers to explore the interesting features of various stellar systems. In
+order to examine the cylindrical distribution, Herrera et al. [31] discussed
+scalar functions in the form of matter variables.
+Using modified Gauss-
+Bonnet gravity, Houndjo et al. [32] constructed a set of seven solutions that
+conform to three different viable models in cylindrical spacetime. To deter-
+mine the complexity of a cylindrical spacetime, Sharif and Butt [33] applied
+Herrera method to decompose the Riemann tensor in the scenario of static
+4
+
+matter configuration. The same authors [34] examined the influence of elec-
+tromagnetic field on the cylindrical geometry using the same procedure. We
+have also investigated the complexity factor in static cylindrical and dynam-
+ical spherical/cylindrical structures in f(G, T) gravity [35].
+This paper studies the effects of charge on the complexity of anisotropic
+cylindrical matter distribution in the formalism of f(G, T) theory. Through-
+out the paper, we have utilized the geometrized units, i.e., G = c = 1. The
+paper is characterized as follows. The modified field equations and physical
+characteristics related to anisotropic matter distribution are derived in sec-
+tion 2. The orthogonal splitting of the Riemann tensor is given in section 3.
+In section 4, the vanishing complexity condition is employed as a constraint
+to compute the possible solutions. The key results are wrapped in section 5.
+2
+f(G, T) Gravity
+In this section, we explore the modified field equations with the inclusion
+of electromagnetic field and examine physical attributes of the anisotropic
+celestial structure. The Einstein-Maxwell action is given as follows
+If(G,T) =
+1
+2κ2
+� √−g[f(G, T) + R]d4x +
+� √−g(Lm + LEM)d4x,
+(1)
+where g denotes determinant of the metric tensor and κ2 represents the cou-
+pling constant. Moreover, the terms Lm and LEM indicate the Lagrangian
+density corresponding to the usual matter and electromagnetic field, respec-
+tively.
+The energy-momentum tensor and matter Lagrangian density are
+associated through the relation
+Tψχ = gψχLm − 2∂Lm
+∂gψχ .
+(2)
+The f(G, T) field equations are obtained by varying the action (1) with
+respect to the metric tensor
+Gψ
+χ
+=
+8π(T ψ
+χ + Sψ
+χ) − (Θψ
+χ + T ψ
+χ )fT(G, T) + 1
+2δψ
+χf(G, T) +
+�
+4RχmRψm
+−
+2RRψ
+χ + 4RmlRψ
+mχl − 2RmlnψRχmln
+�
+fG(G, T) +
+�
+4Rψ
+χ∇2
+−
+4Rψ
+mχl∇m∇l − 2δψ
+χR∇2 + 2R∇ψ∇χ − 4Rψm∇m∇χ − 4Rm
+χ ∇ψ∇m
+5
+
++
+4δψ
+χRml∇m∇l
+�
+fG(G, T),
+(3)
+where the Einstein tensor is expressed as Gψ
+χ = Rψ
+χ − 1
+2Rδψ
+χ and ∇υ∇υ = ∇2
+denotes the d’ Alembert operator. Moreover, the partial derivatives of an ar-
+bitrary f(G, T) function such as fG(G, T) and fT(G, T) represent derivatives
+taken with respect to G and T, respectively, and Θψχ = −2Tψχ + pgψχ.
+The cylindrical geometric distribution surrounded by the hypersurface
+(Σ) is delineated by the line element [33]
+ds2 = −K2(r)dt2 + L2(r)dr2 + r2dθ2 + r2α2dz2,
+(4)
+where the metric coefficients K and L are unknown potential functions while
+α (constant term) has the dimension of inverse length. The stress-energy
+tensor corresponding to anisotropic matter distribution reads
+T ψM
+χ
+= ̺uψuχ + phψ
+χ + Πψ
+χ,
+(5)
+along with
+Πψ
+χ
+=
+Π
+�
+φψφχ − 1
+3hψ
+χ
+�
+,
+Π = pr − p⊥,
+p = pr + 2p⊥
+3
+,
+uψuψ
+=
+−1,
+φψuψ = 0,
+φψφψ = 1
+hψ
+χ = δψ
+χ + uψuχ,
+where p⊥ and pr are the tangential and radial pressures, respectively, and Π
+is the anisotropic factor. The components of four velocity and radial four-
+vector are denoted as uψ =
+� 1
+K, 0, 0, 0
+�
+, φψ =
+�
+0, 1
+L, 0, 0
+�
+, respectively. The
+extra curvature terms of f(G, T) gravity are given as
+T ψGT
+χ
+=
+1
+8π
+�
+{(̺ + p)uψuχ + Πψ
+χ}fT(G, T) + 1
+2δψ
+χf(G, T) +
+�
+4RχmRψm
++
+4RmlRψ
+mχl − 2RRψ
+χ − 2RmlnψRχmln
+�
+fG(G, T) +
+�
+4δψ
+χRml∇m∇l
++
+4Rψ
+χ∇2 + 2R∇ψ∇χ − 2δψ
+χR∇2 − 4Rψm∇m∇χ − 4Rψ
+mχl∇m∇l
+−
+4Rm
+χ ∇ψ∇m
+�
+fG(G, T)
+�
+.
+(6)
+In order to investigate charged matter distribution, the electromagnetic en-
+ergy tensor is defined as
+Sψχ = 1
+4π
+�
+F l
+ψFχl − 1
+4FlmF lmgψχ
+�
+,
+(7)
+6
+
+where Fψχ = γχ,ψ −γψ,χ stands for the Maxwell field tensor and γψ is the four
+potential which becomes γψ = γ(r)δ0
+ψ for the static structure. The Maxwell
+field equations, in the tensorial notation, read
+F ψχ
+;χ = 4πJψ,
+F[ψχ;l] = 0,
+where Jψ = ̟υψ indicates the electromagnetic four-current vector and ̟
+represents the charge density.
+In this framework, the modified field equations are analyzed as
+8π
+�
+̺eff +
+q2
+8πr4
+�
+=
+� 1
+rL2
+�2L′
+L − 1
+r
+��
+,
+(8)
+8π
+�
+peff
+r −
+q2
+8πr4
+�
+=
+� 1
+rL2
+�2K′
+K + 1
+r
+��
+,
+(9)
+8π
+�
+peff
+⊥ +
+q2
+8πr4
+�
+=
+K′
+rKL2 + K′′
+KL2 − K′L′
+KL3 − L′
+rL3,
+(10)
+where derivative with respect to r is signified by prime and the correction
+terms ̺eff, peff
+r and peff
+⊥ are computed as
+̺eff
+=
+̺ + 1
+8π
+�
+(̺ + p)fT − f
+2 + 4K′′fG
+r2KL4 + 12L′f ′
+G
+r2L5
+− 12K′L′fG
+r2KL5
+− 4f ′′
+G
+r2L4
+�
+,
+peff
+r
+=
+pr + 1
+8π
+�2
+3ΠfT + f
+2 + 12K′L′fG
+r2KL5
++ 12K′f ′
+G
+r2KL4 − 4K′′fG
+r2KL4
+�
+,
+peff
+⊥
+=
+p⊥ + 1
+8π
+�
+−Π
+3 fT + f
+2 − 4K′′fG
+r2KL4 − 12K′L′f ′
+G
+rKL5
++ 4K′f ′′
+G
+rKL4 + 12K′L′fG
+r2KL5
++
+4K′′f ′
+G
+rKL4
+�
+.
+The non-conservation of energy-momentum tensor is observed through the
+covariant differentiation of Eq.(3). As a result, an extra force appears that
+pushes the particles to disobey the geodesic trajectory. The corresponding
+non-conservation equation as well as the hydrostatic equilibrium equation,
+respectively, under the effect of electromagnetic field are written as
+∇ψTψχ
+=
+fT(G, T)
+k2 − fT(G, T)
+�
+(Tψχ + Θψχ)∇ψ(ln fT(G, T)) + ∇ψΘψχ
+−
+1
+2gψχ∇ψT
+�
+,
+(11)
+7
+
+(peff
+r
+−
+q2
+8πr4)
+′
+=
+−K′(peff
+r + ̺eff)
+K
++ 2(peff
+⊥ − peff
+r +
+q2
+4πr4)
+r
++ ZL2,
+(12)
+where the curvature terms present in Z are displayed in Appendix A. The
+Tolman-Oppenheimer-Volkoff equation can also be represented by Eq.(12)
+that helps to describe the formation of anisotropic geometry. Equation (9)
+provides the value of K′
+K as
+K′
+K =
+4r2
+r2 − 8mr + 4q2
+�
+4πr(peff
+r −
+q2
+8πr4) − 1
+8r − q2
+2r3 + m
+r2
+�
+.
+(13)
+Utilizing the above expression in Eq.(11), we have
+�
+peff
+r
+−
+q2
+8πr4
+�′
+= −
+4r2
+r2 − 8mr + 4q2
+�
+4πr(peff
+r −
+q2
+8πr4) − 1
+8r − q2
+2r3 + m
+r2
+�
+×
+�
+̺eff + peff
+r
+�
++ 2(peff
+⊥ − peff
+r +
+q2
+4πr4)
+r
++ ZL2,
+(14)
+where m is the mass of the fluid distribution. The mass function of the inte-
+rior geometry can be evaluated through C-energy and Tolman mass. Firstly,
+the C-energy formalism [36] is utilized to compute the mass as
+m(r) ∼= �E = lE =
+�1
+4 − 1
+L2
+� r
+2 + q2
+2r,
+(15)
+Making use of the field Eq.(8), it can be rewritten as
+m =
+� r
+0
+4πr2
+�
+̺eff +
+q2
+8πr4
+�
+dr + q2
+2r + r
+8.
+(16)
+The Weyl tensor calculates the fluctuation in the celestial system caused
+by the gravitational fields in nearby objects. The Riemann tensor is divided
+into the Ricci tensor, curvature scalar and Weyl tensor as
+Rσ
+ψχω = Cσ
+ψχω+ 1
+2Rψωδσ
+χ− 1
+2Rψχδσ
+ω+ 1
+2Rσ
+χgψω− 1
+2Rσ
+ωgψχ− 1
+6R
+�
+δσ
+χgψω−gψχδσ
+ω
+�
+.
+(17)
+The decomposition of the Weyl tensor through the observer’s four velocity
+gives magnetic and electric parts as
+Hψχ = 1
+2ηψλβγCβγ
+χρuλuρ,
+Eψχ = Cψγχδuγuδ,
+8
+
+where
+Cψχλκ = (gψχβαgλκγδ − ηψχβαηλκγδ) uβuγEαδ,
+(18)
+and ηψχβα indicates the Levi-Civita tensor along with gψχβα = gψβgχα −
+gψαgχβ. The magnetic part of the Weyl tensor vanishes in symmetric cylin-
+drical spacetime, while the electric component becomes
+Eψχ = E
+�
+φψφχ − hψχ
+3
+�
+,
+where E and some of its properties are
+E =
+1
+2KL2
+�
+K′′ − K′L′
+L
++ KL′
+rL − K′
+r + K
+r2
+�
+,
+Eψ
+ψ = 0 = Eψγuγ = 0.
+(19)
+The connection between the mass of the cylindrical object with matter
+variables, charge, and Weyl tensor is given by Eqs.(8)-(10), (16) and (19) as
+m(r) = 4πr3
+3
+�
+̺eff + peff
+⊥ − peff
+r + 3q2
+8πr4
+�
++ q2
+2r − Er3
+3
++ r
+8,
+(20)
+and the value of E from the above equation leads to
+E = 4π
+r3
+� r
+0
+r3
+�
+̺eff +
+q2
+8πr4
+�′
+dr − 4π
+�
+peff
+r − peff
+⊥ −
+q2
+4πr4
+�
+.
+(21)
+This shows the relation among the energy density inhomogeneity, pressure
+anisotropy, electromagnetic effect and the Weyl tensor in the presence of
+f(G, T) theory. After inserting this value of E in Eq.(20), we have
+m(r) = q2
+2r + 4πr3
+3
+�
+̺eff +
+q2
+8πr4
+�
+− 4π
+3
+� r
+0
+r3
+�
+̺eff +
+q2
+8πr4
+�′
+dr + r
+8.
+(22)
+The effects of inhomogeneous energy density and charge on the mass can be
+noticed from the above equation.
+For the static symmetric structure, the total mass of the fluid distribution
+is determined by the Tolman formula [37] as
+mT = 4π
+� r
+0
+KLr2 �
+−T 0eff
+0
++ T 1eff
+1
++ 2T 2eff
+2
+�
+dr.
+(23)
+9
+
+Using the field equations in the above equation, it follows that
+mT = K′
+L r2.
+(24)
+Inserting the value of K
+′ from Eq.(13), it follows that
+mT = KL
+�
+m − q2
+2r
+�
++ 4πKL
+�
+peff
+r −
+q2
+8πr4
+�
+r3 − KLr
+8
+.
+The Tolman mass can also be referred to as the effective gravitational mass
+as it interlinks with the gravitational acceleration of test particles by
+a = K′
+KL = mT
+Kr2.
+(25)
+We obtain the expression for mT [38] as
+mT = (mT )Σ
+� r
+rΣ
+�3
+− r3
+� rΣ
+r
+KL
+r4
+�
+8πr3
+�
+peff
+⊥ +
+q2
+8πr4 − peff
+r +
+q2
+8πr4
+�
++
+� r
+0
+4πr3(̺eff +
+q2
+8πr4)
+′dr
+�
+dr.
+(26)
+Alternatively, the Tolman mass is acquired in terms of E as
+mT =
+� r
+rΣ
+�3
+(mT)Σ−r3
+� rΣ
+r
+KL
+r4
+�
+4πr3
+�
+peff
+⊥ +
+q2
+8πr4 − peff
+r +
+q2
+8πr4
+�
++ r3E
+�
+dr.
+(27)
+It is important to notice that inhomogeneous energy density, electromagnetic
+field, correction terms, and anisotropic pressure affect the Tolman mass.
+3
+Structure Scalars
+Bel [39] was the pioneer to propose the orthogonal splitting of the Riemann
+tensor, followed by Herrera [40], who formulated various scalar functions
+known as structure scalars. These scalar functions are defined on account
+of the essential characteristics of fluid distributions such as inhomogeneous
+energy density, anisotropic pressure, electromagnetic field and active grav-
+itational mass.
+These structure scalars are crucial in comprehending the
+10
+
+complexity factor for the self-gravitating systems. The tensors Yψχ, Zψχ and
+Xψχ are defined as
+Yψχ
+=
+Rψγχδuγuδ,
+(28)
+Zψχ
+=
+∗Rψγχδuγuδ = 1
+2ηψγǫβRǫβ
+χδuγuδ,
+(29)
+Xψχ
+=
+∗R∗
+ψγχδuγuδ = 1
+2ηǫβ
+ψγ, R∗
+ǫβχδuγuδ,
+(30)
+where R∗
+ψχγδ = 1
+2ηǫαγδRǫα
+ψχ. Equation (17) can also be rewritten in terms of
+the Riemann tensor as
+Rψγ
+χδ = Cψγ
+χδ + 16πT eff[ψ[χδγ]
+δ] + 8πT eff
+�1
+3δψ
+[χδγ
+δ] − δ[ψ
+[χδγ]
+δ]
+�
+.
+(31)
+The above equation is decomposed into four tensorial quantities after substi-
+tuting Eqs.(5) and (6) as
+Rψγ
+(I)χδ
+=
+16π̺u[ψu[χδγ]
+δ] + 2̺u[ψu[χδγ]
+δ] + 16πph[ψ
+[χδγ]
+δ]
++
+2pu[ψu[χδγ]
+δ] + 8π(̺ + 3p)
+�1
+3δψ
+[χδγ
+δ] − δ[ψ
+[χδγ]
+δ]
+�
+,
+(32)
+Rψγ
+(II)χδ
+=
+16πΠ[ψ
+[χδγ]
+δ] + 2Π[ψ
+[χδγ]
+δ] + δ[ψ
+[χδγ]
+δ] f
++
+8δ[ψ
+[χδγ]
+δ] Rml∇m∇lfG − 4Rδ[ψ
+[χδγ]
+δ] □fG,
+(33)
+Rψγ
+(III)χδ
+=
+4̺χ[ψ
+[χEγ]
+δ] − ǫψγ
+β ǫχδαEβα,
+(34)
+Rψγ
+(IV )χδ
+=
+2(RmχRmψδγ
+δ − RmδRmψδγ
+χ + RmχRmγδψ
+δ + RmδRmγδψ
+χ)fG
++
+2Rml(Rψ
+mχlδγ
+δ − Rψ
+mδlδγ
+χ − Rγ
+mχlδψ
+δ + Rγ
+mδlδψ
+χ)fG − R(Rψ
+χδγ
+δ
+−
+Rψ
+δ δγ
+χ − Rγ
+χδψ
+γ + Rγ
+δδψ
+χ)fG − 2(Rψ
+mχlδγ
+δ − Rψ
+mδlδγ
+χ − Rγ
+mχlδψ
+δ
++
+Rγ
+mδlδψ
+χ)∇m∇lfG + 2(Rψ
+χδγ
+δ − Rψ
+δ δγ
+χ − Rγ
+χδψ
+γ + Rγ
+δδψ
+χ)□fG
++
+R(δγ
+δ ∇ψ
+χ − δγ
+χ∇ψ
+δ − δψ
+δ ∇γ
+χ + δψ
+χ∇γ
+δ)fG − 2(Rmψδγ
+δ ∇χ∇m
+−
+Rmψδγ
+χ∇δ∇m − Rmγδψ
+δ ∇χ∇m + Rmγδψ
+χ∇δ∇m)fG − 2(Rm
+χ
+×
+δγ
+δ ∇ψ∇m − Rm
+δ δγ
+χ∇ψ∇m − Rm
+χ δψ
+δ ∇γ∇m + Rm
+δ δψ
+χ∇γ∇m)fG
+−
+(RχlmnRlmnψδγ
+δ − RδlmnRlmnψδγ
+χ − RχlmnRlmnγδψ
+δ + Rδlmn
+×
+Rlmnγδψ
+χ)fG + 1
+3
+�
+(̺ + p)fT + 2f + 4RlαRmαfG + 4Rlm
+11
+
+×
+Rα
+lαmfG − 2Rβ
+lmnRlmn
+α
+fG − 4Rmβ∇m∇βfG − 4Rα
+lαm∇m∇lfG
+−
+2R□fG + 16Rlm∇m∇lfG − 4Rmα∇α∇lfG − 2R2fG
+�
+−
+q2
+6r4h[ψ
+[χδγ]
+δ] + q2
+2r4u[ψu[χδγ]
+δ] − q2
+r4
+�
+φ[ψφ[χδγ]
+δ] + 1
+3h[ψ
+[χδγ]
+δ]
+�
++
+q2
+12πr4u[ψu[χδγ]
+δ] −
+q2
+8πr4δ[ψ
+[χδγ]
+δ] ,
+(35)
+we have employed the following formulas
+ǫψγχ = uβηβψγχ,
+ǫψγχuχ = 0,
+ǫβγαǫαψχ = δβ
+ψhγ
+χ − δβ
+ψhγ
+χ + uψ(uβδγ
+χ − δβ
+χuγ).
+The following tensors in terms of fluid variables can be written as
+Yψχ
+=
+1
+6(̺ + p)hψχfT + q2hψχ
+3r4
++ 4π
+3 (̺ + 3p)hψχ + Eψχ
+−
+4πΠψχ − q2
+r4(φψφχ − 1
+3hψχ) + Πψχ
+2 fT + DGT
+ψχ ,
+(36)
+Xψχ
+=
+8π̺hψχ
+3
++ q2hψχ
+3r4
++ Πψχ
+2 fT − 4πΠψχ − q2
+r4(φψφχ − 1
+3hψχ)
+−
+Eψχ + MGT
+ψχ ,
+(37)
+Zψχ
+=
+NGT
+ψχ .
+(38)
+The extra curvature terms DGT
+ψχ , NGT
+ψχ and MGT
+ψχ present in the above equations
+are displayed in Appendix A.
+The tensors Xψχ and Yψχ are subdivided into its trace and trace-free parts,
+which are denoted by XT, XTF, YT, YTF, respectively. For the charged static
+cylindrical system, the scalar corresponding to the tensor Zψχ disappears.
+The obtained scalars are represented as
+XT
+=
+8π
+�
+̺ +
+q2
+8πr4
+�
++ OGT,
+(39)
+XTF
+=
+−4πΠ − q2
+r4 + Π
+2 fT − E,
+(40)
+YT
+=
+q2
+r4 + (̺ + p) fT
+2
++ 4π (̺ + 3pr − 2Π) + FGT ,
+(41)
+YTF
+=
+E − q2
+r4 + Π
+2 fT − 4πΠ + IGT,
+(42)
+12
+
+where IGT =
+QGT
+ψχ
+φψφχ− 1
+3 hψχ and the terms OGT, FGT and QGT are given in Ap-
+pendix A. The scalar XT is associated with the contribution of energy den-
+sity, whereas YT controls the anisotropic stresses in the presence of correction
+terms and total charge of the system. The scalars XTF and YTF are rewritten
+by making use of Eq.(21) as
+XTF
+=
+4π
+r3
+� r
+0
+r3(̺eff −
+q2
+8πr4)
+′dr − 3q2
+2r4 + Π
+2 fT − 4πΠGT,
+(43)
+YTF
+=
+8πΠ + Π
+2 fT − 4π
+r3
+� r
+0
+r3�
+̺eff −
+q2
+8πr4
+�′
+dr + IGT + 4πΠGT. (44)
+It should be noted that XTF gives the effects of inhomogeneity caused by the
+energy density, charge and modified terms. On the other hand, the scalar
+YTF includes the pressure anisotropy, inhomogeneous energy density, charge
+and correction terms. The complexity factor for a charged static cylindrical
+configuration is based on the assumption that the factor must incorporate all
+basic physical variables, electromagnetic effects and extra curvature terms.
+As a result, YTF is chosen as the complexity factor. Using Eq.(44) in (26) to
+discuss physical meaning of YTF
+mT = (mT)Σ
+� r
+rΣ
+�3
++r3
+� rΣ
+r
+KL
+r [YTF − 3q2
+2r4 − Π
+2 fT −IGT +4πΠGT]dr. (45)
+When we compare Eqs.(26) and (45), it can be observed that Tolman mass
+is affected by anisotropic pressure, charge terms and inhomogeneous energy
+density due to YTF.
+4
+Zero Complexity Factor Condition
+The behavior of celestial objects is governed by a vast array of interconnected
+physical factors.
+The least complicated structures are those that possess
+isotropic properties, like a cylinder occupying only dust. However, the fac-
+tors responsible for generating complexity in the current setup are pressure
+anisotropy, charge terms, inhomogeneity of energy density and the presence
+of modified terms. The structure scalar YTF is taken as the complexity fac-
+tor on the basis of the aforementioned parameters. It also describes how
+these factors affect the Tolman mass. Substituting YTF = 0, we set the zero
+13
+
+complexity factor constraint as
+Π =
+1
+4π + fT
+2
+�4π
+r3
+� r
+0
+r3�
+̺eff −
+q2
+8πr4
+�′
+dr + q2
+2r4 − IGT − 4πΠGT
+�
+.
+(46)
+In order to construct solution to the modified field equations, this condition
+will be utilized as an extra limitation.
+4.1
+Gokhroo-Mehra Model
+Gokhroo and Mehra [41] addressed a particular form of the energy density to
+evaluate the solution of the modified field equations as well as the behavior of
+anisotropic stellar system. They designed a system that describes both the
+dynamics of neutron stars as well as larger redshifts of numerous quasi-stellar
+configurations. For the considered setup, the variable energy density is
+̺eff = ̺o
+�
+1 − kr2
+r2
+Σ
+�
+.
+(47)
+After making use of this expression in Eq.(16), the mass function will become
+m = q2
+2r + r
+8 − ζr3
+� kr2
+5r2
+Σ
+− 1
+3
+�
++ 4π
+�
+q2
+8πr2,
+(48)
+where k is a constant in the range (0, 1) and ζ = 4π̺o. When we compare
+Eqs.(16) and (48), the expression for the metric function becomes
+L =
+1
+�
+2ζr2
+�
+−1
+3 + kr2
+5r2
+Σ
+�
++ 1
+r
+� q2
+r2
+.
+(49)
+The field equations can also be composed in the following manner
+8π(peff
+r −
+q2
+8πr4 − peff
+⊥ −
+q2
+8πr4) = 1
+L2
+� K′
+rK + 1
+r2 − K′′
+K + K′L′
+KL + L′
+rL
+�
+.
+(50)
+Here, we introduce two new variables (y and z) [21] as
+L−2 = y(r),
+K2 = e
+� (2z(r)−2/r)dr,
+(51)
+14
+
+and Eq.(50) in terms of y and z is given as
+y′ + y
+�
+−6
+r + 4
+zr2 + 2z + 2z′
+z
+�
+= 16π(Πeff −
+q2
+4πr4)
+z
+.
+(52)
+Its solution provides the metric functions (K2 and L2), thus, the respective
+line element turns out to be
+ds2
+=
+−e
+� (2z−2/r)drdt2 +
+z2e
+� (2z(r)−
+4
+r2z(r))dr
+r6
+�
+16π
+� z(Πeff−
+q2
+4πr4 )e
+�
+(2z(r)−
+4
+r2z(r) )dr
+r6
+dr + H
+�dr2
++
+r2(dθ2 + α2dz2),
+(53)
+where the integration constant is denoted by H.
+The field equations are
+rewritten in the light of new variables as
+4π(peff
+r −
+q2
+8πr4) =
+�1
+4 + q2
+r2 − 2m
+r
+� �z
+r − 1
+2r2
+�
+,
+(54)
+4π(̺eff +
+q2
+8πr4) = m′
+r2 − qq
+′
+r3 + q2
+2r4 − 1
+8r2,
+(55)
+8π(peff
+⊥ +
+q2
+8πr4) =
+�1
+4 + q2
+r2 − 2m
+r
+��
+z′ + z2 − z
+�
+4rm′ + 4q2
+r − 4qq
+′ − 4m
+r2 − 8mr + 4q2
+�
+− z
+r + 1
+r2
+�
+.
+(56)
+4.2
+Polytrope with Zero Complexity Constraint
+The interior regime of self-gravitating systems is determined by a variety of
+physical factors (density, temperature, pressure etc.), each of which plays a
+specific function. However, the contribution of some variables over others
+is more important in analyzing the structure. In this regard, it is benefi-
+cial to possess an equation of state that efficiently investigates the behav-
+ior of anisotropic self-gravitating objects. Arias et al. [42] discussed three
+anisotropic models using zero complexity condition (as a constraint) to close
+the system. They also illustrated the graphical analysis of polytropic system.
+Khan et al. [43] studied the generalized polytropes for charged spherical and
+15
+
+cylindrical geometries and solved them through the complexity factor condi-
+tion. The anisotropic polytropic equation of state (relation between energy
+density and pressure) is given as [44]
+peff
+r = ̺eff(ω) = B̺eff(1+1/n),
+(57)
+where B stands for the polytropic constant, ω denotes the polytropic ex-
+ponent and n represents the polytropic index. Here, we utilize the linear
+f(G, T) gravity model [45] and known expression for charge [46] to describe
+the interior configuration, respectively, as
+f(G, T) = G + εT,
+q = ζr3.
+(58)
+Introducing the dimensionless quantities, we have
+σ
+=
+peff
+rc
+̺eff
+c
+,
+r = ξ
+J ,
+J2 =
+4π̺eff
+c
+σ(1 + n),
+Ψn
+=
+̺eff
+̺eff
+c
+,
+ϑ(ξ) = m(r)J3/4π̺eff
+c .
+Substituting these variables in Eqs.(12) and (16), we obtain the Tolman-
+Oppenheimer-Volkoff equation and mass function as
+ξ2dΨ
+dξ
+�
+1 + 4q2J2
+ξ2
+− 8ϑσ(n + 1)/ξ
+1 + σΨ
+�
+− 4
+�
+ξ3(−σΨn+1 −
+q2J4
+̺eff8πξ4) − ϑ
+�
+−
+ξ
+2(n + 1)σ −
+4q2J2
+2σ(n + 1)ξ +
+2(Π − q2J4
+4πξ4 )ξΨ−n
+peff
+rc(n + 1)
+�
+1 + 4q2J2
+ξ2
+− 8ϑσ(n + 1)/ξ
+1 + σΨ
+�
+=
+ξ2
+�
+1 + 4q2J2
+ξ2
+− 8ϑσ(n + 1)/ξ
+Ψnpeff
+rc(1 + σΨ)(n + 1)J
+� �
+ZL2 +
+�q2J4
+8πξ4
+�′�
+,
+(59)
+dϑ
+dξ =
+1
+2σ(n + 1)
+�q2J2
+ξ2
++ J2
+4π +
+�q2J
+ξ
+�′�
++ Ψnξ2.
+(60)
+where the evaluation of these quantities at the center is represented by sub-
+script c. These ordinary differential equations are composed of three un-
+knowns, i.e., Ψ, ϑ and Π that further depend on σ and n. It is noticed that
+the number of unknowns are greater than the number of equations, therefore,
+16
+
+0.0
+0.5
+1.0
+1.5
+2.0
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Ξ
+Ψ�Ξ�
+0.0
+0.5
+1.0
+1.5
+2.0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Ξ
+Ν�Ξ�
+0.0
+0.5
+1.0
+1.5
+2.0
+�2.0
+�1.5
+�1.0
+�0.5
+0.0
+Ξ
+�
+Figure 1: Behavior of density, mass function and anisotropy versus ξ
+we require one more equation to obtain a unique solution. For this reason,
+the complexity free condition can be used as a constraint and takes the form
+3Π
+ξ + dΠ
+dξ
+=
+1
+(4π + ε
+2)
+�
+4π̺eff
+c nΨn−1dΨ
+dξ + qq
+′J3
+4πξ4 − 3
+ξ
+�
+4πΠGT + IGT
+�
+−
+q2J4
+ξ5
+− 4πdΠeff
+dξ
++ qq
+′J3
+ξ4
+− q2J4
+2πξ5 − IGT
+,ξ
+�
+.
+(61)
+Thus a system of three equations (59)-(61) is obtained that will represent
+a unique solution. Using this solution for certain values of σ and n, it is
+possible to study physical characteristics of quantities, such as density and
+mass. Thus, to illustrate the graphical analysis of dimensionless quantities
+(mass, density and anisotropy), we select the values of unknowns as σ = 0.1,
+ε = 0.5, ζ = 0.001, n = 0.2 and ̺eff
+c
+= 2. Figure 1 depicts the behavior of
+density, mass function and anisotropy, respectively. It can be observed that
+density is maximum at the center while goes on decreasing with increasing
+ξ. In the 2nd plot, the mass function exhibits the increasing trend for higher
+values of ξ. On the other hand, the last plot indicates that the anisotropy of
+the system is zero closer to the center and follows a decreasing behavior as ξ
+increases.
+17
+
+5
+Concluding Remarks
+In this paper, we analyze the influence of electromagnetic field to gauge
+the complexity factor of static cylindrical symmetric spacetime within the
+formalism of f(G, T) theory. We have established the charged modified field
+equations together with hydrostatic equilibrium condition corresponding to
+anisotropic matter source. We have calculated mass function of the system
+using C-energy formalism and Tolman mass formula. We have also discussed
+relationship between the mass functions with physical variables as well as the
+Weyl tensor. A system that has isotropic pressure and homogenous energy
+density is thought to be less complicated. In other words, when both terms
+cancel the influence of each other, the system is said to be complexity free.
+The splitting of the Riemann tensor yields four structure scalars XT,XTF,
+YT,YTF) and we have followed Herrera technique to determine the complexity
+factor. The reasons to recognize the scalar function YTF as complexity factor
+are given below.
+1. All physical parameters such as energy density inhomogeneity and pres-
+sure anisotropy along with extra curvature terms are incorporated in
+this factor.
+2. This scalar depicts the influence of pressure anisotropy and inhomoge-
+neous source on the Tolman mass.
+3. It also encompasses the electromagnetic effects for the charged cylin-
+drical distribution.
+In the current setup, the existence of homogeneous energy density and
+isotropic pressure does not correspond to a complex free structure as a result
+of correction terms (opposite to GR). Consequently, the additional curvature
+terms cause to increase the complexity of the cylindrical configuration. It
+is worthy to mention here that the absence of correction terms and charges
+along with homogeneous energy density and isotropic pressure leads to com-
+plexity free system. To achieve a complex free system, we have substituted
+YTF = 0 which gives rise to vanishing complexity condition that contains
+additional terms. We have used this constraint to find solution for the two
+cases.
+In the first case, Gokhroo-Mehra model is used to examine physical as-
+pects of compact entities. It is found that if specific values are assigned to
+18
+
+m, z this model will correspond to distinct solution. For the second case, we
+have considered the polytropic equation of state and have introduced some
+dimensionless quantities. This leads to form a system of three dimensionless
+equations (mass, Tolman-Oppenheimer-Volkoff equation and zero complexity
+condition). For this model, we have investigated the graphical behavior of
+the system through numerical approach by assigning some particular values
+to the involved unknowns. It is seen that the system represents a decline in
+energy density. whereas mass enlarges for larger ξ. However, the system has
+zero anisotropy at the center which becomes negative as ξ increases.
+Appendix A
+The modified term Z is expressed as
+Z
+=
+fT
+k2 − fT
+�
+(−pr − p) f ′
+T
+fT
+− qq
+′
+4πr4 +
+�
+−2prL2 − pL2�′ − L2
+2 (−̺ + 3p)
+′�
+.
+The following describes the impact of additional terms in the scalar structure
+NGT
+ψχ
+=
+�
+2RmδRmγfG − RRγ
+δfG + 2RlmRγ
+lδmfG − 2RδlmnRlmnγfG
++
+2Rγ
+δ□fG − 2Rmγ∇δ∇mfG + R∇γ∇δfG − 2Rm
+δ ∇γ∇mfG
+−
+2Rγ
+lδm∇m∇lfG
+�
+ǫψχγuδ,
+DGT
+ψχ
+=
+2
+�
+RmχRm
+ψ fG − 1
+2RRψχfG + RlmRlχmψfG − 1
+2RχlmnRlmn
+ψ
+fG + Rψχ□fG
++
+1
+2R∇ψ∇χfG − Rm
+ψ ∇χ∇mfG − Rm
+χ ∇ψ∇mfG − Rlχmψ∇m∇lfG
+�
++
+4Rlmhψχ∇m∇lfG − 2Rhψχ□fG + 2
+�
+−RmδRm
+ψ fG − RlmRlδmψfG
++
+1
+2RδlmnRlmn
+ψ
+fG + 1
+2RRψχfG − Rδψ□fG + Rlδmψ∇m∇mfG
+−
+1
+2R∇ψ∇δfG + Rm
+ψ ∇δ∇mfG + Rm
+δ ∇ψ∇mfG
+�
+uχuδ + 2 [−RmχRmγfG
++
+1
+2RχlmnRlmnγfG − RlmRγ
+lχmfG + 1
+2RRγ
+χfG − Rγ
+χ□fG + Rmγ∇χ∇mfG
+−
+1
+2R∇γ∇χfG + Rm
+χ ∇γ∇mfG + Rγ
+mχl∇m∇l
+�
+uψuγ + 2 [RmδRmγfG
+−
+1
+2RδlmnRlmnγfG + RlmRγ
+lδmfG − 1
+2RRγ
+δfG + Rγ
+δ□fG + 1
+2R∇γ∇δfG
+19
+
+−
+Rγ
+lδm∇m∇lfG − Rmγ∇δ∇mfG − Rm
+δ ∇γ∇mfG
+�
+gψχuγuδ − 1
+3
+�
+−2R2fG
+−
+2Rβ
+lmnRlmn
+β
+fG − 2R□fG + 4RmαRmαfG + 4RlmRα
+mαlfG + 16Rlm∇m∇lfG
+−
+4Rmβ∇β∇mfG − 4Rmα∇α∇mfG − 4Rα
+lαm∇m∇lfG
+�
+hψχ − 1
+6fhψχ,
+FGT
+=
+2
+�
+RmχRm
+ψ fG + RlmRmχlψfG − 1
+2RRψχfG − 1
+2RχχlmnRlmn
+ψ
+fG + Rψχ□fG
++
+1
+2R∇ψ∇χfG − Rm
+ψ ∇χ∇mfG − Rm
+χ ∇ψ∇mfG − Rmχlψ∇m∇lfG
+�
+gψχ
++
+12Rlm∇m∇lfG − 6R□fG + 2
+�
+−RmδRm
+ψ fG − RlmRmδlψfG + 1
+2RRψδfG
+−
+1
+2R∇ψ∇δfG − Rδψ□fG + Rmδlψ∇m∇lfG + Rm
+ψ ∇δ∇mfG + Rm
+δ ∇ψ∇mfG
++
+1
+2RδlmnRlmn
+ψ
+fG
+�
+uχuδgψχ + 2
+�
+−RmχRmγfG − RlmRγ
+mχlfG + 1
+2RRγ
+χfG
++
+1
+2RχlmnRlmnγfG − Rγ
+χ□fG + Rmγ∇χ∇mfG + Rm
+χ ∇γ∇mfG + Rγ
+mχl∇m∇lfG
+−
+1
+2R∇γ∇χfG
+�
+uψuγgψχ + 2
+�
+RmδRmγfG + RlmRγ
+mδlfG − Rmγ∇δ∇mfG
+−
+1
+2RδlmnRlmnγfG + Rγ
+δ□fG + 1
+2R∇γ∇δfG − 1
+2RRγ
+δfG − Rm
+δ ∇γ∇mfG
+−
+Rγ
+mδl∇m∇lfG
+�
+gψχuγuδgψχ −
+�
+4RlαRlαfG + 4RlmRα
+lαmfG − 2R2fG
+−
+2Rβ
+lmnRlmn
+β
+fG − 2R□fG + 16Rlm∇m∇lfG − 4Rmα∇α∇mfG
+−
+4Rmβ∇β∇mfG − 4Rα
+lαm∇m∇lfG
+�
+− 1
+2f,
+QGT
+(ψχ)
+=
+�
+2RmdRm
+c fG + 2RlmRldmcfG − RRcdfG − RdlmnRlmn
+c
+fG + 2Rcd□fG
++
+R∇c∇dfG − 2Rm
+c ∇d∇mfG − 2Rm
+d ∇c∇mfG − 2Rldmc∇m∇lfG
+�
+hc
+ψhd
+χ
++
+2
+�
+RmδRmγfG + RlmRγ
+mδlfG − 1
+2RRγ
+δfG − 1
+2RδlmnRlmnγfG
++
+Rγ
+δ□fG + 1
+2R∇γ∇δfG − Rmγ∇δ∇mfG − Rm
+δ ∇γ∇mfG
+−
+Rγ
+mδl∇m∇lfG
+�
+hψχuγuδ − 2
+�
+RmχRm
+ψ fG + RlmRmχlψfG − 1
+2RRψχfG
+−
+1
+2RχlmnRlmn
+ψ
+fG + Rψχ□fG + 1
+2R∇ψ∇χfG − Rm
+ψ ∇χ∇mfG
+20
+
+−
+Rm
+χ ∇ψ∇mfG − Rmχlψ∇m∇lfG
+�
+− 2
+�
+−RmδRm
+ψ fG − RlmRmδlψfG
++
+1
+2RRψδfG + 1
+2RδlmnRlmn
+ψ
+fG − Rδψ□fG + Rmδlψ∇m∇lfG
++
+Rm
+ψ ∇δ∇mfG + Rm
+δ ∇ψ∇mfG − 1
+2R∇ψ∇δfG
+�
+uχuδ − 2 [−RmχRmγfG
+−
+RlmRγ
+mχlfG + 1
+2RRγ
+χfG + 1
+2RχlmnRlmnγfG − Rγ
+χ□fG
++
+Rmγ∇χ∇mfG + Rm
+χ ∇γ∇mfG + Rγ
+mχl∇m∇lfG − 1
+2R∇γ∇χfG
+�
+uψuγ
+−
+2
+�
+RmδRmγfG + RlmRγ
+mδlfG − 1
+2RRγ
+δfG − 1
+2RδlmnRlmnγfG
++
+Rγ
+δ□fG + 1
+2R∇γ∇δfG − Rmγ∇δ∇mfG − Rm
+δ ∇γ∇mfG
+−
+Rγ
+mδl∇m∇lfG
+�
+uγuδgψχ,
+MGT
+ψχ
+=
+�1
+2RmǫRmpfG + 1
+2RlmRp
+mǫlfG − 1
+4RRp
+ǫfG − 1
+4RǫlmnRlmnpfG
++
+1
+2Rp
+ǫ□fG + 1
+4R∇p∇ǫfG − 1
+4Rlp∇ǫ∇mfG − 1
+2Rm
+ǫ ∇p∇mfG
+−
+1
+2Rp
+mǫl∇m∇lfG
+�
+ǫpδχǫǫδ
+ψ +
+�
+−1
+2RmδRlpfG − 1
+2RlmRp
+mδlfG
++
+1
+4RRp
+δfG + 1
+4RδlmnRlmnpfG − 1
+2Rp
+δ□fG − 1
+4R∇p∇δfG
++
+1
+4Rlp∇δ∇mfG + 1
+2Rm
+δ ∇p∇mfG + 1
+2Rp
+mδl∇m∇lfG
+�
+ǫpǫχǫǫδ
+ψ
++
+�
+−1
+2RmǫRmγfG − 1
+2RlmRγ
+mǫlfG + 1
+4RRγ
+ǫ fG + 1
+4RǫlmnRlmnγfG
+−
+1
+2Rγ
+ǫ □fG − 1
+4R∇γ∇ǫfG + 1
+4Rmγ∇ǫ∇mfG + 1
+2Rm
+ǫ ∇γ∇mfG
++
+1
+2Rγ
+mǫl∇m∇lfG
+�
+ǫδγχǫǫδ
+ψ +
+�1
+2RmδRmγfG + 1
+2RlmRγ
+mδlfG
+−
+1
+4RRγ
+δfG − 1
+4RδlmnRlmnγfG + 1
+2Rγ
+δ□fG + 1
+4R∇γ∇δfG
+−
+1
+4Rmγ∇δ∇mfG − 1
+2Rm
+δ ∇γ∇mfG − 1
+2Rγ
+mδl∇m∇lfG
+�
+ǫǫγχǫǫδ
+ψ
+21
+
+−
+4Rlm∇m∇lhψχfG + 2Rhψχ□fG + 1
+3 [(̺ + p) fT + 4RmαRmαfG
++
+4RlmRα
+lαmfG − 2R2fG − 2Rβ
+lmnRlmn
+β
+fG − 2R□fG
++
+16Rlm∇m∇lfG − 4Rmα∇α∇mfG − 4Rmβ∇β∇mfG
+−
+4Rα
+lαm∇m∇lfG
+�
+hψχ + 1
+6fhψχ,
+OGT
+=
+�1
+2RmǫRmpfG + 1
+2RlmRp
+mǫlfG − 1
+4RRp
+ǫfG − 1
+4RǫlmnRlmnpfG
++
+1
+2Rp
+ǫ□fG + 1
+4R∇p∇ǫfG − 1
+4Rlp∇ǫ∇mfG − 1
+2Rm
+ǫ ∇p∇mfG
+−
+1
+2Rp
+mǫl∇m∇lfG
+�
+gψχǫpδχǫǫδ
+ψ +
+�
+−1
+2RmδRlpfG − 1
+2RlmRp
+mδlfG
++
+1
+4RRp
+δfG + 1
+4RδlmnRlmnpfG − 1
+2Rp
+δ□fG − 1
+4R∇p∇δfG
++
+1
+4Rlp∇δ∇mfG + 1
+2Rm
+δ ∇p∇mfG + 1
+2Rp
+mδl∇m∇lfG
+�
+gψχǫpǫχǫǫδ
+ψ
++
+�
+−1
+2RmǫRmγfG − 1
+2RlmRγ
+mǫlfG + 1
+4RRγ
+ǫ fG + 1
+4RǫlmnRlmnγfG
+−
+1
+2Rγ
+ǫ □fG − 1
+4R∇γ∇ǫfG + 1
+4Rmγ∇ǫ∇mfG + 1
+2Rm
+ǫ ∇γ∇mfG
++
+1
+2Rγ
+mǫl∇m∇lfG
+�
+gψχǫδγχǫǫδ
+ψ +
+�1
+2RmδRmγfG + 1
+2RlmRγ
+mδlfG
+−
+1
+4RRγ
+δfG − 1
+4RδlmnRlmnγfG + 1
+2Rγ
+δ□fG + 1
+4R∇γ∇δfG
+−
+1
+4Rmγ∇δ∇mfG − 1
+2Rm
+δ ∇γ∇mfG − 1
+2Rγ
+mδl∇m∇lfG
+�
+gψχǫǫγχǫǫδ
+ψ
+−
+12Rlm∇m∇lhψχfG + 6R□fG +
+�
+(̺ + p) fT + 4RlψRlψfG
++
+4RlmRψ
+lψmfG − 2Rχ
+lmnRlmn
+χ
+fG − 2R2fG − 2R□fG
++
+16Rlm∇l∇mfG − 4Rlχ∇χ∇lfG − 4Rlψ∇ψ∇lfG
+−
+4Rψ
+lψm∇l∇mfG
+�
++ 1
+2f.
+Data Availability Statement: This manuscript has no associated data.
+22
+
+References
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+Eur. Phys. J. C
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+77(2022)2655.
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+24
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+Fuenmayor, E. and Leon, P.: Phys. Rev. D 93(2016)024047.
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+277(1995)L17.
+25
+
diff --git a/pdE2T4oBgHgl3EQf0Qhs/content/tmp_files/load_file.txt b/pdE2T4oBgHgl3EQf0Qhs/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..eaa5a732f4d9999340d4b4bf124e88eaf8abbe96
--- /dev/null
+++ b/pdE2T4oBgHgl3EQf0Qhs/content/tmp_files/load_file.txt
@@ -0,0 +1,1282 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf,len=1281
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='04139v1  [gr-qc]  10 Jan 2023  Electromagnetic Effects on the  Complexity of Static Cylindrical  Object in f(G, T) Gravity  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Sharif1 ∗and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Hassan2 †  1 Department of Mathematics and Statistics, The University of Lahore,  1-KM Defence Road Lahore, Pakistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  2 Department of Mathematics, University of the Punjab,  Quaid-e-Azam Campus, Lahore-54590, Pakistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Abstract  In this paper, we investigate complexity of anisotropic cylindrical  object under the influence of electromagnetic field in f(G, T) the-  ory, where G and T indicate the Gauss-Bonnet term and trace of  the stress-energy tensor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' For this purpose, we calculate  the modified field equations, non-conservation equation and mass dis-  tributions that assist in comprehending the structure of astrophysical  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The Riemann tensor is divided into different structure scalars,  among which one is called the complexity factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' This factor is used  to measure complexity of the system due to the involvement of in-  homogeneous energy density, anisotropic pressure and charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The  vanishing of the complexity factor is employed as a constraint to for-  mulate charged static solutions for the Gokhroo-Mehra model and  polytropic equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' We conclude that the presence of charge  reduces the complexity of the anisotropic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Keywords:  Anisotropic fluid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Self-gravitating systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' f(G, T) gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Complexity factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  PACS: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='-b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Kd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Dg  ∗msharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='math@pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='pk  †komalhassan3@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='com  1  1  Introduction  General theory of relativity (GR) improved the fundamental ideas about our  universe and the notion of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Einstein assumed that the universe was  neither contracting nor expanding, so he considered it as a static epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Later, Hubble confirmed the expansion of the cosmos by relating the reces-  sion velocity of galaxies and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The inexplicable force known as dark  energy is thought to be responsible for the accelerated expansion of the uni-  verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Supernovae and microwave background radiations are among different  astrophysical events that unveil the fast cosmic expansion [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The ΛCDM  model is the best model to explain the current accelerated expansion of the  universe, however, it has two major challenges: fine-tuning and cosmic co-  incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Thus, some modifications in GR (so called modified theories of  gravity) were introduced to explain the expanding universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  In this regard, Nojiri and Odintsov [2] revised the Einstein-Hilbert action  by adding the function of Gauss-Bonnet (GB) invariant, which resulted in  f(G) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' This also helped in explaining how the cosmos transitioned  from a decelerating to an accelerating epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The GB invariant is a com-  bination of the Riemann tensor (Rψχβα), Ricci tensor (Rψχ) and curvature  scalar (R) as  G = −4RψχRψχ + RψχβαRψχβα + R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Bamba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [3] studied the early-time bounce as well as late-time acceler-  ation by considering the scale factor in the form of an exponential function  by reconstructing f(G) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Abbas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [4] examined viable and sta-  ble aspects of stellar objects by utilizing the Krori-Barua metric together  with a power-law model in f(G) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Sharif and Ikram [5] addressed the  inflationary characteristics of homogeneous and isotropic universe using fea-  sible f(G) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' For a plane symmetric structure, Shamir and Saeed [6]  investigated the exponential and power-law solutions in this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Sharif  and Ramzan [7] applied the embedding class-1 method to examine physical  features of anisotropic self-gravitating objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Sharif and Ikram [8] introduced trace of the energy-momentum tensor in  f(G) action and proposed a new modified theory named as f(G, T) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  They discussed energy conditions using FRW metric in this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The same  authors [9] used the perturbation approach to check the stable behavior of  the Einstein universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' For a certain range of the parameters, Hossienkhani et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [10] demonstrated that weak and null energy conditions show consistency  2  with anisotropic f(G, T) universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Sharif and Naeem [11] developed a new  solution to analyze viability and stability of different stellar configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Shamir [12] investigated the oscillating universe for two models, such as linear  and logarithmic trace terms of f(G, T) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  In the evolution and stable configuration of the celestial entities, electro-  magnetic field plays a crucial role as it suppresses the gravitational force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  A significant quantity of charge is required to counteract gravitational at-  traction and maintain the stabilizing behavior of self-gravitating entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' In  astrophysical realm, the first static charged solution of the Einstein-Maxwell  field equations was presented by Reissner and Nordstrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Bondi [13] im-  posed the known cylindrical wave solution of the free space on the static field  and observed the propagation of energy from such waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Ivanov [14] studied  general and regular solutions of the field equations in three different classes  for the charged cylindrical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Esculpi and Aloma [15] investigated the  influence of anisotropy on charged compact objects using linear equation  of state that relates radial and tangential pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Sharif and Naz [16]  used dynamical field equations to investigate the collapsing phenomenon of a  cylindrically charged star and evaluated the relationship between GB terms,  physical variables and Weyl tensor in f(G) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  We have worked on  Tolman IV and Krori-Barua spacetimes to illustrate physical features of the  compact object in uncharged/charged configurations, respectively, in f(G, T)  theory [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The complexity of self-gravitating objects is greatly influenced by feasible  characteristics of astronomical entities, such as temperature, heat, pressure,  energy density, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Therefore, a mathematical formula that includes all sig-  nificant components must be used to determine complexity of cosmic entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The notion of complexity in terms of entropy and information was introduced  by L´opez-Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Initially, this concept was implemented on two  physical objects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', perfect crystal and ideal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The molecules of perfect  crystal are grouped in an ordered manner (due to the symmetric nature),  thus it has zero entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' In contrast, as the molecules of ideal gas are scat-  tered randomly, so it shows maximum entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The probability distribution  around the symmetric component of a perfect crystal adds little information  because of its symmetry, and a small portion is enough to characterize all of  its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' However, maximum amount of information is available while  studying a tiny region of ideal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Both these structures exhibit different  behavior, yet they are allocated zero complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Another appropriate con-  cept of complexity was established on the basis of how different probabilistic  3  states depart from the equiprobable distribution of the structure [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Using  this concept, both ideal gas and perfect crystal are considered to give zero  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The energy density was employed in the revised methodology  (in place of probability distribution) to evaluate the complexity of celestial  objects [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' However, other state variables like heat, pressure, temperature,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' were not included in this benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Recently, Herrera [21] redefined the notion of complexity in terms of in-  homogeneous energy density, pressure anisotropy and Tolman mass for the  case of static anisotropic fluid source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' In this approach, different structure  scalars are obtained through the orthogonal breakdown of the Riemann ten-  sor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A scalar that encompasses all the above-mentioned parameters is called  complexity factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Sharif and Butt [22] studied how the complexity of spher-  ical configuration was affected in the presence of an electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Herrera and his collaborators [23] extended the concept of complexity for  dynamical dissipative configuration and addressed two modes of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The same authors [24] also considered an axially symmetric structure to  calculate its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Contreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [25] utilized the temporal metric  component of Durgapal IV and V solutions along with vanishing complexity  condition for both charged/uncharged cases to discuss feasibility of the con-  structed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Contreras and Stuchlik [26] formulated a simple protocol  to develop the anisotropic interior solutions via vanishing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Abbas  and Nazar [27] calculated the complexity of various structures in the realm  of non-minimally coupled f(R) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Sharif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [28] applied Herrera’s  approach to the spherically symmetric matter source in f(G) gravity and  found an increment in complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A large body of literature is found on  the application of complexity to different geometries in the background of  various modified theories [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Yousaf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [30] inspected the complexity  producing factor for the charged/uncharged spheres in f(G, T) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  When a strong gravitational field is present, the departure from spherical  to non-spherical geometries becomes crucial for assessing the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Levi  Civita developed vacuum cylindrically symmetric spacetime that inspired as-  tronomers to explore the interesting features of various stellar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' In  order to examine the cylindrical distribution, Herrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [31] discussed  scalar functions in the form of matter variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Using modified Gauss-  Bonnet gravity, Houndjo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [32] constructed a set of seven solutions that  conform to three different viable models in cylindrical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' To deter-  mine the complexity of a cylindrical spacetime, Sharif and Butt [33] applied  Herrera method to decompose the Riemann tensor in the scenario of static  4  matter configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The same authors [34] examined the influence of elec-  tromagnetic field on the cylindrical geometry using the same procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' We  have also investigated the complexity factor in static cylindrical and dynam-  ical spherical/cylindrical structures in f(G, T) gravity [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  This paper studies the effects of charge on the complexity of anisotropic  cylindrical matter distribution in the formalism of f(G, T) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Through-  out the paper, we have utilized the geometrized units, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', G = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The  paper is characterized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The modified field equations and physical  characteristics related to anisotropic matter distribution are derived in sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The orthogonal splitting of the Riemann tensor is given in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  In section 4, the vanishing complexity condition is employed as a constraint  to compute the possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The key results are wrapped in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  2  f(G, T) Gravity  In this section, we explore the modified field equations with the inclusion  of electromagnetic field and examine physical attributes of the anisotropic  celestial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The Einstein-Maxwell action is given as follows  If(G,T) =  1  2κ2  � √−g[f(G, T) + R]d4x +  � √−g(Lm + LEM)d4x,  (1)  where g denotes determinant of the metric tensor and κ2 represents the cou-  pling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Moreover, the terms Lm and LEM indicate the Lagrangian  density corresponding to the usual matter and electromagnetic field, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The energy-momentum tensor and matter Lagrangian density are  associated through the relation  Tψχ = gψχLm − 2∂Lm  ∂gψχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (2)  The f(G, T) field equations are obtained by varying the action (1) with  respect to the metric tensor  Gψ  χ  =  8π(T ψ  χ + Sψ  χ) − (Θψ  χ + T ψ  χ )fT(G, T) + 1  2δψ  χf(G, T) +  �  4RχmRψm  −  2RRψ  χ + 4RmlRψ  mχl − 2RmlnψRχmln  �  fG(G, T) +  �  4Rψ  χ∇2  −  4Rψ  mχl∇m∇l − 2δψ  χR∇2 + 2R∇ψ∇χ − 4Rψm∇m∇χ − 4Rm  χ ∇ψ∇m  5  +  4δψ  χRml∇m∇l  �  fG(G, T),  (3)  where the Einstein tensor is expressed as Gψ  χ = Rψ  χ − 1  2Rδψ  χ and ∇υ∇υ = ∇2  denotes the d’ Alembert operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Moreover, the partial derivatives of an ar-  bitrary f(G, T) function such as fG(G, T) and fT(G, T) represent derivatives  taken with respect to G and T, respectively, and Θψχ = −2Tψχ + pgψχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The cylindrical geometric distribution surrounded by the hypersurface  (Σ) is delineated by the line element [33]  ds2 = −K2(r)dt2 + L2(r)dr2 + r2dθ2 + r2α2dz2,  (4)  where the metric coefficients K and L are unknown potential functions while  α (constant term) has the dimension of inverse length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The stress-energy  tensor corresponding to anisotropic matter distribution reads  T ψM  χ  = ̺uψuχ + phψ  χ + Πψ  χ,  (5)  along with  Πψ  χ  =  Π  �  φψφχ − 1  3hψ  χ  �  ,  Π = pr − p⊥,  p = pr + 2p⊥  3  ,  uψuψ  =  −1,  φψuψ = 0,  φψφψ = 1  hψ  χ = δψ  χ + uψuχ,  where p⊥ and pr are the tangential and radial pressures, respectively, and Π  is the anisotropic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The components of four velocity and radial four-  vector are denoted as uψ =  � 1  K, 0, 0, 0  �  , φψ =  �  0, 1  L, 0, 0  �  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The  extra curvature terms of f(G, T) gravity are given as  T ψGT  χ  =  1  8π  �  {(̺ + p)uψuχ + Πψ  χ}fT(G, T) + 1  2δψ  χf(G, T) +  �  4RχmRψm  +  4RmlRψ  mχl − 2RRψ  χ − 2RmlnψRχmln  �  fG(G, T) +  �  4δψ  χRml∇m∇l  +  4Rψ  χ∇2 + 2R∇ψ∇χ − 2δψ  χR∇2 − 4Rψm∇m∇χ − 4Rψ  mχl∇m∇l  −  4Rm  χ ∇ψ∇m  �  fG(G, T)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (6)  In order to investigate charged matter distribution, the electromagnetic en-  ergy tensor is defined as  Sψχ = 1  4π  �  F l  ψFχl − 1  4FlmF lmgψχ  �  ,  (7)  6  where Fψχ = γχ,ψ −γψ,χ stands for the Maxwell field tensor and γψ is the four  potential which becomes γψ = γ(r)δ0  ψ for the static structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The Maxwell  field equations, in the tensorial notation, read  F ψχ  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ = 4πJψ,  F[ψχ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='l] = 0,  where Jψ = ̟υψ indicates the electromagnetic four-current vector and ̟  represents the charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  In this framework,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' the modified field equations are analyzed as  8π  �  ̺eff +  q2  8πr4  �  =  � 1  rL2  �2L′  L − 1  r  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (8)  8π  �  peff  r −  q2  8πr4  �  =  � 1  rL2  �2K′  K + 1  r  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (9)  8π  �  peff  ⊥ +  q2  8πr4  �  =  K′  rKL2 + K′′  KL2 − K′L′  KL3 − L′  rL3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (10)  where derivative with respect to r is signified by prime and the correction  terms ̺eff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' peff  r and peff  ⊥ are computed as  ̺eff  =  ̺ + 1  8π  �  (̺ + p)fT − f  2 + 4K′′fG  r2KL4 + 12L′f ′  G  r2L5  − 12K′L′fG  r2KL5  − 4f ′′  G  r2L4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  peff  r  =  pr + 1  8π  �2  3ΠfT + f  2 + 12K′L′fG  r2KL5  + 12K′f ′  G  r2KL4 − 4K′′fG  r2KL4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  peff  ⊥  =  p⊥ + 1  8π  �  −Π  3 fT + f  2 − 4K′′fG  r2KL4 − 12K′L′f ′  G  rKL5  + 4K′f ′′  G  rKL4 + 12K′L′fG  r2KL5  +  4K′′f ′  G  rKL4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The non-conservation of energy-momentum tensor is observed through the  covariant differentiation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' As a result, an extra force appears that  pushes the particles to disobey the geodesic trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The corresponding  non-conservation equation as well as the hydrostatic equilibrium equation,  respectively, under the effect of electromagnetic field are written as  ∇ψTψχ  =  fT(G, T)  k2 − fT(G, T)  �  (Tψχ + Θψχ)∇ψ(ln fT(G, T)) + ∇ψΘψχ  −  1  2gψχ∇ψT  �  ,  (11)  7  (peff  r  −  q2  8πr4)  ′  =  −K′(peff  r + ̺eff)  K  + 2(peff  ⊥ − peff  r +  q2  4πr4)  r  + ZL2,  (12)  where the curvature terms present in Z are displayed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The  Tolman-Oppenheimer-Volkoff equation can also be represented by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (12)  that helps to describe the formation of anisotropic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Equation (9)  provides the value of K′  K as  K′  K =  4r2  r2 − 8mr + 4q2  �  4πr(peff  r −  q2  8πr4) − 1  8r − q2  2r3 + m  r2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (13)  Utilizing the above expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (11), we have  �  peff  r  −  q2  8πr4  �′  = −  4r2  r2 − 8mr + 4q2  �  4πr(peff  r −  q2  8πr4) − 1  8r − q2  2r3 + m  r2  �  ×  �  ̺eff + peff  r  �  + 2(peff  ⊥ − peff  r +  q2  4πr4)  r  + ZL2,  (14)  where m is the mass of the fluid distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The mass function of the inte-  rior geometry can be evaluated through C-energy and Tolman mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Firstly,  the C-energy formalism [36] is utilized to compute the mass as  m(r) ∼= �E = lE =  �1  4 − 1  L2  � r  2 + q2  2r,  (15)  Making use of the field Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (8), it can be rewritten as  m =  � r  0  4πr2  �  ̺eff +  q2  8πr4  �  dr + q2  2r + r  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (16)  The Weyl tensor calculates the fluctuation in the celestial system caused  by the gravitational fields in nearby objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The Riemann tensor is divided  into the Ricci tensor, curvature scalar and Weyl tensor as  Rσ  ψχω = Cσ  ψχω+ 1  2Rψωδσ  χ− 1  2Rψχδσ  ω+ 1  2Rσ  χgψω− 1  2Rσ  ωgψχ− 1  6R  �  δσ  χgψω−gψχδσ  ω  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (17)  The decomposition of the Weyl tensor through the observer’s four velocity  gives magnetic and electric parts as  Hψχ = 1  2ηψλβγCβγ  χρuλuρ,  Eψχ = Cψγχδuγuδ,  8  where  Cψχλκ = (gψχβαgλκγδ − ηψχβαηλκγδ) uβuγEαδ,  (18)  and ηψχβα indicates the Levi-Civita tensor along with gψχβα = gψβgχα −  gψαgχβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The magnetic part of the Weyl tensor vanishes in symmetric cylin-  drical spacetime, while the electric component becomes  Eψχ = E  �  φψφχ − hψχ  3  �  ,  where E and some of its properties are  E =  1  2KL2  �  K′′ − K′L′  L  + KL′  rL − K′  r + K  r2  �  ,  Eψ  ψ = 0 = Eψγuγ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (19)  The connection between the mass of the cylindrical object with matter  variables, charge, and Weyl tensor is given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (8)-(10), (16) and (19) as  m(r) = 4πr3  3  �  ̺eff + peff  ⊥ − peff  r + 3q2  8πr4  �  + q2  2r − Er3  3  + r  8,  (20)  and the value of E from the above equation leads to  E = 4π  r3  � r  0  r3  �  ̺eff +  q2  8πr4  �′  dr − 4π  �  peff  r − peff  ⊥ −  q2  4πr4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (21)  This shows the relation among the energy density inhomogeneity, pressure  anisotropy, electromagnetic effect and the Weyl tensor in the presence of  f(G, T) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' After inserting this value of E in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (20), we have  m(r) = q2  2r + 4πr3  3  �  ̺eff +  q2  8πr4  �  − 4π  3  � r  0  r3  �  ̺eff +  q2  8πr4  �′  dr + r  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (22)  The effects of inhomogeneous energy density and charge on the mass can be  noticed from the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  For the static symmetric structure, the total mass of the fluid distribution  is determined by the Tolman formula [37] as  mT = 4π  � r  0  KLr2 �  −T 0eff  0  + T 1eff  1  + 2T 2eff  2  �  dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (23)  9  Using the field equations in the above equation, it follows that  mT = K′  L r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (24)  Inserting the value of K  ′ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (13), it follows that  mT = KL  �  m − q2  2r  �  + 4πKL  �  peff  r −  q2  8πr4  �  r3 − KLr  8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The Tolman mass can also be referred to as the effective gravitational mass  as it interlinks with the gravitational acceleration of test particles by  a = K′  KL = mT  Kr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (25)  We obtain the expression for mT [38] as  mT = (mT )Σ  � r  rΣ  �3  − r3  � rΣ  r  KL  r4  �  8πr3  �  peff  ⊥ +  q2  8πr4 − peff  r +  q2  8πr4  �  +  � r  0  4πr3(̺eff +  q2  8πr4)  ′dr  �  dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (26)  Alternatively, the Tolman mass is acquired in terms of E as  mT =  � r  rΣ  �3  (mT)Σ−r3  � rΣ  r  KL  r4  �  4πr3  �  peff  ⊥ +  q2  8πr4 − peff  r +  q2  8πr4  �  + r3E  �  dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (27)  It is important to notice that inhomogeneous energy density, electromagnetic  field, correction terms, and anisotropic pressure affect the Tolman mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  3  Structure Scalars  Bel [39] was the pioneer to propose the orthogonal splitting of the Riemann  tensor, followed by Herrera [40], who formulated various scalar functions  known as structure scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' These scalar functions are defined on account  of the essential characteristics of fluid distributions such as inhomogeneous  energy density, anisotropic pressure, electromagnetic field and active grav-  itational mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  These structure scalars are crucial in comprehending the  10  complexity factor for the self-gravitating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The tensors Yψχ, Zψχ and  Xψχ are defined as  Yψχ  =  Rψγχδuγuδ,  (28)  Zψχ  =  ∗Rψγχδuγuδ = 1  2ηψγǫβRǫβ  χδuγuδ,  (29)  Xψχ  =  ∗R∗  ψγχδuγuδ = 1  2ηǫβ  ψγ, R∗  ǫβχδuγuδ,  (30)  where R∗  ψχγδ = 1  2ηǫαγδRǫα  ψχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Equation (17) can also be rewritten in terms of  the Riemann tensor as  Rψγ  χδ = Cψγ  χδ + 16πT eff[ψ[χδγ]  δ] + 8πT eff  �1  3δψ  [χδγ  δ] − δ[ψ  [χδγ]  δ]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (31)  The above equation is decomposed into four tensorial quantities after substi-  tuting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (5) and (6) as  Rψγ  (I)χδ  =  16π̺u[ψu[χδγ]  δ] + 2̺u[ψu[χδγ]  δ] + 16πph[ψ  [χδγ]  δ]  +  2pu[ψu[χδγ]  δ] + 8π(̺ + 3p)  �1  3δψ  [χδγ  δ] − δ[ψ  [χδγ]  δ]  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (32)  Rψγ  (II)χδ  =  16πΠ[ψ  [χδγ]  δ] + 2Π[ψ  [χδγ]  δ] + δ[ψ  [χδγ]  δ] f  +  8δ[ψ  [χδγ]  δ] Rml∇m∇lfG − 4Rδ[ψ  [χδγ]  δ] □fG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (33)  Rψγ  (III)χδ  =  4̺χ[ψ  [χEγ]  δ] − ǫψγ  β ǫχδαEβα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='(34)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Rψγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='(IV )χδ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2(RmχRmψδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ − RmδRmψδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ + RmχRmγδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ + RmδRmγδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ)fG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2Rml(Rψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mχlδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ − Rψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mδlδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ − Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mχlδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ + Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mδlδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ)fG − R(Rψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Rψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ δγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ − Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='γ + Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ)fG − 2(Rψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mχlδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ − Rψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mδlδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ − Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mχlδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mδlδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ)∇m∇lfG + 2(Rψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ − Rψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ δγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ − Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='γ + Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ)□fG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='R(δγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ ∇ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ − δγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ∇ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ − δψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ ∇γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ + δψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ∇γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ)fG − 2(Rmψδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ ∇χ∇m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Rmψδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ∇δ∇m − Rmγδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ ∇χ∇m + Rmγδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ∇δ∇m)fG − 2(Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ ∇ψ∇m − Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ δγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ∇ψ∇m − Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ δψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ ∇γ∇m + Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ δψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ∇γ∇m)fG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='(RχlmnRlmnψδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ − RδlmnRlmnψδγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ − RχlmnRlmnγδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ + Rδlmn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Rlmnγδψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ)fG + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='(̺ + p)fT + 2f + 4RlαRmαfG + 4Rlm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Rα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='lαmfG − 2Rβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='lmnRlmn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='fG − 4Rmβ∇m∇βfG − 4Rα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='lαm∇m∇lfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2R□fG + 16Rlm∇m∇lfG − 4Rmα∇α∇lfG − 2R2fG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='q2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='6r4h[ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='[χδγ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ] + q2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2r4u[ψu[χδγ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ] − q2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='r4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='φ[ψφ[χδγ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ] + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='3h[ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='[χδγ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='q2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='12πr4u[ψu[χδγ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ] −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='q2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='8πr4δ[ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='[χδγ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (35)  we have employed the following formulas  ǫψγχ = uβηβψγχ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  ǫψγχuχ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  ǫβγαǫαψχ = δβ  ψhγ  χ − δβ  ψhγ  χ + uψ(uβδγ  χ − δβ  χuγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The following tensors in terms of fluid variables can be written as  Yψχ  =  1  6(̺ + p)hψχfT + q2hψχ  3r4  + 4π  3 (̺ + 3p)hψχ + Eψχ  −  4πΠψχ − q2  r4(φψφχ − 1  3hψχ) + Πψχ  2 fT + DGT  ψχ ,  (36)  Xψχ  =  8π̺hψχ  3  + q2hψχ  3r4  + Πψχ  2 fT − 4πΠψχ − q2  r4(φψφχ − 1  3hψχ)  −  Eψχ + MGT  ψχ ,  (37)  Zψχ  =  NGT  ψχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (38)  The extra curvature terms DGT  ψχ , NGT  ψχ and MGT  ψχ present in the above equations  are displayed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The tensors Xψχ and Yψχ are subdivided into its trace and trace-free parts,  which are denoted by XT, XTF, YT, YTF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' For the charged static  cylindrical system, the scalar corresponding to the tensor Zψχ disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The obtained scalars are represented as  XT  =  8π  �  ̺ +  q2  8πr4  �  + OGT,  (39)  XTF  =  −4πΠ − q2  r4 + Π  2 fT − E,  (40)  YT  =  q2  r4 + (̺ + p) fT  2  + 4π (̺ + 3pr − 2Π) + FGT ,  (41)  YTF  =  E − q2  r4 + Π  2 fT − 4πΠ + IGT,  (42)  12  where IGT =  QGT  ψχ  φψφχ− 1  3 hψχ and the terms OGT, FGT and QGT are given in Ap-  pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The scalar XT is associated with the contribution of energy den-  sity, whereas YT controls the anisotropic stresses in the presence of correction  terms and total charge of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The scalars XTF and YTF are rewritten  by making use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (21) as  XTF  =  4π  r3  � r  0  r3(̺eff −  q2  8πr4)  ′dr − 3q2  2r4 + Π  2 fT − 4πΠGT,  (43)  YTF  =  8πΠ + Π  2 fT − 4π  r3  � r  0  r3�  ̺eff −  q2  8πr4  �′  dr + IGT + 4πΠGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (44)  It should be noted that XTF gives the effects of inhomogeneity caused by the  energy density, charge and modified terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' On the other hand, the scalar  YTF includes the pressure anisotropy, inhomogeneous energy density, charge  and correction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The complexity factor for a charged static cylindrical  configuration is based on the assumption that the factor must incorporate all  basic physical variables, electromagnetic effects and extra curvature terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  As a result, YTF is chosen as the complexity factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (44) in (26) to  discuss physical meaning of YTF  mT = (mT)Σ  � r  rΣ  �3  +r3  � rΣ  r  KL  r [YTF − 3q2  2r4 − Π  2 fT −IGT +4πΠGT]dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (45)  When we compare Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (26) and (45), it can be observed that Tolman mass  is affected by anisotropic pressure, charge terms and inhomogeneous energy  density due to YTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  4  Zero Complexity Factor Condition  The behavior of celestial objects is governed by a vast array of interconnected  physical factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The least complicated structures are those that possess  isotropic properties, like a cylinder occupying only dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' However, the fac-  tors responsible for generating complexity in the current setup are pressure  anisotropy, charge terms, inhomogeneity of energy density and the presence  of modified terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The structure scalar YTF is taken as the complexity fac-  tor on the basis of the aforementioned parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' It also describes how  these factors affect the Tolman mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Substituting YTF = 0, we set the zero  13  complexity factor constraint as  Π =  1  4π + fT  2  �4π  r3  � r  0  r3�  ̺eff −  q2  8πr4  �′  dr + q2  2r4 − IGT − 4πΠGT  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (46)  In order to construct solution to the modified field equations, this condition  will be utilized as an extra limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='1  Gokhroo-Mehra Model  Gokhroo and Mehra [41] addressed a particular form of the energy density to  evaluate the solution of the modified field equations as well as the behavior of  anisotropic stellar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' They designed a system that describes both the  dynamics of neutron stars as well as larger redshifts of numerous quasi-stellar  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' For the considered setup, the variable energy density is  ̺eff = ̺o  �  1 − kr2  r2  Σ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (47)  After making use of this expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (16), the mass function will become  m = q2  2r + r  8 − ζr3  � kr2  5r2  Σ  − 1  3  �  + 4π  �  q2  8πr2,  (48)  where k is a constant in the range (0, 1) and ζ = 4π̺o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' When we compare  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (16) and (48), the expression for the metric function becomes  L =  1  �  2ζr2  �  −1  3 + kr2  5r2  Σ  �  + 1  r  � q2  r2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (49)  The field equations can also be composed in the following manner  8π(peff  r −  q2  8πr4 − peff  ⊥ −  q2  8πr4) = 1  L2  � K′  rK + 1  r2 − K′′  K + K′L′  KL + L′  rL  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (50)  Here, we introduce two new variables (y and z) [21] as  L−2 = y(r),  K2 = e  � (2z(r)−2/r)dr,  (51)  14  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (50) in terms of y and z is given as  y′ + y  �  −6  r + 4  zr2 + 2z + 2z′  z  �  = 16π(Πeff −  q2  4πr4)  z  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (52)  Its solution provides the metric functions (K2 and L2), thus, the respective  line element turns out to be  ds2  =  −e  � (2z−2/r)drdt2 +  z2e  � (2z(r)−  4  r2z(r))dr  r6  �  16π  � z(Πeff−  q2  4πr4 )e  �  (2z(r)−  4  r2z(r) )dr  r6  dr + H  �dr2  +  r2(dθ2 + α2dz2),  (53)  where the integration constant is denoted by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The field equations are  rewritten in the light of new variables as  4π(peff  r −  q2  8πr4) =  �1  4 + q2  r2 − 2m  r  � �z  r − 1  2r2  �  ,  (54)  4π(̺eff +  q2  8πr4) = m′  r2 − qq  ′  r3 + q2  2r4 − 1  8r2,  (55)  8π(peff  ⊥ +  q2  8πr4) =  �1  4 + q2  r2 − 2m  r  ��  z′ + z2 − z  �  4rm′ + 4q2  r − 4qq  ′ − 4m  r2 − 8mr + 4q2  �  − z  r + 1  r2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (56)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2  Polytrope with Zero Complexity Constraint  The interior regime of self-gravitating systems is determined by a variety of  physical factors (density, temperature, pressure etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' ), each of which plays a  specific function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' However, the contribution of some variables over others  is more important in analyzing the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' In this regard, it is benefi-  cial to possess an equation of state that efficiently investigates the behav-  ior of anisotropic self-gravitating objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Arias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [42] discussed three  anisotropic models using zero complexity condition (as a constraint) to close  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' They also illustrated the graphical analysis of polytropic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' [43] studied the generalized polytropes for charged spherical and  15  cylindrical geometries and solved them through the complexity factor condi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The anisotropic polytropic equation of state (relation between energy  density and pressure) is given as [44]  peff  r = ̺eff(ω) = B̺eff(1+1/n),  (57)  where B stands for the polytropic constant, ω denotes the polytropic ex-  ponent and n represents the polytropic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Here, we utilize the linear  f(G, T) gravity model [45] and known expression for charge [46] to describe  the interior configuration, respectively, as  f(G, T) = G + εT,  q = ζr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (58)  Introducing the dimensionless quantities, we have  σ  =  peff  rc  ̺eff  c  ,  r = ξ  J ,  J2 =  4π̺eff  c  σ(1 + n),  Ψn  =  ̺eff  ̺eff  c  ,  ϑ(ξ) = m(r)J3/4π̺eff  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Substituting these variables in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' (12) and (16), we obtain the Tolman-  Oppenheimer-Volkoff equation and mass function as  ξ2dΨ  dξ  �  1 + 4q2J2  ξ2  − 8ϑσ(n + 1)/ξ  1 + σΨ  �  − 4  �  ξ3(−σΨn+1 −  q2J4  ̺eff8πξ4) − ϑ  �  −  ξ  2(n + 1)σ −  4q2J2  2σ(n + 1)ξ +  2(Π − q2J4  4πξ4 )ξΨ−n  peff  rc(n + 1)  �  1 + 4q2J2  ξ2  − 8ϑσ(n + 1)/ξ  1 + σΨ  �  =  ξ2  �  1 + 4q2J2  ξ2  − 8ϑσ(n + 1)/ξ  Ψnpeff  rc(1 + σΨ)(n + 1)J  � �  ZL2 +  �q2J4  8πξ4  �′�  ,  (59)  dϑ  dξ =  1  2σ(n + 1)  �q2J2  ξ2  + J2  4π +  �q2J  ξ  �′�  + Ψnξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (60)  where the evaluation of these quantities at the center is represented by sub-  script c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' These ordinary differential equations are composed of three un-  knowns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', Ψ, ϑ and Π that further depend on σ and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' It is noticed that  the number of unknowns are greater than the number of equations, therefore,  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  Ξ  Ψ�Ξ�  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='25  Ξ  Ν�Ξ�  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='0  Ξ  �  Figure 1: Behavior of density, mass function and anisotropy versus ξ  we require one more equation to obtain a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' For this reason,  the complexity free condition can be used as a constraint and takes the form  3Π  ξ + dΠ  dξ  =  1  (4π + ε  2)  �  4π̺eff  c nΨn−1dΨ  dξ + qq  ′J3  4πξ4 − 3  ξ  �  4πΠGT + IGT  �  −  q2J4  ξ5  − 4πdΠeff  dξ  + qq  ′J3  ξ4  − q2J4  2πξ5 − IGT  ,ξ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  (61)  Thus a system of three equations (59)-(61) is obtained that will represent  a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Using this solution for certain values of σ and n, it is  possible to study physical characteristics of quantities, such as density and  mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Thus, to illustrate the graphical analysis of dimensionless quantities  (mass, density and anisotropy), we select the values of unknowns as σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='1,  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='5, ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='001, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2 and ̺eff  c  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Figure 1 depicts the behavior of  density, mass function and anisotropy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' It can be observed that  density is maximum at the center while goes on decreasing with increasing  ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' In the 2nd plot, the mass function exhibits the increasing trend for higher  values of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' On the other hand, the last plot indicates that the anisotropy of  the system is zero closer to the center and follows a decreasing behavior as ξ  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  17  5  Concluding Remarks  In this paper, we analyze the influence of electromagnetic field to gauge  the complexity factor of static cylindrical symmetric spacetime within the  formalism of f(G, T) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' We have established the charged modified field  equations together with hydrostatic equilibrium condition corresponding to  anisotropic matter source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' We have calculated mass function of the system  using C-energy formalism and Tolman mass formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' We have also discussed  relationship between the mass functions with physical variables as well as the  Weyl tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A system that has isotropic pressure and homogenous energy  density is thought to be less complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' In other words, when both terms  cancel the influence of each other, the system is said to be complexity free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The splitting of the Riemann tensor yields four structure scalars XT,XTF,  YT,YTF) and we have followed Herrera technique to determine the complexity  factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' The reasons to recognize the scalar function YTF as complexity factor  are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' All physical parameters such as energy density inhomogeneity and pres-  sure anisotropy along with extra curvature terms are incorporated in  this factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' This scalar depicts the influence of pressure anisotropy and inhomoge-  neous source on the Tolman mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' It also encompasses the electromagnetic effects for the charged cylin-  drical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  In the current setup, the existence of homogeneous energy density and  isotropic pressure does not correspond to a complex free structure as a result  of correction terms (opposite to GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Consequently, the additional curvature  terms cause to increase the complexity of the cylindrical configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' It  is worthy to mention here that the absence of correction terms and charges  along with homogeneous energy density and isotropic pressure leads to com-  plexity free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' To achieve a complex free system, we have substituted  YTF = 0 which gives rise to vanishing complexity condition that contains  additional terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' We have used this constraint to find solution for the two  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  In the first case, Gokhroo-Mehra model is used to examine physical as-  pects of compact entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' It is found that if specific values are assigned to  18  m, z this model will correspond to distinct solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' For the second case, we  have considered the polytropic equation of state and have introduced some  dimensionless quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' This leads to form a system of three dimensionless  equations (mass, Tolman-Oppenheimer-Volkoff equation and zero complexity  condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' For this model, we have investigated the graphical behavior of  the system through numerical approach by assigning some particular values  to the involved unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' It is seen that the system represents a decline in  energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' whereas mass enlarges for larger ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' However, the system has  zero anisotropy at the center which becomes negative as ξ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Appendix A  The modified term Z is expressed as  Z  =  fT  k2 − fT  �  (−pr − p) f ′  T  fT  − qq  ′  4πr4 +  �  −2prL2 − pL2�′ − L2  2 (−̺ + 3p)  ′�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  The following describes the impact of additional terms in the scalar structure  NGT  ψχ  =  �  2RmδRmγfG − RRγ  δfG + 2RlmRγ  lδmfG − 2RδlmnRlmnγfG  +  2Rγ  δ□fG − 2Rmγ∇δ∇mfG + R∇γ∇δfG − 2Rm  δ ∇γ∇mfG  −  2Rγ  lδm∇m∇lfG  �  ǫψχγuδ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='DGT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψχ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='RmχRm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψ fG − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RRψχfG + RlmRlχmψfG − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RχlmnRlmn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='fG + Rψχ□fG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2R∇ψ∇χfG − Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψ ∇χ∇mfG − Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ ∇ψ∇mfG − Rlχmψ∇m∇lfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='4Rlmhψχ∇m∇lfG − 2Rhψχ□fG + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−RmδRm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψ fG − RlmRlδmψfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RδlmnRlmn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='fG + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RRψχfG − Rδψ□fG + Rlδmψ∇m∇mfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2R∇ψ∇δfG + Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψ ∇δ∇mfG + Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ ∇ψ∇mfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='uχuδ + 2 [−RmχRmγfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RχlmnRlmnγfG − RlmRγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='lχmfG + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RRγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χfG − Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ□fG + Rmγ∇χ∇mfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2R∇γ∇χfG + Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ ∇γ∇mfG + Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mχl∇m∇l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='uψuγ + 2 [RmδRmγfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RδlmnRlmnγfG + RlmRγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='lδmfG − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RRγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δfG + Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ□fG + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2R∇γ∇δfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='lδm∇m∇lfG − Rmγ∇δ∇mfG − Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='δ ∇γ∇mfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='gψχuγuδ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−2R2fG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2Rβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='lmnRlmn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='fG − 2R□fG + 4RmαRmαfG + 4RlmRα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mαlfG + 16Rlm∇m∇lfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='4Rmβ∇β∇mfG − 4Rmα∇α∇mfG − 4Rα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='lαm∇m∇lfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='hψχ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='6fhψχ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='FGT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='RmχRm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψ fG + RlmRmχlψfG − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RRψχfG − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RχχlmnRlmn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='fG + Rψχ□fG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2R∇ψ∇χfG − Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψ ∇χ∇mfG − Rm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='χ ∇ψ∇mfG − Rmχlψ∇m∇lfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='2RδlmnRlmn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='fG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='uχuδgψχ + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='−RmχRmγfG − RlmRγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='uψuγgψχ + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='RmδRmγfG + RlmRγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='Rγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='mδl∇m∇lfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='gψχuγuδgψχ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='4RlαRlαfG + 4RlmRα  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='fG − 2R□fG + 16Rlm∇m∇lfG − 4Rmα∇α∇mfG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='QGT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='(ψχ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='2RmdRm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='hc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='ψhd  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='  Data Availability Statement: This manuscript has no associated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  22  References  [1] Bergstrom, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 63(2000)793;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Pietrobon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', Balbi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  and Marinucci, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' D 74(2006)043524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [2] Nojiri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Odintsov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' B 631(2005)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [3] Bamba, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' B 732(2014)349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [4] Abbas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 357(2015)158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [5] Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Ikram, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=': J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='  [8] Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=' Plus 132(2017)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=': Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='  [14] Ivanov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Quantum Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 19(2002)5131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [15] Esculpi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Aloma, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' C 67(2010)521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [16] Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Naz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A 35(2020)1950340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [17] Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Hassan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Plus 137(2022)997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Methods Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 19(2022)2250150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [18] Lopez-Ruiz,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=',  Mancini,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Calbet,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=':  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A  209(1995)321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [19] Calbet, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Lopez-Ruiz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' E 63(2001)066116;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Catalan,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', Garay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and L´opez-Ruiz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' E 66(2002)011102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  23  [20] Sanudo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Lopez-Ruiz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A 372(2008)5283;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Sanudo,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Pacheco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A 373(2009)807;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' De Avellar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A 378(2014)3481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [21] Herrera, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' D 97(2018)044010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [22] Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Butt, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' C 78(2018)688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [23] Herrera, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', Di Prisco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Ospino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' D 98(2018)104059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [24] Herrera, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', Di Prisco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Ospino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' D 99(2019)044049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=', Fuenmayor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Abell´an, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=':  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' C  82(2022)187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=': Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' C 82(2022)706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=': Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=': Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 364(2019)1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  and Majid, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 61(2019)38;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' C 80(2020)1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Indian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 95(2021)769;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Zubair, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Azmat, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Dark  Universe 28(2020)100531;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 420(2020)168267;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  and Naseer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=':  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Plus 137(2022)947;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  77(2022)2655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [30] Yousaf,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=',  Bhatti,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Hassan,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=':  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Plus  135(2020)397;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Yousaf, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : New Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 84(2021)101541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [31] Herrera, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', Di Prisco, A, and Ospino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=':  Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Gravit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  44(2012)2645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [32] Houndjo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 92(2014)1528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [33] Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Butt, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' C 78(2018)850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [34] Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Butt, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 61(2019)238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [35] Sharif, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Hassan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Pramana 96(2022)50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A  37(2022)2250027;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys 77(2022)1479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  24  [36] Thorne, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 138(1965)B251;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 139(1965)B244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [37] Tolman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='C: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 35(1930)875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [38] Herrera, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 237(1998)113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+page_content=': Annales de l’institut Henri Poincar´e 17(1961)37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [40] Herrera, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' D 79(2009)064025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [41] Gokhroo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Mehra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Gravit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 26(1994)75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [42] Arias, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 436(2022)168671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [43] Khan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', Mardan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Rehman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' :  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Plus  136(2021)404;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' C 81(2021)831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [44] Herrera, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Barreto, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' : Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' 88(2013)084022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Herrera, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=',  Fuenmayor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Leon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' D 93(2016)024047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [45] Shamir, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Ahmad, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A 34(2019)1950038;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  Shamir, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Uzair, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=': Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' A 34(2019)1950215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  [46] de Felice, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=', Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' and Fang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=':  Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Aston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  277(1995)L17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
+page_content='  25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE2T4oBgHgl3EQf0Qhs/content/2301.04139v1.pdf'}
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+arXiv:2301.04365v1  [math.NT]  11 Jan 2023
+LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+ABSTRACT. Two infinite sets A and B of non-negative integers are called perfect additive comple-
+ments of non-negative integers, if every non-negative integer can be uniquely expressed as the sum
+of elements from A and B. In this paper, we define a Lagrange-like spectrum of the perfect additive
+complements (LSPAC for short). As a main result, we obtain the smallest accumulation point of the
+set LSPAC and prove that the set LSPAC is closed. Other related results and problems are also
+contained.
+1. Introduction
+Let Z be the set of integers. For nonempty sets A, B of integers and an integer n, let rA,Bpnq
+be the number of representations of n as a ` b, where a P A and b P B. Two infinite sets A
+and B of non-negative integers are called perfect additive complements of non-negative integers,
+if rA,Bpnq “ 1 for every non-negative integer n. For a non-negative integer m, denote by Zěm the
+set of non-negative integers no less than m. For simplicity, we also denote Zě1 by Z`.
+In [5], Fang and S´andor characterized the perfect additive complements A, B of non-negative
+integers.
+Theorem A. [5, Theorem 1.1] The infinite sets A, B of the non-negative integers form perfect
+additive complements if and only if
+A “ tǫ0 ` ǫ2m1m2 ` ¨ ¨ ¨ ` ǫ2k´2m1 . . . m2k´2 ` ¨ ¨ ¨ : ǫ2i “ 0, 1, . . . , m2i`1 ´ 1u and
+B “ tǫ1m1 ` ǫ3m1m2m3 ` ¨ ¨ ¨ ` ǫ2k´1m1 . . . m2k´1 ` ¨ ¨ ¨ : ǫ2i´1 “ 0, 1, . . . , m2i ´ 1u
+(1.1)
+(or A, B interchanged), where mi P Zě2 for every i P Z`.
+Let S be a set of non-negative integers. Its counting function is defined by Spxq “ |S X r0, xs|
+for every x P Zě0. It is easy to see that if A, B Ď Zě0 form perfect additive complements then
+ApxqBpxq ě x ` 1 for every non-negative integer x. In particular, Fang and S´andor showed that
+Date: January 12, 2023.
+2010 Mathematics Subject Classification. Primary 11B34, Secondary 11J06.
+Key words and phrases. additive complements, Lagrange spectrum, Lebesgue-measure.
+* Corresponding author.
+J.H Fang is supported by the National Natural Science Foundation of China, Grant No. 12171246 and the Natural
+Science Foundation of Jiangsu Province, Grant No. BK20211282. B. B´ar´any acknowledges support from the grant
+NKFI FK134251. B. B´ar´any and Cs. S´andor was supported by the grant NKFI KKP144059 ”Fractal geometry and
+applications”.
+1
+
+2
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+lim inf
+xÑ8
+ApxqBpxq
+x
+“ 1, see [5, Theorem 1.5]. Recently, Ma [12] determined the lim sup
+xÑ8
+ApxqBpxq
+x
+for the sets A and B with the form (1.1).
+Theorem B. [12, Lemma 2.1] Let m1, m2, . . . be arbitrary integers no less than two. Then the
+sets A and B with the form (1.1) are perfect additive complements of non-negative integers such
+that
+lim sup
+xÑ8
+ApxqBpxq
+x
+“ lim sup
+kÑ8
+2
+1 ` Dk
+,
+where
+(1.2)
+Dk “ 1
+mk
+´
+1
+mkmk´1
+`
+1
+mkmk´1mk´2
+´ ¨ ¨ ¨ ` p´1qk´1
+1
+mkmk´1 . . . m1
+.
+In this paper, we consider the properties of the set called Lagrange spectrum of perfect additive
+components
+LSPAC :“
+"
+lim sup
+kÑ8
+2
+1 ` Dk
+: pmiq P ZZ`
+ě2
+*
+,
+where Dk is defined in (1.2). In 2011, Chen and Fang [1, Theorem 1] obtained that
+2a ` 2
+a ` 2 P LSPAC for any integer a with a ě 2.
+In 2016, Liu and Fang [10, Theorem 1.1] extended this result by showing that
+2
+a´1
+ab´1 ` 1 P LSPAC for any integers a, b with 2 ď a ď b.
+Recently, Ma [12, Theorem 1.1 and Theorem 1.2] proved that
+2 P LSPAC and
+ˆˆ16
+9 , 2
+˙
+zQ
+˙
+X LSPAC ‰ H,
+where Q denotes the set of rationals. Fang and S´andor [5, Theorem 1.5] showed that
+LSPAC Ď
+„3
+2, 2
+
+.
+The main theorem of this paper can be summarized as follows:
+Theorem 1.1.
+(1) The set LSPAC is closed.
+(2) There exists γ0 P LSPAC, which is the smallest accumulation point of LSPAC. Fur-
+thermore, the set r 3
+2, γ0q X LSPAC “ tγ1, γ2, . . .u, where γn is a monotone increasing
+sequence converging to γ0.
+(3)
+“ 7
+4, 2
+‰
+Ď LSPAC but
+“ 12
+7 ´ δ, 2
+‰
+Ę LSPAC for any δ ą 0.
+
+LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS
+3
+We will determine the constant γ0 and the set r 3
+2, γ0s X LSPAC in Theorem 1.1(2) precisely
+later, see Section 2.
+It follows from Theorem 1.1 that the set LSPAC has some similar properties to the so-called
+Lagrange spectrum LS. Let α be a positive irrational number. Define kpαq “ lim sup
+n,mÑ8
+1
+|n2α ´ nm|.
+Hurvitz [7] proved that kpαq ě
+?
+5 for every positive irrational number α. The Lagrange spectrum
+LS :“ tkpαq : α is a positive irrational numberu.
+For results related to Lagrange spectrum, one may refer to [2], [3], [6], [11], [13] and [14].
+It is well known that the Lagrange spectrum is closed, see [2, Theorem 3.2], furthermore, the
+least accumulation point of the Lagrange spectrum is 3 and l P L, l ă 3 if and only if l “
+b
+9 ´ 4
+z2n,
+where zn’s are the Markov integers, see [11]. The corresponding phenomena for the Lagrange-like
+spectrum of perfect additive complement follows by Theorem 1.1(1) and Theorem 1.1(2).
+Furthermore, Freiman’s constant F “
+2221564096`283748
+?
+463
+491993569
+“ 4.527 . . . is the name of the
+supremum of the set RzLS, that is rF, 8q Ă LS, but for any δ ą 0, rF ´ δ, 8q Ć LS, see [6].
+In point of view of Theorem 1.1(3), the LSPAC has also a Freiman-like constant, namely, there
+exists 12
+7 ď c0 ď 7
+4 such that
+c0 “ inftc P R : rc, 2s Ă LSPACu.
+Problem 1.2. Determine the exact value of c0. Is it true that c0 “ 7{4?
+There is another important similarity between the sets LS and LSPAC, namely, both can be
+represented by using infinite iterated function systems (IFS). It is well known that every α can be
+written as a simple infinite continued fraction
+α “ m0 `
+1
+m1 `
+1
+m2`...
+“: rm0; m1, m2, . . . s,
+where mi P Z`. On the other hand if mi P Z`, then the above continued fraction defines a positive
+irrational number. Let us define a map Gmpxq “
+1
+m`x for every integer m P Z`. Then
+rm0; m1, m2, . . . s “ m0 ` lim
+kÑ8 Gm1 ˝ ¨ ¨ ¨ ˝ Gmkp0q.
+If
+1
+|n2α´nm| ą 2, then there exists a k such that m
+n “ pk
+qk “ rm0; m1, . . . , mks, see for example [9,
+Theorem 19]. Hence kpαq “ lim supkÑ8
+1
+|p2
+kα´pkqk|. In 1921, Perron [15] proved
+1
+|p2
+kα ´ pkqk| “ r0; mk, mk´1, . . . , m1s ` rmk`1; mk`2, . . . s.
+In particular,
+LS “
+"
+lim sup
+kÑ8
+´
+Gmk ˝ ¨ ¨ ¨ ˝ Gm1p0q ` mk`1 ` lim
+ℓÑ8 Gmk`2 ˝ ¨ ¨ ¨ ˝ Gmℓp0q
+¯
+|pmiq P ZZ`
+ě1
+*
+.
+
+4
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+Now, let us define the maps pGmpxq “
+2mx
+pm`2qx´2. By Theorem B, we will show later that
+(1.3)
+LSPAC “
+"
+lim sup
+kÑ8
+pGmk ˝ ¨ ¨ ¨ ˝ pGm1p2q|pmiq P ZZ`
+ě2
+*
+.
+Moreira [13] showed that the map α ÞÑ dimH
+`
+r
+?
+5, αs X LS
+˘
+“ dimB
+`
+r
+?
+5, αs X LS
+˘
+is
+monotone increasing and continuous on r
+?
+5, 8q, where dimH denotes the Hausdorff dimension
+and dimB denotes the upper box-counting dimension. For the definition and basic properties of the
+Hausdorff- and box-counting dimension we refer to [4].
+Problem 1.3. Is dimH
+`
+r 3
+2, αs X LSPAC
+˘
+“
+dimB
+`
+r 3
+2, αs X LSPAC
+˘
+?
+Is the map
+α ÞÑ dimH
+`
+r 3
+2, αs X LSPAC
+˘
+continuous?
+2. Preliminaries
+In this section, we summarize some basic facts in the theory of iterated function systems relevant
+for our later calculations. We say that a map f : R ÞÑ R is contracting if there exists a constant
+0 ă c ă 1 such that |fpxq ´ fpyq| ď c|x ´ y|. By Banach’s fixed point theorem, every contractive
+map f has a unique fixed point x “ fpxq. For a contractive map f, let us denote its unique fixed
+point by Fixpfq.
+Let Ψ “ tf1, . . . , fnu be a finite collection of contractions, which we call iterated function
+system (IFS). Hutchinson [8] showed that there exists a unique non-empty compact set Λ such that
+(2.1)
+Λ “
+nď
+i“1
+fipΛq.
+The set Λ is called the attractor of the IFS Ψ. In particular, if B Ă R is a compact set such that
+fipBq Ď B for every i “ 1, . . . , n then
+(2.2)
+Λ “
+8
+č
+k“1
+ď
+pi1,...,ikqPt1,...,nuk
+fi1 ˝ ¨ ¨ ¨ ˝ fikpBq Ă B.
+Using (2.2), one can prove the following simple observation.
+Lemma 2.1. Let Ψ “ tf1, . . . , fnu be a finite collection of contractions such that the contracting
+ratio of fi is ci. If řn
+i“1 ci ă 1 then λpΛq “ 0, where λ denotes the Lebesgue measure on the real
+line.
+Proof. Since |fipxq ´ fipyq| ď ci|x ´ y| then λpfi1 ˝ ¨ ¨ ¨ ˝ fikpBqq ď ci1 ¨ ¨ ¨ cinλpBq and so, by
+(2.2),
+λpΛq ď
+ÿ
+pi1,...,ikqPt1,...,nuk
+λpfi1 ˝ ¨ ¨ ¨ ˝ fikpBqq “
+˜ nÿ
+i“1
+ci
+¸k
+λpBq Ñ 0 as k Ñ 8.
+□
+
+LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS
+5
+Let us denote the distance between sets by dist,
+that is,
+for A, B
+Ď
+R,
+let
+distpA, Bq “ inft|x ´ y|
+ˇˇx P A, y P Bu. With a slight abuse of notation, we write distpx, Aq “
+distptxu, Aq for the distance of a point x P R and a set A Ď R.
+Lemma 2.2. Let Ψ “ tf1, . . . , fnu be a finite collection of contractions such that the contracting
+ratio of every fi is at most c P p0, 1q. For every sequence pi1, i2, . . .q P t1, . . . , nuZ` and every
+x P R, lim infkÑ8 fik ˝¨ ¨ ¨˝fi1pxq P Λ, where Λ is the attractor of Ψ. In particular, for every open
+set U Ą Λ, for every x P R and for every sufficiently large k, fik ˝ ¨ ¨ ¨ ˝ fi1pxq P U.
+Proof. By (2.1)
+distpfik ˝ ¨ ¨ ¨ ˝ fi1pxq, Λq ď distpfik ˝ ¨ ¨ ¨ ˝ fi1pxq, fik ˝ ¨ ¨ ¨ ˝ fi1pΛqq ď ckdistpx, Λq Ñ 0 as k Ñ 8,
+where 0 ă c ă 1 is chosen such that |fipxq´fipyq| ď c|x´y| for every i “ 1, . . . , n and x, y P R.
+The claim then follows by the compactness of Λ.
+□
+For every point x P Λ, there exists an infinite sequence i “ pi1, i2, . . .q P t1, . . . , nuZ` such that
+x “ lim
+kÑ8 fi1 ˝ ¨ ¨ ¨ ˝ fikp0q.
+Observe that the limit on the right-hand side exists since the maps fi are contractions. One can
+define a map Π: t1, . . . , nuZ` ÞÑ Λ by
+Πpiq :“ lim
+kÑ8 fi1 ˝ ¨ ¨ ¨ ˝ fikp0q
+called the natural projection. Let σ: t1, . . . , nuZ` ÞÑ t1, . . . , nuZ` be the left-shift operator, that
+is,
+σpi1, i2, . . .q “ pi2, i3, . . .q.
+Hence, by using the definition of the natural projection Π it is easy to see that
+Πpiq “ fi1pΠpσiqq.
+Now, let us define a specific family of contractive maps on R as Tmpxq “ 1´x
+m for m P Zě2.
+Then clearly for every pmiq P ZZ`
+ě2
+Tmk ˝ Tmk´1 ˝ ¨ ¨ ¨ ˝ Tm1p0q “ 1
+mk
+´
+1
+mkmk´1
+`
+1
+mkmk´1mk´2
+´ ¨ ¨ ¨ ` p´1qk´1
+1
+mkmk´1 . . . m1
+,
+which corresponds to (1.2). Let
+L “
+!
+lim inf
+kÑ8 Tmk ˝ ¨ ¨ ¨ ˝ Tm1p0q|pmiq P ZZ`
+ě2
+)
+.
+Hence,
+(2.3)
+LSPAC “ gpLq,
+where gpxq “
+2
+1`x. Furthermore, pGmpxq “ g ˝ Tm ˝ g´1, thus, (1.3) follows. Hence, our main
+theorem will follow from the following theorems.
+Theorem 2.1. The set L is closed.
+
+6
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+Theorem 2.2. r0, 1
+7s Ă L.
+Theorem 2.3.
+L X
+8
+ď
+n“0
+ˆ1
+6 ` 1
+93
+1
+4n, 1
+6 ` 1
+84
+1
+4n
+˙
+“ H.
+Let S Ă R be a Lebesgue-measurable set. The Lebesgue-measure of S will be denoted by λpSq.
+Theorem 2.4. λ
+`
+L X r 3
+17, 1
+3s
+˘
+“ 0.
+We introduce the following notations. Let i “ pi1, . . . , inq P Zn
+ě2 be a finite word, then denote
+by Ti the map
+Ti “ Ti1 ˝ ¨ ¨ ¨ ˝ Tin.
+Let u, v
+be positive integers and m
+“
+pmiq
+P
+ZZ`
+ě2.
+If u
+ď
+v
+then let
+Tmru,vspxq
+“
+pTmu
+˝
+Tmu`1
+˝
+¨ ¨ ¨
+˝
+Tmvqpxq,
+and
+if
+u
+ą
+v
+then
+let
+Tmru,vspxq “ pTmu ˝ Tmu´1 ˝ ¨ ¨ ¨ ˝ Tmvqpxq. Finally, let us introduce the notation that for any
+sequence m P ZZ`
+ě2
+(2.4)
+Πpmq “ lim
+kÑ8 Tm1 ˝ ¨ ¨ ¨ Tmkp0q “ lim
+kÑ8 Tmr1,ksp0q “
+8
+ÿ
+k“1
+p´1qk´1
+m1 ¨ ¨ ¨ mk
+.
+Let us define the sequences Mpnq recursively. Let Mp1q “ 2, Mp2q “ 3 and let Mpnq be the con-
+catenation Mpnq “ Mpn´1qMpn´2qMpn´2q for n ě 3, that is Mp3q “ p3, 2, 2q, Mp4q “ p3, 2, 2, 3, 3q
+and so on. By the definition of Mpnq, it is easy to see that the length of the finite sequence Mpnq
+is ln “ 2n´p´1qn
+3
+, and Mpnq starts with Mpn´1q. Thus, it is possible to define the limiting infinite
+sequence M “ limnÑ8 Mpnq as
+M “ p3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 2, 3, 2, . . .q “: pM1, M2, . . .q
+such that pM1, M2, . . . , Mlnq “ pMpnqq for every positive integer n ě 2. Let
+λn “ FixpTMpnqq “ TMpnqp0q
+M1M2 . . . Mln
+M1M2 . . . Mln ` 1,
+and
+λ0 “
+8
+ÿ
+l“1
+p´1ql´1
+1
+M1M2 . . . Ml
+“ 0.2293 . . ..
+We will prove that λn is a strictly increasing sequence, λn ą λ0.
+Theorem 2.5.
+λ P L,
+λ ą λ0 if and only if λ “ λn for some n ě 1.
+Proof of Theorem 1.1. The first claim follows by (2.3), the fact the map gpxq “
+2
+1`x is continuous
+on R` and Theorem 2.1. The second claim follows by Theorem 2.5 with the choices γn “ gpλnq
+for n ě 0. Finally, the last claim follows by the combination of Theorem 2.2 and Theorem 2.3.
+□
+
+LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS
+7
+3. Closedness of the spectrum
+Proof of Theorem 2.1. Let αn P L be a sequence such that lim
+nÑ8 αn “ α. Hence, for every n ě 1
+there exists mpnq P ZZ`
+ě2, mpnq “ pmpnq
+1 , mpnq
+2 , . . . q such that lim inf
+kÑ8 Tmpnq
+rk,1sp0q “ αn. Let εn “
+|α ´ αn|. Without loss of generality we may assume that εn Œ 0.
+Let l1 “ 0 and let us choose k1 such that |Tmp1q
+rk1,1sp0q ´ α1| ă ε1.
+For n ě 2, let 0 ă ln ă kn be such that |Tmpnq
+rln,1sp0q ´ αn| ă εn, |Tmpnq
+rkn,1sp0q ´ αn| ă εn,
+T p1q
+mpnq
+rl,1sp0q ą αn ´ εn for every l ě ln and 5εn´1
+εn
+ă 2kn´ln. Let
+m “ pmp1q
+l1`1, . . . , mp1q
+k1 , mp2q
+l2`1, . . . , mp2q
+k2 , mp3q
+l3`1, . . . , mp3q
+k3 , . . . q “ pm1, m2, . . . q.
+We will show that
+(3.1)
+α “ lim inf
+kÑ8 Tmrk,1sp0q.
+Let aN “ řN
+n“1pkn ´ lnq. To verify (3.1), it is enough to prove that
+(3.2)
+|TmraN ,1sp0q ´ αN| ă 2εN for every N ě 1
+and
+(3.3)
+Tmrl,1sp0q ą αN ´ 3εN for every aN ă l ď aN`1.
+Indeed, in this case
+lim
+NÑ8 TmraN ,1sp0q “ α and lim inf
+lÑ8 Tmrl,1sp0q ě lim
+NÑ8 αN ´ 3εN “ α.
+To prove (3.2) we argue by induction. Clearly,
+|Tmra1,1sp0q ´ α1| “ |Tmp1q
+rk1,1sp0q ´ α1| ă ε1 ă 2ε1.
+Suppose that (3.2) holds for N ´ 1. Then
+|TmraN ,1sp0q ´ αN| ď |TmraN ,1sp0q ´ TmpNq
+rkN ,1sp0q| ` |TmpNq
+rkN ,1sp0q ´ αN|
+“
+1
+mpNq
+kN . . . mpNq
+lN`1
+|TmraN´1,1sp0q ´ TmpNq
+rlN ,1sp0q| ` |TmpNq
+rkN ,1sp0q ´ αN|
+ď
+1
+2kN´lN
+ˆ
+|TmraN´1,1sp0q ´ αN´1| ` |αN´1 ´ α| ` |α ´ αN| ` |TmpNq
+rlN ,1sp0q ´ αN|
+˙
+` εN
+ă
+1
+2kN´lN p2εN´1 ` εN´1 ` εN ` εNq ` εN ă
+1
+2kN´lN 5εN´1 ` εN ă 2εN.
+To prove (3.3) we write
+Tmrl,1sp0q “ Tmrl,aN `1s ˝ TmraN ,1sp0q “ TmpNq
+rl´aN `lN ,lN `1s ˝ TmraN ,1sp0q
+“ TmpNq
+rl´aN `lN ,lN `1s ˝ TmraN ,1sp0q ´ TmpNq
+rl´aN `lN ,lN `1s ˝ TmpNq
+rlN ,1sp0q ` TmpNq
+rl´aN `lN ,lN `1s ˝ TmpNq
+rlN ,1sp0qq,
+
+8
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+where
+ˇˇˇˇTmpNq
+rl´aN `lN ,lN `1s ˝ TmraN ,1sp0q ´ TmpNq
+rl´aN `lN ,lN `1s ˝ TmpNq
+rlN ,1sp0q
+ˇˇˇˇ
+“
+1
+mpNq
+l´aN `lN . . . mpNq
+lN`1
+ˇˇˇˇTmraN ,1sp0q ´ TmpNq
+rlN ,1sp0q
+ˇˇˇˇ
+ď
+1
+2l´aN p|TmraN ,1sp0q ´ αN| ` |αN ´ TmpNq
+rlN ,1sp0q|q ă 1
+2p2εN ` εNq ă 2εN
+and
+TmpNq
+rl´aN `lN ,lN `1s ˝ TmpNq
+rlN ,1sp0q “ TmpNq
+rl´aN `lN ,1sp0q ą αN ´ εN.
+Hence,
+T p1q
+mrl,1sp0q ą αN ´ εN ´ 2εN “ αN ´ 3εN,
+which completes the proof.
+□
+4. Estimates on the Freiman-like constant
+Let us consider the finite IFS Ψ4 “ tT2, T3, T4u. Let I “
+“ 1
+7, 3
+7
+‰
+. For m ě 2, TmpIq “ r 4
+7m,
+6
+7ms,
+and
+I “
+4ď
+m“2
+TmpIq.
+(4.1)
+Thus, by the uniqueness the attractor of Ψ4 is I “
+“ 1
+7, 3
+7
+‰
+. By (4.1) and direct calculation, we obtain
+the following statements.
+Lemma 4.1. For every z P r 1
+7, 3
+7s and K P t2, 3, 4u, we have 1
+K ´ 1
+K z P r 1
+7, 3
+7s.
+Lemma 4.2. For every y P r 1
+7, 3
+7s, there exist K P t2, 3, 4u and z P r 1
+7, 3
+7s such that y “ 1
+K ´ 1
+Kz.
+In particular, it follows from Lemma 4.2 that for every y P r 1
+7, 3
+7s, there exists an infinite sequence
+pK1, K2, . . .q P t2, 3, 4uZ` such that
+ΠpK1, K2, . . .q “ y,
+where Π is defined in (2.4).
+Proof of Theorem 2.2. It infers from
+4
+7m ď
+6
+7pm`1q for m ě 6 that
+8
+ď
+m“6
+TmpIq “
+ˆ
+0, 1
+7
+
+.
+(4.2)
+Suppose that 0 ă x ă 1
+7. By (4.2), we know that the real x can be written as x “
+1
+m ´ 1
+my,
+where m ě 6 and y P r 1
+7, 3
+7s. It follows that there exist sequences K1, K2, . . . , Kk, ¨ ¨ ¨ P t2, 3, 4u
+
+LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS
+9
+and z1, z2, . . . , zk, ¨ ¨ ¨ P r 1
+7, 3
+7s such that
+y
+“
+ΠpK1, K2, . . .q “
+8
+ÿ
+k“1
+p´1qk´1
+K1K2 . . . Kk
+.
+Now, let
+m “ pm1, m2, . . . q “ p3, K1, m, 3, K2, K1, m, 3, K3, K2, K1, m, 3, K4, K3, K2, K1, m, . . . q.
+We will prove that
+x “ lim inf
+nÑ8 Tmrn,1sp0q.
+First, observe that Tmrn,1sp0q P
+“
+0, 1
+2
+‰
+for every n ě 1. Indeed, Tmp
+“
+0, 1
+2
+‰
+q Ă
+“
+0, 1
+2
+‰
+for every
+m ě 2. On the other hand, since T3
+`“
+0, 1
+2
+‰˘
+Ă
+“ 1
+6, 1
+3
+‰
+Ă
+“ 1
+7, 3
+7
+‰
+, we have that Tmrn,1sp0q P
+“ 1
+7, 3
+7
+‰
+for every n ě 1 with mn “ 3. Hence, it follows from Lemma 4.1 and (4.1) that if mn ‰ m, then
+Tmrn,1sp0q P r 1
+7, 3
+7s.
+Simple calculations shows that mk “ m if and only if k “ n2`5n
+2
+for some n P Z`. Furthermore,
+it is easy to see that
+Tm
+r n2`5n
+2
+,1sp0q
+“
+1
+m ´ 1
+m
+ˆ 1
+K1
+´
+1
+K1K2
+` ¨ ¨ ¨ `
+p´1qn´1
+K1K2 . . . Kn
+`
+p´1qn
+K1 ¨ ¨ ¨ Kn
+Tm
+r pn´1q2`5pn´1q
+2
+`1,1sp0q
+˙
+“
+1
+m ´ 1
+my ` Op 1
+2nq “ x ` Op 1
+2nq.
+This completes the proof of Theorem 2.2.
+□
+Before we continue, we state a lemma on the position of the possible smallest accumulation
+points depending on the defining sequence.
+Lemma 4.3.
+(1) Let pmiq P ZZ`
+ě2 be such that mi P t2, 3, . . . , Ku except at most finitely many i. Then
+1
+2K ´ 1 ď lim inf
+nÑ8 Tmrn,1sp0q ď lim sup
+nÑ8 Tmrn,1sp0q ď K ´ 1
+2K ´ 1.
+(2) Let K P Zě2 and pmiq P ZZ`
+ě2 such that mi ě K for infinitely many integer i. Then
+lim inf
+kÑ8 Tmrk,1sp0q ď
+1
+K ` 1.
+Proof. To prove the first claim, it is enough to show that
+(4.3)
+Tm
+ˆ„
+1
+2K ´ 1, K ´ 1
+2K ´ 1
+˙
+Ď
+„
+1
+2K ´ 1, K ´ 1
+2K ´ 1
+
+for every 2 ď m ď K.
+Indeed,
+Tm
+ˆ„
+1
+2K ´ 1, K ´ 1
+2K ´ 1
+˙
+“
+„
+K
+mp2K ´ 1q,
+2K ´ 2
+mp2K ´ 1q
+
+,
+where
+1
+2K´1 ď
+K
+mp2K´1q if and only if m ď K and
+2K´2
+mp2K´1q ď
+K´1
+2K´1 if and only if m ě 2.
+
+10
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+Then by (2.2), (4.3) and Lemma 2.2, we get that lim infnÑ8 Tmrn,1sp0q P
+“
+1
+2K´1, K´1
+2K´1
+‰
+.
+To show the last claim, let us argue by contradiction. If lim infkÑ8 Tmrk,1sp0q ą
+1
+K`1, then there
+exists a δ ą 0 such that Tmrn,1sp0q ą
+1
+K`1 ` δ for every sufficiently large n. Then
+Tmrn,1sp0q “ 1
+mn
+´ 1
+mn
+Tmrn´1,1sp0q ă 1
+mn
+´ 1
+mn
+p
+1
+K ` 1 ` δq.
+Hence, for every sufficiently large n we have that
+1
+K`1 ` δ ă
+1
+mn ´
+1
+mnp
+1
+K`1 ` δq, equivalently
+1
+K`1 ` δ ă
+1
+mn`1 for sufficiently large n. Thus, mn ď K ´ 1 for every sufficiently large n, which
+is a contradiction.
+□
+Finally, let us state a technical lemma.
+Lemma 4.4. Let pmiq P t2, 3, 4uZ` such that if pmi, mi´1q “ p4, 2q then mi´2 “ 2. Then
+lim supnÑ8 Tmrn,1sp0q ď 13
+31.
+Proof. Observe that
+min T2 ˝ T4 ˝ T2
+ˆ„1
+7, 3
+7
+˙
+ě max
+¨
+˚
+˚
+˝
+ď
+i,j,kPt2,3,4u3
+pi,j,kq‰p2,4,2q
+Ti ˝ Tj ˝ Tk
+ˆ„1
+7, 3
+7
+˙
+˛
+‹‹‚,
+where we recall that r 1
+7, 3
+7s is the attractor of the IFS tT2, T3, T4u. Thus, if pmi, mi´1, mi´2, mi´3q “
+p2, 4, 2, 2q only for finitely many i then
+lim sup
+nÑ8 Tmrn,1sp0q ď T2 ˝ T4 ˝ T2 ˝ T2
+ˆ1
+7
+˙
+“ 23
+56 ă 13
+31.
+On the other hand, if pmi, mi´1, mi´2, mi´3q “ p2, 4, 2, 2q for infinitely many i then
+lim sup
+nÑ8 Tmrn,1sp0q ď T2 ˝ T4 ˝ T2 ˝ T2
+ˆ
+lim sup
+nÑ8 Tmrn,1sp0q
+˙
+,
+which implies after some algebraic manipulations that lim supnÑ8 Tmrn,1sp0q ď 13
+31.
+□
+Proof of Theorem 2.3. It follows from Lemma 4.3(2) that if mn ě 5 for infinitely many n, then
+lim inf
+kÑ8 Tmrk,1sp0q ď 1
+6.
+So we may assume without loss of generality that mn P t2, 3, 4u for every n P Z`.
+Direct computations show that
+ˆ1
+6, 1
+6 ` 1
+84
+˙
+X
+ď
+pk,lqPt2,3,4u2
+pk,lq‰p4,2q
+Tk ˝ Tl
+ˆ„1
+7, 3
+7
+˙
+“ H.
+
+LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS
+11
+Thus, by Lemma 2.2 and the fact that
+“ 1
+7, 3
+7
+‰
+is the attractor of the IFS tT2, T3, T4u we get that if
+lim inf
+kÑ8 Tmrk,1sp0q P L X
+ˆ1
+6, 1
+6 ` 1
+84
+˙
+with the sequence pmiq P t2, 3, 4uZ` then pmi, mi´1q “ p4, 2q for infinitely many i P Z`, and in
+particular,
+(4.4)
+lim inf
+kÑ8 Tmrk,1sp0q “ lim inf
+ℓÑ8 Tmrkℓ,1sp0q,
+where k1 “ minti ě 2 : pmi, mi´1q “ p4, 2qu and kℓ “ minti ą kℓ´1 : pmi, mi´1q “ p4, 2qu for
+all ℓ ě 2
+If pmkℓ, mkℓ´1, mkℓ´2q “ p4, 2, bq for some b P t3, 4u for infinitely many i then 1
+6 ě Tmrkℓ,kℓ´2sp0q ě
+Tmrkℓ,1sp0q. So we may assume that if
+(4.5)
+pmi, mi´1q “ p4, 2q then mi´2 “ 2 for every i.
+Furthermore, if for every N there exist infinitely many k ě 2N ` 2 such that
+pmk, mk´1, . . . , mk´2N´1q “ p4, 2, 2, . . ., 2q
+then
+lim inf
+kÑ8 Tmrk,1sp0q ď lim
+NÑ8 T4 ˝
+2N`1-times
+hkkkkkikkkkkj
+T2 ˝ ¨ ¨ ¨ ˝ T2p0q “ 1
+6.
+Hence, we may assume that there exists a non-negative integer N0 such that
+pmkℓ, mkℓ´1, . . . , mkℓ´2N0´1q “ p4,
+2N0 ` 1-times
+hkkkkikkkkj
+2, 2, . . . , 2q for infinitely many ℓ but
+pmkℓ, mkℓ´1, . . . , mkℓ´2N0´3q “ p4,
+2N0 ` 3-times
+hkkkkikkkkj
+2, 2, . . . , 2q only for a finite number of ℓ.
+(4.6)
+Let us suppose that (4.6) holds. For short, let pN0 “ pmkℓ, mkℓ´1, . . . , mkℓ´2N0´1q. Then by
+Lemma 4.4 and the fact that the maps Tm are orientation reversing we get
+lim inf
+ℓÑ8 Tmrkℓ,1sp0q ď TpN0plim sup
+kÑ8
+Tmrk´2N0´2,1sp0qq
+“ 1
+8
+1 ´
+1
+4N0`1
+1 ´ 1
+4
+`
+1
+8 ¨ 4N0 ¨ 13
+31 “ 1
+6 ` 1
+93
+1
+4N0 .
+(4.7)
+If pmkℓ, mkℓ´1, . . . , mkℓ´2N0´2q “ p4, 2, 2, . . ., 2, aq, where a P t3, 4u then
+Tmrkℓ,1sp0q ď Tmrkℓ,kℓ´2N0´2sp0q “ TpN0 ˝ Tap0q ď TpN0p1
+3q “ 1
+6.
+
+12
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+Hence, we may suppose that pmkℓ´2N0´2, mkℓ´2N0´3q P tp2, 3q, p2, 4qu. Thus, by Lemma 4.3(1)
+and the fact that the maps Tm are orientation reversing we get
+lim inf
+ℓÑ8 Tmrkℓ,1sp0q ě TpN0 ˝ T2 ˝ Taplim inf
+kÑ8 Tmrk´2N0´3,1sp0qq
+ě TpN0 ˝ T2 ˝ Tap1
+7q ě TpN0 ˝ T2p2
+7q “ 1
+6 ` 1
+84
+1
+4N0`1.
+(4.8)
+Finally, (4.8) with (4.4) and (4.7) implies that
+1
+6 ` 1
+84
+1
+4N0`1. ď lim inf
+kÑ8 Tmrk,1sp0q ď 1
+6 ` 1
+93
+1
+4N0 ,
+and so lim infkÑ8 T p1q
+mrk,1sp0q R Ť8
+n“0
+`1
+6 ` 1
+93
+1
+4n, 1
+6 ` 1
+84
+1
+4n
+˘
+.
+□
+5. Computation of the Markov-like constant
+Throughout this section, we will consider the set L X
+`1
+5, 1
+2
+˘
+. By Lemma 4.3(2), for every
+x P L X
+` 1
+5, 1
+2
+˘
+if x “ lim infnÑ8 Tmrn,1sp0q “ x then pmiq P t2, 3uZ`. By (4.3),
+T2
+ˆ„1
+5, 2
+5
+˙
+Y T3
+ˆ„1
+5, 2
+5
+˙
+Ď
+„1
+5, 2
+5
+
+,
+and so by denoting the attractor of the IFS tT2, T3u by Λ Ă
+“ 1
+5, 2
+5
+‰
+, we get by Lemma 2.2 that
+(5.1)
+L X
+„1
+5, 2
+5
+
+Ă Λ.
+Furthermore, direct computations show that
+(5.2)
+T2
+ˆ„1
+5, 2
+5
+˙
+X T3
+ˆ„1
+5, 2
+5
+˙
+“ H.
+Hence, by choosing δ “ dist
+`
+T2
+`“ 1
+5, 2
+5
+‰˘
+, T3
+`“ 1
+5, 2
+5
+‰˘˘
+{3 ą 0, we get
+(5.3)
+T2 pJq X T3 pJq “ H and T2 pJq Y T3 pJq Ď J,
+where J “
+“ 1
+5 ´ δ, 2
+5 ` δ
+‰
+.
+Lemma 5.1.
+(1) Let pi1, . . . , i2k`1q P ZZ`
+ě2 be for some non-negative integer k, and let pmiq P ZZ`
+ě2 be such
+that pmj, . . . , mj´2kq “ pi1, . . . , i2k`1q for infinitely many j. Then
+lim inf
+nÑ8 Tmrn,1sp0q ď FixpTi1 ˝ ¨ ¨ ¨ ˝ Ti2k`1q.
+(2) Let x P L X
+`1
+5, 2
+5
+˘
+and let pi1, . . . , i2kq P t2, 3u2k be such that x P Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k
+`“ 1
+5, 2
+5
+‰˘
+.
+Then
+x ě FixpTi1 ˝ ¨ ¨ ¨ ˝ Ti2kq.
+
+LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS
+13
+Proof. Let us show the first claim. Let jℓ be the sequence such that pmjℓ, . . . , mjℓ´2kq “ pi1, . . . , i2k`1q.
+Since the maps Tm are orientation reversing
+lim inf
+nÑ8 Tmrn,1sp0q ď lim inf
+ℓÑ8 Tmrjℓ,1sp0q
+“ Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k`1plim inf
+ℓÑ8 Tmrjℓ´2k´1,1sp0qq
+ď Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k`1plim inf
+nÑ8 Tmrn,1sp0qq.
+Let us denote the fixed point of Ti1 ˝ ¨ ¨ ¨˝ Ti2k`1 by x0 and, for short, let x “ lim infnÑ8 Tmrn,1sp0q.
+Since the maps Tm are linear and contracting, we get
+x ´ x0 ď Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k`1pxq ´ Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k`1px0q “
+´1
+i1 ¨ ¨ ¨ i2k`1
+px ´ x0q,
+thus the claim follows.
+Now we turn to the second claim. Let x P L X
+` 1
+5, 2
+5
+˘
+be such that x P Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k
+`“ 1
+5, 2
+5
+‰˘
+.
+Suppose that x “ lim infnÑ8 Tmrn,1sp0q. By (5.3)
+(5.4)
+dist
+¨
+˚
+˚
+˝Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k pJq ,
+ď
+pj1,...,j2kqPt2,3u2k
+pj1,...,j2kq‰pi1,...,i2kq
+Tj1 ˝ ¨ ¨ ¨ ˝ Tj2k pJq
+˛
+‹‹‚ą 0.
+By Lemma 2.2, for every sufficiently large n, Tmrn,1sp0q P J. Then by (5.3), for every sufficiently
+large n, Tmrn,1sp0q P Tmrn,n´2k`1spJq. Hence, by (5.4), Tmrn,1sp0q P Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k pJq if and only if
+pmn, . . . , mn´2k`1q “ pi1, . . . , i2kq, and so by our assumption on x P L X Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k
+`“ 1
+5, 2
+5
+‰˘
+lim inf
+nÑ8 Tmrn,1sp0q “ lim inf
+ℓÑ8 Tmrnℓ,1sp0q,
+where nℓ is the sequence such that pmnℓ, . . . , mnℓ´2k`1q “ pi1, . . . , i2kq. Since Tm is orientation
+reversing
+lim inf
+nÑ8 Tmrn,1sp0q “ lim inf
+ℓÑ8 Tmrnℓ,1sp0q “ Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k
+´
+lim inf
+ℓÑ8 Tmrnℓ´2k,1sp0q
+¯
+ě Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k
+´
+lim inf
+nÑ8 Tmrn,1sp0q
+¯
+.
+Thus the statement follows similarly than the first claim.
+□
+Let us recall the definition of the sequences Mpnq. Let Mp1q “ 2, Mp2q “ 3 and let Mpnq be the
+concatenation
+(5.5)
+Mpnq “ Mpn´1qMpn´2qMpn´2q for n ě 3.
+By definition, the length ln of Mpnq satisfies the equation ln “ ln´1 ` 2ln´2 for every n ě 3 with
+l1 “ l2 “ 1, which implies a standard calculation that ln “ 2n´p´1qn
+3
+.
+We say for any two compact intervals ra, bs and rc, ds that ra, bs ă rc, ds if b ă c.
+For any two closed intervals ra, bs and rc, ds with ra, bs ă rc, ds, let midpra, bs, rc, dsq “ rb, cs.
+
+14
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+Lemma
+5.2.
+For
+every
+n
+ě
+3,
+TMpnqpJq
+Ă
+TMpn´1qpJq.
+Furthermore,
+TMpnqpJq
+X
+TMpn´1qMpn´1qpJq
+“
+H,
+TMpnqpJq
+ă
+TMpn´1qMpn´1qpJq
+and
+Λ X midpTMpnqpJq, TMpn´1qMpn´1qpJqq “ H for every n ě 2. That is, there is no element of Λ
+in-between TMpnqpJq and TMpn´1qMpn´1qpJq for every n ě 2.
+Proof. By (5.5) and (5.3), clearly TMpnqpJq Ă TMpn´1qpJq. We prove the second claim by induc-
+tion. Clearly, by (5.3) T3pJq X T2,2pJq “ H, furthermore, since
+Λ X
+ˆ„1
+5, 2
+5
+
+z
+ˆ
+T2p
+„1
+5, 2
+5
+
+q Y T3p
+„1
+5, 2
+5
+
+q
+˙˙
+“ H
+and
+„1
+5, 2
+5
+
+z
+ˆ
+T2p
+„1
+5, 2
+5
+
+q Y T3p
+„1
+5, 2
+5
+
+q
+˙
+Ą midpT2,2pJq, T3pJqq
+the claim holds for n “ 2.
+Let us suppose that the claim holds for n. Then
+TMpn`1qpJq X TMpnqMpnqpJq “ TMpnq pTMpn´1qMpn´1qpJq X TMpnqpJqq “ H,
+moreover, since TMpnq is orientation reversing, TMpnqpJq ă TMpn´1qMpn´1qpJq implies that
+TMpn`1qpJq “ TMpnq pTMpn´1qMpn´1qpJqq ă TMpnq pTMpnqpJqq .
+Observe that
+midpTMpn`1qpJq, TMpnqMpnqpJqq “ TMpnq pmidpTMpn´1qMpn´1qpJq, TMpnqpJqqq Ă TMpnqpJq
+and so by (5.3)
+ΛXmidpTMpn`1qpJq, TMpnqMpnqpJqq “ TMpnqpΛqXTMpnq pmidpTMpn´1qMpn´1qpJq, TMpnqpJqqq “ H.
+□
+Let us recall the definition of the sequence λn and λ0. For every n ě 1, let λn “ FixpTMpnqq.
+Since FixpTMpnqq P TMpnqMpnqpJq Ă TMpnqpJq, by Lemma 5.2 we get
+λn`1 ă λn.
+Thus, the sequence λn is convergent. Let us denote the limit limnÑ8 λn by λ0. Then by TMpnqpJq Ă
+TMpn´1qpJq, we get that
+λ0 “ ΠpMq,
+where M
+“
+limnÑ8 Mpnq
+“
+pM1, M2, . . .q is the limiting sequence defined such that
+pM1, M2, . . . , Mlnq “ Mpnq for every positive integer n ě 2. So
+λ0 “
+8
+ÿ
+l“1
+p´1ql´1
+1
+M1M2 . . . Ml
+“ 0.2293 . . ..
+First, we show the following proposition:
+Proposition 5.3. tλ1, λ2, . . .u Ă L X
+` 1
+5, 2
+5
+˘
+. In particular, λ0 P L.
+Before we turn to its proof, we require the following technical lemmas.
+
+LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS
+15
+Lemma 5.4. Let a “ pa1, . . . , akq and b “ pb1, . . . , bnq be finite sequences formed by the integers
+t2, 3u. Suppose that there exist a prefix a1 “ pa1, . . . , ak1q of a with k1 ď k and a prefix b1 “
+pb1, . . . , bn1q of b with n1 ď n such that Ta1pJq ă Tb1pJq. Then TapJq ă TbpJq.
+Proof. Observe that for every compact intervals A, B, if C Ă A and D Ă B are compact intervals
+then A ă B implies C ă D. Thus, the claim follows by TapJq Ă Ta1pJq and TbpJq Ă Tb1pJq.
+□
+For the finite sequence Mpnq “ pMpnq
+1 , . . . , Mpnq
+ln q and 1 ď ℓ ď ln´1, let σℓMpnq be the ln-length
+word such that
+σℓMpnq “ pMpnq
+ℓ`1, . . . , Mpnq
+ln , Mpnq
+1 , . . . , Mpnq
+ℓ
+q,
+with the convention that σlnMpnq “ Mpnq. Thus, σℓMpnq can be defined for every ℓ P Z` in a
+natural, periodic way.
+Lemma 5.5. For every n ě 3 and 1 ď ℓ ď ln ´ 1, TMpnqpJq ă TσℓMpnqpJq.
+Proof. Simple calculations show that
+(5.6)
+T3,2,2pJq ă T3,3,3pJq ă T3,3,2pJq ă T2,2,3pJq ă T2,3,3pJq ă T2,3,2pJq.
+Clearly for Mp3q “ p3, 2, 2q, we have σ1Mp3q “ p2, 2, 3q and σ2Mp3q “ p2, 3, 2q, and for Mp4q “
+Mp3qMp2qMp2q “ p3, 2, 2, 3, 3q we have
+σ1Mp4q “ p2, 2, 3, 3, 3q, σ2Mp4q “ p2, 3, 3, 3, 2q, σ3Mp4q “ p3, 3, 3, 2, 2q, σ4Mp4q “ p3, 3, 2, 2, 3q.
+Thus, by Lemma 5.4 and (5.6), the claim follows for n “ 3 and n “ 4.
+Let us prove the rest by induction. So suppose that the claim holds for n ě 4.
+First, assume that 1 ď ℓ ď ln´1. Then by
+(5.7)
+Mpn`1q “ Mpn´1qMpn´2qMpn´2qMpn´1qMpn´1q
+and Mpnq “ Mpn´1qMpn´2qMpn´2q, we get that Mpn`1q
+k
+“ Mpn´1q
+k
+“ Mpnq
+k
+for every 1 ď k ď ln´1,
+and so σℓMpnq is a prefix of σℓMpn`1q. Since Mpnq is a prefix of Mpn`1q, we get by the induction
+condition TMpnqpJq ă TσℓMpnqpJq that TMpn`1qpJq ă TσℓMpn`1qpJq by Lemma 5.4.
+Now, assume that ln´1 ă ℓ ă ln but ℓ ‰ ln´1 ` ln´2. Then by
+(5.8)
+Mpn`1q “ Mpn´1qMpn´2qMpn´2qMpn´2qMpn´3qMpn´3qMpn´1q,
+we get that σℓ´ln´1Mpn´2q is a prefix of σℓMpn`1q. Hence, again by the fact that Mpn´2q is a prefix
+of Mpn`1q and the assumption that TMpn´2qpJq ă TσkMpn´2qpJq for every k R tln´2, 2ln´2, . . .u
+integer, the claim follows by Lemma 5.4.
+If ℓ “ ln´1 ` ln´2 then by (5.8), we get that Mpn´2qMpn´2q is a prefix of σℓMpn`1q, meanwhile,
+if ℓ “ ln´1 ` 2ln´2 “ ln then by (5.7), we get that Mpn´1qMpn´1q is a prefix of σℓMpn`1q, thus, the
+claim follows by Lemma 5.2.
+Now, suppose that ln ă ℓ ă ln ` ln´1 then by (5.5) we get that σℓ´lnMpn´1q is the prefix
+of σℓMpn`1q, and since Mpn´1q is a prefix of Mpn`1q, the claim follows by Lemma 5.4 and the
+induction condition.
+If ℓ “ ln ` ln´1 then by (5.5) Mpn´1qMpn´1q is a prefix of σℓMpn`1q, thus, the claim again
+follows by Lemma 5.2.
+
+16
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+Finally, if ln ` ln´1 ă ℓ ă ln ` 2ln´1 “ ln`1 then by (5.5), σℓ´ln´ln´1Mpn´1q is a prefix of
+σℓMpn`1q, so the claim follows again by the induction condition and Lemma 5.4.
+□
+Proof of Proposition 5.3. For n “ 1 and n “ 2, let
+mp1q “ p2, 2, . . .q and mp2q “ p3, 3, . . .q.
+Since the maps Tm are contractions, we get
+lim inf
+kÑ8 Tmpnq
+rk,1sp0q “ lim
+kÑ8 Tmpnq
+rk,1sp0q “ λn,
+and so, tλ1, λ2u Ă L.
+For every n ě 3 integer, let us define the following sequence:
+mpnq “ pMpnq
+ln , . . . , Mpnq
+1 , Mpnq
+ln , . . . , Mpnq
+1 , . . .q.
+Then by Lemma 2.2, Tmpnq
+rk,1sp0q P Ťln´1
+ℓ“0 TσℓMpnqpJq for every sufficiently large k. Since the maps
+Tm are contractions, we get
+lim
+kÑ8 Tmpnq
+rkln,1sp0q “ λn P TMpnqpJq,
+furthermore, by Lemma 5.5, TMpnqpJq ă TσℓMpnqpJq for every 1 ď ℓ ď ln ´ 1 and so
+λn ă Tmpnq
+rk,1sp0q for every k R tln, 2ln, . . .u.
+Hence, lim infkÑ8 Tmpnq
+rk,1sp0q “ λn.
+The last claim follows by Theorem 2.1 and the fact that λn converges to λ0 as n Ñ 8.
+□
+Proposition 5.6. pλ1, 8q X L “ H, and for every n ě 1, pλn`1, λnq X L “ H.
+Proof. First, observe that max L “ λ1. Indeed, this follows by Lemma 4.3(2) with K “ 2, which
+implies the first claim.
+Let us show that pλn`1, λnq X L “ H. Contrary, let us assume that there exists an integer n ě 1
+and x P pλn`1, λnq X L.
+Since λn is the fixed point of TMpnq, we have λn P TMpnqMpnqpJq Ă TMpnqpJq. Since L Ă Λ,
+where Λ is the attractor of tT2, T3u, and midpTMpn`1qpJq, TMpnqMpnqpJqq Ă pλn`1, λnq, we get
+by Lemma 5.2 that either x P TMpnqMpnqpJq or x P TMpn`1qpJq. But by Lemma 5.1(2), if x P
+TMpnqMpnqpJq then x ě λn, while if x P TMpn`1qpJq then x ď λn`1 by Lemma 5.1(1), which is a
+contradiction.
+□
+Proof of Theorem 2.5. The statement follows by combining Proposition 5.3 and Proposition 5.6.
+□
+
+LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS
+17
+6. Proof of Theorem 2.4
+Proof. It follows from Lemma 4.3(2) that if mn ě 5 for infinitely many n, then lim infkÑ8 Tmrk,1sp0q
+ď 1
+6, thus we may assume without loss of generality that pmiq P t2, 3, 4uZ`.
+If pmk, mk´1q “ p4, 2q for infinitely many k, then pmk, mk´1, mk´2q “ p4, 2, aq for an a P
+t2, 3, 4u and for infinitely many k. By Lemma 5.1(1)
+lim inf
+kÑ8 T p1q
+mrk,1sp0q ď maxtFixpT4 ˝ T2 ˝ Taq : a P t2, 3, 4uu “ 3
+17,
+so we may assume that pmk, mk´1q ‰ p4, 2q for every k. Thus,
+L X
+ˆ 3
+17, 1
+3
+
+Ă tlim inf
+nÑ8 Tmrn,1sp0q : mi P t2, 3, 4u, pmi, mi´1q ‰ p4, 2qu “: S.
+If mi P t2, 3, 4u and pmi, mi´1q ‰ p4, 2q, then there are 55 possibilities for pmi, mi´1, mi´2, mi´3q:
+A “ tp2, 2, 2, 2q, p2, 2, 2, 3q, p2, 2, 2, 4q, p2, 2, 3, 2q, p2, 2, 3, 3q, p2, 2, 3, 4q, p2, 2, 4, 3q, p2, 2, 4, 4q,
+p2, 3, 2, 2q, p2, 3, 2, 3q, p2, 3, 2, 4q, p2, 3, 3, 2q, p2, 3, 3, 3q, p2, 3, 3, 4q, p2, 3, 4, 3q, p2, 3, 4, 4q,
+p2, 4, 3, 2q, p2, 4, 3, 3q, p2, 4, 3, 4q, p2, 4, 4, 3q, p2, 4, 4, 4q, p3, 2, 2, 2q, p3, 2, 2, 3q, p3, 2, 2, 4q,
+p3, 2, 3, 2q, p3, 2, 3, 3q,p3, 2, 3, 4q, p3, 2, 4, 3q, p3, 2, 4, 4q, p3, 3, 2, 2q, p3, 3, 2, 3q, p3, 3, 2, 4q,
+p3, 3, 3, 2q, p3, 3, 3, 3q, p3, 3, 3, 4q, p3, 3, 4, 3q p3, 3, 4, 4q, p3, 4, 3, 2q, p3, 4, 3, 3q, p3, 4, 3, 4q,
+p3, 4, 4, 3q, p3, 4, 4, 4q, p4, 3, 2, 2q, p4, 3, 2, 3q, p4, 3, 2, 4q, p4, 3, 3, 2q, p4, 3, 3, 3q, p4, 3, 3, 4q,
+p4, 3, 4, 3q, p4, 3, 4, 4q, p4, 4, 3, 2q, p4, 4, 3, 3q, p4, 4, 3, 4q, p4, 4, 4, 3q, p4, 4, 4, 4qu.
+Let us consider the IFS Φ “ tTa ˝Tb ˝Tc ˝Td|pa, b, c, dq P Au and let Λ1 be the attractor of Φ. Then
+by Lemma 2.2, S Ă Λ1. Moreover, it is easy to check that the sum of the 55 contractions strictly
+less than 1, therefore by Lemma 2.1, λpΛ1q “ 0, and so λpL X r 3
+17, 1
+3sq “ 0.
+□
+REFERENCES
+[1] Y.G. Chen, J.H. Fang, On additive complements. II, Proc. Amer. Math. Soc. 139 (2011), 881-883.
+[2] T. W. Cusick and M. E. Flahive, The Markoff and Lagrange spectra. Mathematical Surveys and Monographs, 30.
+American Mathematical Society, Providence, RI, 1989.
+[3] D. Lima, C. Matheus, C.G. Moreira, S. Vieira, MzL is not closed. Int. Math. Res. Not. IMRN 2022(1) (2022),
+265-311.
+[4] K. Falconer, Fractal geometry. Mathematical foundations and applications. Third edition. John Wiley & Sons,
+Ltd., Chichester, 2014.
+[5] J.H. Fang, C. S´andor, On sets with sum and difference structure, arXiv:2205.06553.
+[6] G.A. Freiman, Non-coincidence of the spectra of Markov and of Lagrange. (Russian) Mat. Zametki 3 (1968),
+195-200.
+[7] A. Hurwitz, ¨Uber die angen¨aherte Darstellung der Irrationalzahlen dureh rationale Br¨uche, Math. Ann. 39 (1891),
+279-284.
+[8] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30(5) (1981), 713-747.
+[9] A. Y. Khinchin, Continued fractions. With a preface by B. V. Gnedenko. Translated from the third (1961) Russian
+edition. Reprint of the 1964 translation. Dover Publications, Inc., Mineola, NY, 1997.
+[10] X.Y. Liu, J.H. Fang, A note on additive complements, Adv. Math. (China) 45 (2016), 533-536.
+
+18
+BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚
+[11] A. Markov, Sur les formes quadratiques binaires ind´efinies, Math. Ann. 15 (1879), 381-406.
+[12] F.Y. Ma, A note on additive complements, arXiv:2205.04128.
+[13] C. G. Moreira, Geometric properties of the Markov and Lagrange spectra, Annals Math. 188(1) (2018), 145-170.
+[14] J. Parkkonen, F. Paulin, On the closedness of approximation spectra. J. Th´eor. Nombres Bordeaux 21 (2009),
+701-710.
+[15] O. Perron, ¨Uber diophantische Approximationen. Math. Ann. 83 (1921), 77-84.
+1DEPARTMENT OF STOCHASTICS, INSTITUTE OF MATHEMATICS, BUDAPEST UNIVERSITY OF TECHNOLOGY
+AND ECONOMICS, M ˝UEGYETEM RKP. 3., H-1111 BUDAPEST, HUNGARY
+Email address: balubs@math.bme.hu
+Email address: csandor@math.bme.hu
+2SCHOOL OF MATHEMATICAL SCIENCES, NANJING NORMAL UNIVERSITY, NANJING 210023, PR CHINA
+Email address: fangjinhui1114@163.com
+
diff --git a/pdE3T4oBgHgl3EQfLgnu/content/tmp_files/load_file.txt b/pdE3T4oBgHgl3EQfLgnu/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..58b862baea98e87a5ee7e79fc306f0d049f7966c
--- /dev/null
+++ b/pdE3T4oBgHgl3EQfLgnu/content/tmp_files/load_file.txt
@@ -0,0 +1,1106 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf,len=1105
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='04365v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='NT]  11 Jan 2023  LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Two infinite sets A and B of non-negative integers are called perfect additive comple-  ments of non-negative integers, if every non-negative integer can be uniquely expressed as the sum  of elements from A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' In this paper, we define a Lagrange-like spectrum of the perfect additive  complements (LSPAC for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' As a main result, we obtain the smallest accumulation point of the  set LSPAC and prove that the set LSPAC is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Other related results and problems are also  contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Introduction  Let Z be the set of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For nonempty sets A, B of integers and an integer n, let rA,Bpnq  be the number of representations of n as a ` b, where a P A and b P B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Two infinite sets A  and B of non-negative integers are called perfect additive complements of non-negative integers,  if rA,Bpnq “ 1 for every non-negative integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For a non-negative integer m, denote by Zěm the  set of non-negative integers no less than m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For simplicity, we also denote Zě1 by Z`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  In [5], Fang and S´andor characterized the perfect additive complements A, B of non-negative  integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1] The infinite sets A, B of the non-negative integers form perfect  additive complements if and only if  A “ tǫ0 ` ǫ2m1m2 ` ¨ ¨ ¨ ` ǫ2k´2m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' m2k´2 ` ¨ ¨ ¨ : ǫ2i “ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , m2i`1 ´ 1u and  B “ tǫ1m1 ` ǫ3m1m2m3 ` ¨ ¨ ¨ ` ǫ2k´1m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' m2k´1 ` ¨ ¨ ¨ : ǫ2i´1 “ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , m2i ´ 1u  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1)  (or A, B interchanged), where mi P Zě2 for every i P Z`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let S be a set of non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Its counting function is defined by Spxq “ |S X r0, xs|  for every x P Zě0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' It is easy to see that if A, B Ď Zě0 form perfect additive complements then  ApxqBpxq ě x ` 1 for every non-negative integer x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' In particular, Fang and S´andor showed that  Date: January 12, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Primary 11B34, Secondary 11J06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' additive complements, Lagrange spectrum, Lebesgue-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='H Fang is supported by the National Natural Science Foundation of China, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 12171246 and the Natural  Science Foundation of Jiangsu Province, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' BK20211282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' B´ar´any acknowledges support from the grant  NKFI FK134251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' B´ar´any and Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' S´andor was supported by the grant NKFI KKP144059 ”Fractal geometry and  applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  1  2  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  lim inf  xÑ8  ApxqBpxq  x  “ 1, see [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Recently, Ma [12] determined the lim sup  xÑ8  ApxqBpxq  x  for the sets A and B with the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' [12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1] Let m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' be arbitrary integers no less than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then the  sets A and B with the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1) are perfect additive complements of non-negative integers such  that  lim sup  xÑ8  ApxqBpxq  x  “ lim sup  kÑ8  2  1 ` Dk  ,  where  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2)  Dk “ 1  mk  ´  1  mkmk´1  `  1  mkmk´1mk´2  ´ ¨ ¨ ¨ ` p´1qk´1  1  mkmk´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' m1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  In this paper, we consider the properties of the set called Lagrange spectrum of perfect additive  components  LSPAC :“  "  lim sup  kÑ8  2  1 ` Dk  : pmiq P ZZ`  ě2 ,  where Dk is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' In 2011, Chen and Fang [1, Theorem 1] obtained that  2a ` 2  a ` 2 P LSPAC for any integer a with a ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  In 2016, Liu and Fang [10, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1] extended this result by showing that  2  a´1  ab´1 ` 1 P LSPAC for any integers a, b with 2 ď a ď b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Recently, Ma [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2] proved that  2 P LSPAC and  ˆˆ16  9 , 2  ˙  zQ  ˙  X LSPAC ‰ H,  where Q denotes the set of rationals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Fang and S´andor [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5] showed that  LSPAC Ď  „3  2, 2  \uf6be  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  The main theorem of this paper can be summarized as follows:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (1) The set LSPAC is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (2) There exists γ0 P LSPAC, which is the smallest accumulation point of LSPAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Fur-  thermore, the set r 3  2, γ0q X LSPAC “ tγ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='u, where γn is a monotone increasing  sequence converging to γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (3)  “ 7  4, 2  ‰  Ď LSPAC but  “ 12  7 ´ δ, 2  ‰  Ę LSPAC for any δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS  3  We will determine the constant γ0 and the set r 3  2, γ0s X LSPAC in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1(2) precisely  later, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  It follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1 that the set LSPAC has some similar properties to the so-called  Lagrange spectrum LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let α be a positive irrational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Define kpαq “ lim sup  n,mÑ8  1  |n2α ´ nm|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Hurvitz [7] proved that kpαq ě  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  5 for every positive irrational number α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' The Lagrange spectrum  LS :“ tkpαq : α is a positive irrational numberu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  For results related to Lagrange spectrum, one may refer to [2], [3], [6], [11], [13] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  It is well known that the Lagrange spectrum is closed, see [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2], furthermore, the  least accumulation point of the Lagrange spectrum is 3 and l P L, l ă 3 if and only if l “  b  9 ´ 4  z2n,  where zn’s are the Markov integers, see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' The corresponding phenomena for the Lagrange-like  spectrum of perfect additive complement follows by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1(1) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Furthermore, Freiman’s constant F “  2221564096`283748  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  463  491993569  “ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='527 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' is the name of the  supremum of the set RzLS, that is rF, 8q Ă LS, but for any δ ą 0, rF ´ δ, 8q Ć LS, see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  In point of view of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1(3), the LSPAC has also a Freiman-like constant, namely, there  exists 12  7 ď c0 ď 7  4 such that  c0 “ inftc P R : rc, 2s Ă LSPACu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Determine the exact value of c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Is it true that c0 “ 7{4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  There is another important similarity between the sets LS and LSPAC, namely, both can be  represented by using infinite iterated function systems (IFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' It is well known that every α can be  written as a simple infinite continued fraction  α “ m0 `  1  m1 `  1  m2`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  “: rm0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' s,  where mi P Z`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' On the other hand if mi P Z`, then the above continued fraction defines a positive  irrational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let us define a map Gmpxq “  1  m`x for every integer m P Z`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then  rm0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' s “ m0 ` lim  kÑ8 Gm1 ˝ ¨ ¨ ¨ ˝ Gmkp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  If  1  |n2α´nm| ą 2, then there exists a k such that m  n “ pk  qk “ rm0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mks, see for example [9,  Theorem 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Hence kpαq “ lim supkÑ8  1  |p2  kα´pkqk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' In 1921, Perron [15] proved  1  |p2  kα ´ pkqk| “ r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' mk, mk´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , m1s ` rmk`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' mk`2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  In particular,  LS “  "  lim sup  kÑ8  ´  Gmk ˝ ¨ ¨ ¨ ˝ Gm1p0q ` mk`1 ` lim  ℓÑ8 Gmk`2 ˝ ¨ ¨ ¨ ˝ Gmℓp0q  ¯  |pmiq P ZZ`  ě1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  4  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  Now, let us define the maps pGmpxq “  2mx  pm`2qx´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By Theorem B, we will show later that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3)  LSPAC “  "  lim sup  kÑ8  pGmk ˝ ¨ ¨ ¨ ˝ pGm1p2q|pmiq P ZZ`  ě2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Moreira [13] showed that the map α ÞÑ dimH  `  r  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  5, αs X LS  ˘  “ dimB  `  r  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  5, αs X LS  ˘  is  monotone increasing and continuous on r  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  5, 8q, where dimH denotes the Hausdorff dimension  and dimB denotes the upper box-counting dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For the definition and basic properties of the  Hausdorff- and box-counting dimension we refer to [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Is dimH  `  r 3  2, αs X LSPAC  ˘  “  dimB  `  r 3  2, αs X LSPAC  ˘  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Is the map  α ÞÑ dimH  `  r 3  2, αs X LSPAC  ˘  continuous?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Preliminaries  In this section, we summarize some basic facts in the theory of iterated function systems relevant  for our later calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' We say that a map f : R ÞÑ R is contracting if there exists a constant  0 ă c ă 1 such that |fpxq ´ fpyq| ď c|x ´ y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By Banach’s fixed point theorem, every contractive  map f has a unique fixed point x “ fpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For a contractive map f, let us denote its unique fixed  point by Fixpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let Ψ “ tf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , fnu be a finite collection of contractions, which we call iterated function  system (IFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Hutchinson [8] showed that there exists a unique non-empty compact set Λ such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1)  Λ “  nď  i“1  fipΛq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  The set Λ is called the attractor of the IFS Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' In particular, if B Ă R is a compact set such that  fipBq Ď B for every i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , n then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2)  Λ “  8  č  k“1  ď  pi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=',ikqPt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=',nuk  fi1 ˝ ¨ ¨ ¨ ˝ fikpBq Ă B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2), one can prove the following simple observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let Ψ “ tf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , fnu be a finite collection of contractions such that the contracting  ratio of fi is ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' If řn  i“1 ci ă 1 then λpΛq “ 0, where λ denotes the Lebesgue measure on the real  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Since |fipxq ´ fipyq| ď ci|x ´ y| then λpfi1 ˝ ¨ ¨ ¨ ˝ fikpBqq ď ci1 ¨ ¨ ¨ cinλpBq and so, by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2),  λpΛq ď  ÿ  pi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=',ikqPt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=',nuk  λpfi1 ˝ ¨ ¨ ¨ ˝ fikpBqq “  ˜ nÿ  i“1  ci  ¸k  λpBq Ñ 0 as k Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS  5  Let us denote the distance between sets by dist,  that is,  for A, B  Ď  R,  let  distpA, Bq “ inft|x ´ y|  ˇˇx P A, y P Bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' With a slight abuse of notation, we write distpx, Aq “  distptxu, Aq for the distance of a point x P R and a set A Ď R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let Ψ “ tf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , fnu be a finite collection of contractions such that the contracting  ratio of every fi is at most c P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For every sequence pi1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , nuZ` and every  x P R, lim infkÑ8 fik ˝¨ ¨ ¨˝fi1pxq P Λ, where Λ is the attractor of Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' In particular, for every open  set U Ą Λ, for every x P R and for every sufficiently large k, fik ˝ ¨ ¨ ¨ ˝ fi1pxq P U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1)  distpfik ˝ ¨ ¨ ¨ ˝ fi1pxq, Λq ď distpfik ˝ ¨ ¨ ¨ ˝ fi1pxq, fik ˝ ¨ ¨ ¨ ˝ fi1pΛqq ď ckdistpx, Λq Ñ 0 as k Ñ 8,  where 0 ă c ă 1 is chosen such that |fipxq´fipyq| ď c|x´y| for every i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , n and x, y P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  The claim then follows by the compactness of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  For every point x P Λ, there exists an infinite sequence i “ pi1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , nuZ` such that  x “ lim  kÑ8 fi1 ˝ ¨ ¨ ¨ ˝ fikp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Observe that the limit on the right-hand side exists since the maps fi are contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' One can  define a map Π: t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , nuZ` ÞÑ Λ by  Πpiq :“ lim  kÑ8 fi1 ˝ ¨ ¨ ¨ ˝ fikp0q  called the natural projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let σ: t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , nuZ` ÞÑ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , nuZ` be the left-shift operator, that  is,  σpi1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q “ pi2, i3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Hence, by using the definition of the natural projection Π it is easy to see that  Πpiq “ fi1pΠpσiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Now, let us define a specific family of contractive maps on R as Tmpxq “ 1´x  m for m P Zě2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Then clearly for every pmiq P ZZ`  ě2  Tmk ˝ Tmk´1 ˝ ¨ ¨ ¨ ˝ Tm1p0q “ 1  mk  ´  1  mkmk´1  `  1  mkmk´1mk´2  ´ ¨ ¨ ¨ ` p´1qk´1  1  mkmk´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' m1  ,  which corresponds to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let  L “  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  lim inf  kÑ8 Tmk ˝ ¨ ¨ ¨ ˝ Tm1p0q|pmiq P ZZ`  ě2  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Hence,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3)  LSPAC “ gpLq,  where gpxq “  2  1`x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Furthermore, pGmpxq “ g ˝ Tm ˝ g´1, thus, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Hence, our main  theorem will follow from the following theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' The set L is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  6  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' r0, 1  7s Ă L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  L X  8  ď  n“0  ˆ1  6 ` 1  93  1  4n, 1  6 ` 1  84  1  4n  ˙  “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let S Ă R be a Lebesgue-measurable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' The Lebesgue-measure of S will be denoted by λpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' λ  `  L X r 3  17, 1  3s  ˘  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  We introduce the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let i “ pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , inq P Zn  ě2 be a finite word, then denote  by Ti the map  Ti “ Ti1 ˝ ¨ ¨ ¨ ˝ Tin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let u, v  be positive integers and m  “  pmiq  P  ZZ`  ě2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  If u  ď  v  then let  Tmru,vspxq  “  pTmu  ˝  Tmu`1  ˝  ¨ ¨ ¨  ˝  Tmvqpxq,  and  if  u  ą  v  then  let  Tmru,vspxq “ pTmu ˝ Tmu´1 ˝ ¨ ¨ ¨ ˝ Tmvqpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Finally, let us introduce the notation that for any  sequence m P ZZ`  ě2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4)  Πpmq “ lim  kÑ8 Tm1 ˝ ¨ ¨ ¨ Tmkp0q “ lim  kÑ8 Tmr1,ksp0q “  8  ÿ  k“1  p´1qk´1  m1 ¨ ¨ ¨ mk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let us define the sequences Mpnq recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let Mp1q “ 2, Mp2q “ 3 and let Mpnq be the con-  catenation Mpnq “ Mpn´1qMpn´2qMpn´2q for n ě 3, that is Mp3q “ p3, 2, 2q, Mp4q “ p3, 2, 2, 3, 3q  and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By the definition of Mpnq, it is easy to see that the length of the finite sequence Mpnq  is ln “ 2n´p´1qn  3  , and Mpnq starts with Mpn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Thus, it is possible to define the limiting infinite  sequence M “ limnÑ8 Mpnq as  M “ p3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 2, 3, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q “: pM1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q  such that pM1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , Mlnq “ pMpnqq for every positive integer n ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let  λn “ FixpTMpnqq “ TMpnqp0q  M1M2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Mln  M1M2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Mln ` 1,  and  λ0 “  8  ÿ  l“1  p´1ql´1  1  M1M2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Ml  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2293 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='.  We will prove that λn is a strictly increasing sequence, λn ą λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  λ P L,  λ ą λ0 if and only if λ “ λn for some n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' The first claim follows by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3), the fact the map gpxq “  2  1`x is continuous  on R` and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' The second claim follows by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5 with the choices γn “ gpλnq  for n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Finally, the last claim follows by the combination of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Closedness of the spectrum  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let αn P L be a sequence such that lim  nÑ8 αn “ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Hence, for every n ě 1  there exists mpnq P ZZ`  ě2, mpnq “ pmpnq  1 , mpnq  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' q such that lim inf  kÑ8 Tmpnq  rk,1sp0q “ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let εn “  |α ´ αn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Without loss of generality we may assume that εn Œ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let l1 “ 0 and let us choose k1 such that |Tmp1q  rk1,1sp0q ´ α1| ă ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  For n ě 2, let 0 ă ln ă kn be such that |Tmpnq  rln,1sp0q ´ αn| ă εn, |Tmpnq  rkn,1sp0q ´ αn| ă εn,  T p1q  mpnq  rl,1sp0q ą αn ´ εn for every l ě ln and 5εn´1  εn  ă 2kn´ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let  m “ pmp1q  l1`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mp1q  k1 , mp2q  l2`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mp2q  k2 , mp3q  l3`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mp3q  k3 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' q “ pm1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  We will show that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1)  α “ lim inf  kÑ8 Tmrk,1sp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let aN “ řN  n“1pkn ´ lnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' To verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1), it is enough to prove that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2)  |TmraN ,1sp0q ´ αN| ă 2εN for every N ě 1  and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3)  Tmrl,1sp0q ą αN ´ 3εN for every aN ă l ď aN`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Indeed, in this case  lim  NÑ8 TmraN ,1sp0q “ α and lim inf  lÑ8 Tmrl,1sp0q ě lim  NÑ8 αN ´ 3εN “ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2) we argue by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Clearly,  |Tmra1,1sp0q ´ α1| “ |Tmp1q  rk1,1sp0q ´ α1| ă ε1 ă 2ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Suppose that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2) holds for N ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then  |TmraN ,1sp0q ´ αN| ď |TmraN ,1sp0q ´ TmpNq  rkN ,1sp0q| ` |TmpNq  rkN ,1sp0q ´ αN|  “  1  mpNq  kN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' mpNq  lN`1  |TmraN´1,1sp0q ´ TmpNq  rlN ,1sp0q| ` |TmpNq  rkN ,1sp0q ´ αN|  ď  1  2kN´lN  ˆ  |TmraN´1,1sp0q ´ αN´1| ` |αN´1 ´ α| ` |α ´ αN| ` |TmpNq  rlN ,1sp0q ´ αN|  ˙  ` εN  ă  1  2kN´lN p2εN´1 ` εN´1 ` εN ` εNq ` εN ă  1  2kN´lN 5εN´1 ` εN ă 2εN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3) we write  Tmrl,1sp0q “ Tmrl,aN `1s ˝ TmraN ,1sp0q “ TmpNq  rl´aN `lN ,lN `1s ˝ TmraN ,1sp0q  “ TmpNq  rl´aN `lN ,lN `1s ˝ TmraN ,1sp0q ´ TmpNq  rl´aN `lN ,lN `1s ˝ TmpNq  rlN ,1sp0q ` TmpNq  rl´aN `lN ,lN `1s ˝ TmpNq  rlN ,1sp0qq,  8  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  where  ˇˇˇˇTmpNq  rl´aN `lN ,lN `1s ˝ TmraN ,1sp0q ´ TmpNq  rl´aN `lN ,lN `1s ˝ TmpNq  rlN ,1sp0q  ˇˇˇˇ  “  1  mpNq  l´aN `lN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' mpNq  lN`1  ˇˇˇˇTmraN ,1sp0q ´ TmpNq  rlN ,1sp0q  ˇˇˇˇ  ď  1  2l´aN p|TmraN ,1sp0q ´ αN| ` |αN ´ TmpNq  rlN ,1sp0q|q ă 1  2p2εN ` εNq ă 2εN  and  TmpNq  rl´aN `lN ,lN `1s ˝ TmpNq  rlN ,1sp0q “ TmpNq  rl´aN `lN ,1sp0q ą αN ´ εN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Hence,  T p1q  mrl,1sp0q ą αN ´ εN ´ 2εN “ αN ´ 3εN,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Estimates on the Freiman-like constant  Let us consider the finite IFS Ψ4 “ tT2, T3, T4u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let I “  “ 1  7, 3  7  ‰  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For m ě 2, TmpIq “ r 4  7m,  6  7ms,  and  I “  4ď  m“2  TmpIq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1)  Thus, by the uniqueness the attractor of Ψ4 is I “  “ 1  7, 3  7  ‰  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1) and direct calculation, we obtain  the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For every z P r 1  7, 3  7s and K P t2, 3, 4u, we have 1  K ´ 1  K z P r 1  7, 3  7s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For every y P r 1  7, 3  7s, there exist K P t2, 3, 4u and z P r 1  7, 3  7s such that y “ 1  K ´ 1  Kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  In particular, it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2 that for every y P r 1  7, 3  7s, there exists an infinite sequence  pK1, K2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q P t2, 3, 4uZ` such that  ΠpK1, K2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q “ y,  where Π is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' It infers from  4  7m ď  6  7pm`1q for m ě 6 that  8  ď  m“6  TmpIq “  ˆ  0, 1  7  \uf6be  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2)  Suppose that 0 ă x ă 1  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2), we know that the real x can be written as x “  1  m ´ 1  my,  where m ě 6 and y P r 1  7, 3  7s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' It follows that there exist sequences K1, K2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , Kk, ¨ ¨ ¨ P t2, 3, 4u  LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS  9  and z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , zk, ¨ ¨ ¨ P r 1  7, 3  7s such that  y  “  ΠpK1, K2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q “  8  ÿ  k“1  p´1qk´1  K1K2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Kk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Now, let  m “ pm1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' q “ p3, K1, m, 3, K2, K1, m, 3, K3, K2, K1, m, 3, K4, K3, K2, K1, m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  We will prove that  x “ lim inf  nÑ8 Tmrn,1sp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  First, observe that Tmrn,1sp0q P  “  0, 1  2  ‰  for every n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Indeed, Tmp  “  0, 1  2  ‰  q Ă  “  0, 1  2  ‰  for every  m ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' On the other hand, since T3  `“  0, 1  2  ‰˘  Ă  “ 1  6, 1  3  ‰  Ă  “ 1  7, 3  7  ‰  , we have that Tmrn,1sp0q P  “ 1  7, 3  7  ‰  for every n ě 1 with mn “ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Hence, it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1) that if mn ‰ m, then  Tmrn,1sp0q P r 1  7, 3  7s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Simple calculations shows that mk “ m if and only if k “ n2`5n  2  for some n P Z`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Furthermore,  it is easy to see that  Tm  r n2`5n  2  ,1sp0q  “  1  m ´ 1  m  ˆ 1  K1  ´  1  K1K2  ` ¨ ¨ ¨ `  p´1qn´1  K1K2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Kn  `  p´1qn  K1 ¨ ¨ ¨ Kn  Tm  r pn´1q2`5pn´1q  2  `1,1sp0q  ˙  “  1  m ´ 1  my ` Op 1  2nq “ x ` Op 1  2nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  This completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  Before we continue, we state a lemma on the position of the possible smallest accumulation  points depending on the defining sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (1) Let pmiq P ZZ`  ě2 be such that mi P t2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , Ku except at most finitely many i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then  1  2K ´ 1 ď lim inf  nÑ8 Tmrn,1sp0q ď lim sup  nÑ8 Tmrn,1sp0q ď K ´ 1  2K ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (2) Let K P Zě2 and pmiq P ZZ`  ě2 such that mi ě K for infinitely many integer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then  lim inf  kÑ8 Tmrk,1sp0q ď  1  K ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' To prove the first claim, it is enough to show that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3)  Tm  ˆ„  1  2K ´ 1, K ´ 1  2K ´ 1  \uf6be˙  Ď  „  1  2K ´ 1, K ´ 1  2K ´ 1  \uf6be  for every 2 ď m ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Indeed,  Tm  ˆ„  1  2K ´ 1, K ´ 1  2K ´ 1  \uf6be˙  “  „  K  mp2K ´ 1q,  2K ´ 2  mp2K ´ 1q  \uf6be  ,  where  1  2K´1 ď  K  mp2K´1q if and only if m ď K and  2K´2  mp2K´1q ď  K´1  2K´1 if and only if m ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  10  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  Then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2, we get that lim infnÑ8 Tmrn,1sp0q P  “  1  2K´1, K´1  2K´1  ‰  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  To show the last claim, let us argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' If lim infkÑ8 Tmrk,1sp0q ą  1  K`1, then there  exists a δ ą 0 such that Tmrn,1sp0q ą  1  K`1 ` δ for every sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then  Tmrn,1sp0q “ 1  mn  ´ 1  mn  Tmrn´1,1sp0q ă 1  mn  ´ 1  mn  p  1  K ` 1 ` δq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Hence, for every sufficiently large n we have that  1  K`1 ` δ ă  1  mn ´  1  mnp  1  K`1 ` δq, equivalently  1  K`1 ` δ ă  1  mn`1 for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Thus, mn ď K ´ 1 for every sufficiently large n, which  is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  Finally, let us state a technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let pmiq P t2, 3, 4uZ` such that if pmi, mi´1q “ p4, 2q then mi´2 “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then  lim supnÑ8 Tmrn,1sp0q ď 13  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Observe that  min T2 ˝ T4 ˝ T2  ˆ„1  7, 3  7  \uf6be˙  ě max  ¨  ˚  ˚  ˝  ď  i,j,kPt2,3,4u3  pi,j,kq‰p2,4,2q  Ti ˝ Tj ˝ Tk  ˆ„1  7, 3  7  \uf6be˙  ˛  ‹‹‚,  where we recall that r 1  7, 3  7s is the attractor of the IFS tT2, T3, T4u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Thus, if pmi, mi´1, mi´2, mi´3q “  p2, 4, 2, 2q only for finitely many i then  lim sup  nÑ8 Tmrn,1sp0q ď T2 ˝ T4 ˝ T2 ˝ T2  ˆ1  7  ˙  “ 23  56 ă 13  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  On the other hand, if pmi, mi´1, mi´2, mi´3q “ p2, 4, 2, 2q for infinitely many i then  lim sup  nÑ8 Tmrn,1sp0q ď T2 ˝ T4 ˝ T2 ˝ T2  ˆ  lim sup  nÑ8 Tmrn,1sp0q  ˙  ,  which implies after some algebraic manipulations that lim supnÑ8 Tmrn,1sp0q ď 13  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3(2) that if mn ě 5 for infinitely many n, then  lim inf  kÑ8 Tmrk,1sp0q ď 1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  So we may assume without loss of generality that mn P t2, 3, 4u for every n P Z`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Direct computations show that  ˆ1  6, 1  6 ` 1  84  ˙  X  ď  pk,lqPt2,3,4u2  pk,lq‰p4,2q  Tk ˝ Tl  ˆ„1  7, 3  7  \uf6be˙  “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS  11  Thus, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2 and the fact that  “ 1  7, 3  7  ‰  is the attractor of the IFS tT2, T3, T4u we get that if  lim inf  kÑ8 Tmrk,1sp0q P L X  ˆ1  6, 1  6 ` 1  84  ˙  with the sequence pmiq P t2, 3, 4uZ` then pmi, mi´1q “ p4, 2q for infinitely many i P Z`, and in  particular,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4)  lim inf  kÑ8 Tmrk,1sp0q “ lim inf  ℓÑ8 Tmrkℓ,1sp0q,  where k1 “ minti ě 2 : pmi, mi´1q “ p4, 2qu and kℓ “ minti ą kℓ´1 : pmi, mi´1q “ p4, 2qu for  all ℓ ě 2  If pmkℓ, mkℓ´1, mkℓ´2q “ p4, 2, bq for some b P t3, 4u for infinitely many i then 1  6 ě Tmrkℓ,kℓ´2sp0q ě  Tmrkℓ,1sp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' So we may assume that if  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5)  pmi, mi´1q “ p4, 2q then mi´2 “ 2 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Furthermore, if for every N there exist infinitely many k ě 2N ` 2 such that  pmk, mk´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mk´2N´1q “ p4, 2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=', 2q  then  lim inf  kÑ8 Tmrk,1sp0q ď lim  NÑ8 T4 ˝  2N`1-times  hkkkkkikkkkkj  T2 ˝ ¨ ¨ ¨ ˝ T2p0q “ 1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Hence, we may assume that there exists a non-negative integer N0 such that  pmkℓ, mkℓ´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mkℓ´2N0´1q “ p4,  2N0 ` 1-times  hkkkkikkkkj  2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , 2q for infinitely many ℓ but  pmkℓ, mkℓ´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mkℓ´2N0´3q “ p4,  2N0 ` 3-times  hkkkkikkkkj  2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , 2q only for a finite number of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='6)  Let us suppose that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For short, let pN0 “ pmkℓ, mkℓ´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mkℓ´2N0´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then by  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4 and the fact that the maps Tm are orientation reversing we get  lim inf  ℓÑ8 Tmrkℓ,1sp0q ď TpN0plim sup  kÑ8  Tmrk´2N0´2,1sp0qq  “ 1  8  1 ´  1  4N0`1  1 ´ 1  4  `  1  8 ¨ 4N0 ¨ 13  31 “ 1  6 ` 1  93  1  4N0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='7)  If pmkℓ, mkℓ´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mkℓ´2N0´2q “ p4, 2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=', 2, aq, where a P t3, 4u then  Tmrkℓ,1sp0q ď Tmrkℓ,kℓ´2N0´2sp0q “ TpN0 ˝ Tap0q ď TpN0p1  3q “ 1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  12  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  Hence, we may suppose that pmkℓ´2N0´2, mkℓ´2N0´3q P tp2, 3q, p2, 4qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3(1)  and the fact that the maps Tm are orientation reversing we get  lim inf  ℓÑ8 Tmrkℓ,1sp0q ě TpN0 ˝ T2 ˝ Taplim inf  kÑ8 Tmrk´2N0´3,1sp0qq  ě TpN0 ˝ T2 ˝ Tap1  7q ě TpN0 ˝ T2p2  7q “ 1  6 ` 1  84  1  4N0`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='8)  Finally, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='8) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='7) implies that  1  6 ` 1  84  1  4N0`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' ď lim inf  kÑ8 Tmrk,1sp0q ď 1  6 ` 1  93  1  4N0 ,  and so lim infkÑ8 T p1q  mrk,1sp0q R Ť8  n“0  `1  6 ` 1  93  1  4n, 1  6 ` 1  84  1  4n  ˘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Computation of the Markov-like constant  Throughout this section, we will consider the set L X  `1  5, 1  2  ˘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3(2), for every  x P L X  ` 1  5, 1  2  ˘  if x “ lim infnÑ8 Tmrn,1sp0q “ x then pmiq P t2, 3uZ`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3),  T2  ˆ„1  5, 2  5  \uf6be˙  Y T3  ˆ„1  5, 2  5  \uf6be˙  Ď  „1  5, 2  5  \uf6be  ,  and so by denoting the attractor of the IFS tT2, T3u by Λ Ă  “ 1  5, 2  5  ‰  , we get by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2 that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1)  L X  „1  5, 2  5  \uf6be  Ă Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Furthermore, direct computations show that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2)  T2  ˆ„1  5, 2  5  \uf6be˙  X T3  ˆ„1  5, 2  5  \uf6be˙  “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Hence, by choosing δ “ dist  `  T2  `“ 1  5, 2  5  ‰˘  , T3  `“ 1  5, 2  5  ‰˘˘  {3 ą 0, we get  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3)  T2 pJq X T3 pJq “ H and T2 pJq Y T3 pJq Ď J,  where J “  “ 1  5 ´ δ, 2  5 ` δ  ‰  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (1) Let pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , i2k`1q P ZZ`  ě2 be for some non-negative integer k, and let pmiq P ZZ`  ě2 be such  that pmj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mj´2kq “ pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , i2k`1q for infinitely many j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then  lim inf  nÑ8 Tmrn,1sp0q ď FixpTi1 ˝ ¨ ¨ ¨ ˝ Ti2k`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  (2) Let x P L X  `1  5, 2  5  ˘  and let pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , i2kq P t2, 3u2k be such that x P Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k  `“ 1  5, 2  5  ‰˘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Then  x ě FixpTi1 ˝ ¨ ¨ ¨ ˝ Ti2kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let us show the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let jℓ be the sequence such that pmjℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mjℓ´2kq “ pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , i2k`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Since the maps Tm are orientation reversing  lim inf  nÑ8 Tmrn,1sp0q ď lim inf  ℓÑ8 Tmrjℓ,1sp0q  “ Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k`1plim inf  ℓÑ8 Tmrjℓ´2k´1,1sp0qq  ď Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k`1plim inf  nÑ8 Tmrn,1sp0qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let us denote the fixed point of Ti1 ˝ ¨ ¨ ¨˝ Ti2k`1 by x0 and, for short, let x “ lim infnÑ8 Tmrn,1sp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Since the maps Tm are linear and contracting, we get  x ´ x0 ď Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k`1pxq ´ Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k`1px0q “  ´1  i1 ¨ ¨ ¨ i2k`1  px ´ x0q,  thus the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Now we turn to the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let x P L X  ` 1  5, 2  5  ˘  be such that x P Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k  `“ 1  5, 2  5  ‰˘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Suppose that x “ lim infnÑ8 Tmrn,1sp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4)  dist  ¨  ˚  ˚  ˝Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k pJq ,  ď  pj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=',j2kqPt2,3u2k  pj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=',j2kq‰pi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=',i2kq  Tj1 ˝ ¨ ¨ ¨ ˝ Tj2k pJq  ˛  ‹‹‚ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2, for every sufficiently large n, Tmrn,1sp0q P J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3), for every sufficiently  large n, Tmrn,1sp0q P Tmrn,n´2k`1spJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Hence, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4), Tmrn,1sp0q P Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k pJq if and only if  pmn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mn´2k`1q “ pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , i2kq, and so by our assumption on x P L X Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k  `“ 1  5, 2  5  ‰˘  lim inf  nÑ8 Tmrn,1sp0q “ lim inf  ℓÑ8 Tmrnℓ,1sp0q,  where nℓ is the sequence such that pmnℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , mnℓ´2k`1q “ pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , i2kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Since Tm is orientation  reversing  lim inf  nÑ8 Tmrn,1sp0q “ lim inf  ℓÑ8 Tmrnℓ,1sp0q “ Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k  ´  lim inf  ℓÑ8 Tmrnℓ´2k,1sp0q  ¯  ě Ti1 ˝ ¨ ¨ ¨ ˝ Ti2k  ´  lim inf  nÑ8 Tmrn,1sp0q  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Thus the statement follows similarly than the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  Let us recall the definition of the sequences Mpnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let Mp1q “ 2, Mp2q “ 3 and let Mpnq be the  concatenation  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5)  Mpnq “ Mpn´1qMpn´2qMpn´2q for n ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  By definition, the length ln of Mpnq satisfies the equation ln “ ln´1 ` 2ln´2 for every n ě 3 with  l1 “ l2 “ 1, which implies a standard calculation that ln “ 2n´p´1qn  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  We say for any two compact intervals ra, bs and rc, ds that ra, bs ă rc, ds if b ă c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  For any two closed intervals ra, bs and rc, ds with ra, bs ă rc, ds, let midpra, bs, rc, dsq “ rb, cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  14  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  Lemma  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  For  every  n  ě  3,  TMpnqpJq  Ă  TMpn´1qpJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Furthermore,  TMpnqpJq  X  TMpn´1qMpn´1qpJq  “  H,  TMpnqpJq  ă  TMpn´1qMpn´1qpJq  and  Λ X midpTMpnqpJq, TMpn´1qMpn´1qpJqq “ H for every n ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' That is, there is no element of Λ  in-between TMpnqpJq and TMpn´1qMpn´1qpJq for every n ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3), clearly TMpnqpJq Ă TMpn´1qpJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' We prove the second claim by induc-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Clearly, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3) T3pJq X T2,2pJq “ H, furthermore, since  Λ X  ˆ„1  5, 2  5  \uf6be  z  ˆ  T2p  „1  5, 2  5  \uf6be  q Y T3p  „1  5, 2  5  \uf6be  q  ˙˙  “ H  and  „1  5, 2  5  \uf6be  z  ˆ  T2p  „1  5, 2  5  \uf6be  q Y T3p  „1  5, 2  5  \uf6be  q  ˙  Ą midpT2,2pJq, T3pJqq  the claim holds for n “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let us suppose that the claim holds for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then  TMpn`1qpJq X TMpnqMpnqpJq “ TMpnq pTMpn´1qMpn´1qpJq X TMpnqpJqq “ H,  moreover, since TMpnq is orientation reversing, TMpnqpJq ă TMpn´1qMpn´1qpJq implies that  TMpn`1qpJq “ TMpnq pTMpn´1qMpn´1qpJqq ă TMpnq pTMpnqpJqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Observe that  midpTMpn`1qpJq, TMpnqMpnqpJqq “ TMpnq pmidpTMpn´1qMpn´1qpJq, TMpnqpJqqq Ă TMpnqpJq  and so by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3)  ΛXmidpTMpn`1qpJq, TMpnqMpnqpJqq “ TMpnqpΛqXTMpnq pmidpTMpn´1qMpn´1qpJq, TMpnqpJqqq “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  Let us recall the definition of the sequence λn and λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For every n ě 1, let λn “ FixpTMpnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Since FixpTMpnqq P TMpnqMpnqpJq Ă TMpnqpJq, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2 we get  λn`1 ă λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Thus, the sequence λn is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let us denote the limit limnÑ8 λn by λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then by TMpnqpJq Ă  TMpn´1qpJq, we get that  λ0 “ ΠpMq,  where M  “  limnÑ8 Mpnq  “  pM1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q is the limiting sequence defined such that  pM1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , Mlnq “ Mpnq for every positive integer n ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' So  λ0 “  8  ÿ  l“1  p´1ql´1  1  M1M2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Ml  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2293 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='.  First, we show the following proposition:  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' tλ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='u Ă L X  ` 1  5, 2  5  ˘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' In particular, λ0 P L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Before we turn to its proof, we require the following technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS  15  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Let a “ pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , akq and b “ pb1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , bnq be finite sequences formed by the integers  t2, 3u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Suppose that there exist a prefix a1 “ pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , ak1q of a with k1 ď k and a prefix b1 “  pb1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , bn1q of b with n1 ď n such that Ta1pJq ă Tb1pJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then TapJq ă TbpJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Observe that for every compact intervals A, B, if C Ă A and D Ă B are compact intervals  then A ă B implies C ă D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Thus, the claim follows by TapJq Ă Ta1pJq and TbpJq Ă Tb1pJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  For the finite sequence Mpnq “ pMpnq  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , Mpnq  ln q and 1 ď ℓ ď ln´1, let σℓMpnq be the ln-length  word such that  σℓMpnq “ pMpnq  ℓ`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , Mpnq  ln , Mpnq  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , Mpnq  ℓ  q,  with the convention that σlnMpnq “ Mpnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Thus, σℓMpnq can be defined for every ℓ P Z` in a  natural, periodic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For every n ě 3 and 1 ď ℓ ď ln ´ 1, TMpnqpJq ă TσℓMpnqpJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Simple calculations show that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='6)  T3,2,2pJq ă T3,3,3pJq ă T3,3,2pJq ă T2,2,3pJq ă T2,3,3pJq ă T2,3,2pJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Clearly for Mp3q “ p3, 2, 2q, we have σ1Mp3q “ p2, 2, 3q and σ2Mp3q “ p2, 3, 2q, and for Mp4q “  Mp3qMp2qMp2q “ p3, 2, 2, 3, 3q we have  σ1Mp4q “ p2, 2, 3, 3, 3q, σ2Mp4q “ p2, 3, 3, 3, 2q, σ3Mp4q “ p3, 3, 3, 2, 2q, σ4Mp4q “ p3, 3, 2, 2, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Thus, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='6), the claim follows for n “ 3 and n “ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let us prove the rest by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' So suppose that the claim holds for n ě 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  First, assume that 1 ď ℓ ď ln´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then by  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='7)  Mpn`1q “ Mpn´1qMpn´2qMpn´2qMpn´1qMpn´1q  and Mpnq “ Mpn´1qMpn´2qMpn´2q, we get that Mpn`1q  k  “ Mpn´1q  k  “ Mpnq  k  for every 1 ď k ď ln´1,  and so σℓMpnq is a prefix of σℓMpn`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Since Mpnq is a prefix of Mpn`1q, we get by the induction  condition TMpnqpJq ă TσℓMpnqpJq that TMpn`1qpJq ă TσℓMpn`1qpJq by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Now, assume that ln´1 ă ℓ ă ln but ℓ ‰ ln´1 ` ln´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then by  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='8)  Mpn`1q “ Mpn´1qMpn´2qMpn´2qMpn´2qMpn´3qMpn´3qMpn´1q,  we get that σℓ´ln´1Mpn´2q is a prefix of σℓMpn`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Hence, again by the fact that Mpn´2q is a prefix  of Mpn`1q and the assumption that TMpn´2qpJq ă TσkMpn´2qpJq for every k R tln´2, 2ln´2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='u  integer, the claim follows by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  If ℓ “ ln´1 ` ln´2 then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='8), we get that Mpn´2qMpn´2q is a prefix of σℓMpn`1q, meanwhile,  if ℓ “ ln´1 ` 2ln´2 “ ln then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='7), we get that Mpn´1qMpn´1q is a prefix of σℓMpn`1q, thus, the  claim follows by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Now, suppose that ln ă ℓ ă ln ` ln´1 then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5) we get that σℓ´lnMpn´1q is the prefix  of σℓMpn`1q, and since Mpn´1q is a prefix of Mpn`1q, the claim follows by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4 and the  induction condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  If ℓ “ ln ` ln´1 then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5) Mpn´1qMpn´1q is a prefix of σℓMpn`1q, thus, the claim again  follows by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  16  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  Finally, if ln ` ln´1 ă ℓ ă ln ` 2ln´1 “ ln`1 then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5), σℓ´ln´ln´1Mpn´1q is a prefix of  σℓMpn`1q, so the claim follows again by the induction condition and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' For n “ 1 and n “ 2, let  mp1q “ p2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q and mp2q “ p3, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Since the maps Tm are contractions, we get  lim inf  kÑ8 Tmpnq  rk,1sp0q “ lim  kÑ8 Tmpnq  rk,1sp0q “ λn,  and so, tλ1, λ2u Ă L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  For every n ě 3 integer, let us define the following sequence:  mpnq “ pMpnq  ln , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , Mpnq  1 , Mpnq  ln , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' , Mpnq  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2, Tmpnq  rk,1sp0q P Ťln´1  ℓ“0 TσℓMpnqpJq for every sufficiently large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Since the maps  Tm are contractions, we get  lim  kÑ8 Tmpnq  rkln,1sp0q “ λn P TMpnqpJq,  furthermore, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5, TMpnqpJq ă TσℓMpnqpJq for every 1 ď ℓ ď ln ´ 1 and so  λn ă Tmpnq  rk,1sp0q for every k R tln, 2ln, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Hence, lim infkÑ8 Tmpnq  rk,1sp0q “ λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  The last claim follows by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1 and the fact that λn converges to λ0 as n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' pλ1, 8q X L “ H, and for every n ě 1, pλn`1, λnq X L “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' First, observe that max L “ λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Indeed, this follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3(2) with K “ 2, which  implies the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let us show that pλn`1, λnq X L “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Contrary, let us assume that there exists an integer n ě 1  and x P pλn`1, λnq X L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Since λn is the fixed point of TMpnq, we have λn P TMpnqMpnqpJq Ă TMpnqpJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Since L Ă Λ,  where Λ is the attractor of tT2, T3u, and midpTMpn`1qpJq, TMpnqMpnqpJqq Ă pλn`1, λnq, we get  by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2 that either x P TMpnqMpnqpJq or x P TMpn`1qpJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' But by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1(2), if x P  TMpnqMpnqpJq then x ě λn, while if x P TMpn`1qpJq then x ď λn`1 by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1(1), which is a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' The statement follows by combining Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  LAGRANGE-LIKE SPECTRUM OF PERFECT ADDITIVE COMPLEMENTS  17  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='3(2) that if mn ě 5 for infinitely many n, then lim infkÑ8 Tmrk,1sp0q  ď 1  6, thus we may assume without loss of generality that pmiq P t2, 3, 4uZ`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  If pmk, mk´1q “ p4, 2q for infinitely many k, then pmk, mk´1, mk´2q “ p4, 2, aq for an a P  t2, 3, 4u and for infinitely many k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1(1)  lim inf  kÑ8 T p1q  mrk,1sp0q ď maxtFixpT4 ˝ T2 ˝ Taq : a P t2, 3, 4uu “ 3  17,  so we may assume that pmk, mk´1q ‰ p4, 2q for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Thus,  L X  ˆ 3  17, 1  3  \uf6be  Ă tlim inf  nÑ8 Tmrn,1sp0q : mi P t2, 3, 4u, pmi, mi´1q ‰ p4, 2qu “: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  If mi P t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4u and pmi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' mi´1q ‰ p4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' then there are 55 possibilities for pmi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' mi´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' mi´2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' mi´3q:  A “ tp2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' p4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' p4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 4qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  Let us consider the IFS Φ “ tTa ˝Tb ˝Tc ˝Td|pa, b, c, dq P Au and let Λ1 be the attractor of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Then  by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='2, S Ă Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Moreover, it is easy to check that the sum of the 55 contractions strictly  less than 1, therefore by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='1, λpΛ1q “ 0, and so λpL X r 3  17, 1  3sq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  □  REFERENCES  [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' II, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' Cusick and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content='  American Mathematical Society, Providence, RI, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Lima, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' Vieira, MzL is not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' IMRN 2022(1) (2022),  265-311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Falconer, Fractal geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Mathematical foundations and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Third edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' John Wiley & Sons,  Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=', Chichester, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Freiman, Non-coincidence of the spectra of Markov and of Lagrange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' (Russian) Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' Hutchinson, Fractals and self-similarity, Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Khinchin, Continued fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' With a preface by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Gnedenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Translated from the third (1961) Russian  edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Reprint of the 1964 translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Dover Publications, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=', Mineola, NY, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  [10] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Fang, A note on additive complements, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content='  18  BAL ´AZS B ´AR ´ANY1, JIN-HUI FANG2, AND CSABA S ´ANDOR1,˚  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Markov, Sur les formes quadratiques binaires ind´efinies, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 15 (1879), 381-406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Ma, A note on additive complements, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='04128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' Moreira, Geometric properties of the Markov and Lagrange spectra, Annals Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+page_content=' Parkkonen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Paulin, On the closedness of approximation spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Th´eor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Nombres Bordeaux 21 (2009),  701-710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  [15] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Perron, ¨Uber diophantische Approximationen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 83 (1921), 77-84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='  1DEPARTMENT OF STOCHASTICS, INSTITUTE OF MATHEMATICS, BUDAPEST UNIVERSITY OF TECHNOLOGY  AND ECONOMICS, M ˝UEGYETEM RKP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content=', H-1111 BUDAPEST, HUNGARY  Email address: balubs@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='bme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='hu  Email address: csandor@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='bme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='hu  2SCHOOL OF MATHEMATICAL SCIENCES, NANJING NORMAL UNIVERSITY, NANJING 210023, PR CHINA  Email address: fangjinhui1114@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE3T4oBgHgl3EQfLgnu/content/2301.04365v1.pdf'}
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+arXiv:2301.11727v1  [physics.gen-ph]  25 Jan 2023
+TETRIONS: A DISCRETE APPROACH TO THE STANDARD
+MODEL
+ROBERT ARNOTT WILSON
+Abstract. I show how the symmetry-breaking of a recently proposed embed-
+ding of the standard model of particle physics in E8 can be explained in terms
+of the representation theory of the binary tetrahedral group. This finite group
+provides a link between various types of spin and isospin that can be exploited
+to ‘explain’ the chirality of the weak interaction, and the existence of three
+generations of fermions. Two apparently small technical differences between
+the finite group model and the standard model turn out to have profound
+consequences for the ways in which the weak and strong forces create mass.
+1. The mathematical model
+1.1. Introduction. In a recent paper [1], Manogue et al. propose a new E8 model
+of particle physics, that differs from other similar models [2, 3, 4] in one important
+respect, namely that the gauge group of the strong force appears in its split rather
+than compact real form. In order to reproduce the standard model, a complex
+structure is then imposed on the representations, rather than the group itself, and
+it is argued that this is sufficient for all practical purposes. The actual group that
+is used is then
+(SL(2, C) × U(1))/Z2 × U(1) × SU(2) × SL(3, R),
+(1)
+compared to the standard model:
+SL(2, C) × (U(1) × SU(2) × SU(3))/Z6.
+(2)
+The technical differences in the scalars in the two cases are of no deep significance
+for the underlying symmetries, as they merely select different representations of
+the group for use in the physical theory. But the difference between SU(3) and
+U(1)× SL(3, R) is, I shall argue in this paper, not merely technical, as proposed in
+[1], but deeply significant, and has far-reaching consequences for the understanding
+of mass and the quantisation of gravity.
+As is usual in grand unified theories, the key question of why the symmetries of
+E8 break down to this group is not addressed in [1]. Regardless of the actual physical
+mechanism of this symmetry-breaking, it is unsatisfactory from the theoretical point
+of view that there is no mathematical reason for it. A Lie algebra [5, 6] does not
+naturally break up in this way. A Clifford algebra [7], often used in models of
+physics, can break up as a sum of two matrix algebras, but not more, and when it
+does break in this way, the two factors are isomorphic. Hence neither of these types
+of algebra can provide a convincing mathematical explanation for the breaking of
+symmetry down to the standard model.
+Date: First draft: 20th January 2023. This version: 25th January 2023.
+1
+
+2
+ROBERT ARNOTT WILSON
+The most obvious examples of algebras that do break up naturally as direct
+sums of matrix algebras are the real and complex group algebras of finite groups
+[8]. In order to obtain the Lie group (1), in which real and complex matrix algebras
+both appear, a real group algebra is required. Moreover, the finite group must
+have irreducible representations of complex dimensions 1 and 2, and another of
+real dimension 3. Assuming that at least one of the 2-dimensional representations
+is faithful, there are essentially just three possible groups [9, 10, 11], namely the
+binary tetrahedral, binary octahedral and binary icosahedral groups, of orders 24,
+48 and 120 respectively. These are precisely the three finite subgroups of SU(2)
+than act irreducibly on the spin 1 (three-dimensional) representation, but they can
+also be modified by arbitrary mixing with finite groups of complex scalars.
+The representation theory of these groups is studied in detail in [12, 13, 14, 15, 16]
+with a view to using the group algebras in potential new models of fundamental
+physics. The arbitrary scalars have minimal effect on the representation theory, so
+they can be omitted without any real loss to the model, and may not contribute
+anything tangible to physics. In this paper I present an updated version of the
+tetrahedral case [12], and explain how it can be related to the E8 model [1] and to
+the standard model.
+1.2. The binary tetrahedral group. It is worth remarking at this point that the
+use of the binary tetrahedral group as a finite version of the weak gauge group SU(2)
+goes back to the original work of Yang and others [17] from the 1950s. More recent
+work [18, 19, 20] shows that this use of the finite group has real predictive power
+for the measured values of mixing angles in the standard model. It is therefore
+of interest to try to turn this phenomenological approach into a deeper theoretical
+basis for the properties of neutrinos and quarks, in order to explore where the
+various masses and mixing angles ultimately come from. Where the present paper
+goes beyond these earlier papers is in proposing a theoretical basis in the binary
+tetrahedral group, not only for the weak force, but also for the strong force and
+electromagnetism.
+It turns out, and is well-known [12], that the real group algebra of the binary
+tetrahedral group is isomorphic to the following sum of full matrix algebras
+R1×1 + C1×1 + R3×3 + H1×1 + C2×2,
+(3)
+where H denotes the algebra of Hamiltonian quaternions. By taking the matrices
+of real determinant 1 in each factor, we obtain exactly the group (1). This remark-
+able coincidence demands some explanation. But it is possible that it is purely a
+coincidence, driven by the intuition of the authors of [1], rather than any real math-
+ematical or physical connection between the group algebra and the Lie algebra. If
+so, then the binary tetrahedral group might be capable of providing a much more
+fundamental explanation for the standard model than any E8 model could ever do.
+It is, of course, possible to derive the standard model of particle physics from the
+group algebra by following the recipe in [1]. However, it should be clear that the
+structure of E8 itself plays only a small part in this construction. It will therefore be
+enlightening, if possible, to remove this complicating step, and derive the standard
+model directly from the representation theory of the binary tetrahedral group.
+The finite group itself provides an extra ingredient that is not in any Lie group
+model, and enables us to link the different gauge groups together in order to ‘ex-
+plain’ the observed ‘mixing’ of the different fundamental forces at the quantum
+
+TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL
+3
+level, but without introducing any new unobserved degrees of freedom. In partic-
+ular, it turns out that the somewhat ad hoc formalism of electro-weak mixing in
+the standard model can be explained by the fact that two different spinor represen-
+tations of the finite group are treated as if they were the same. Separating these
+two representations again enables us to revise the formalism in such a way as to
+incorporate the three generations of fermions.
+1.3. Real and complex representations. The binary tetrahedral group is most
+easily described as the multiplicative group of the 24 quaternions
+±1, ±i, ±j, ±k, (±1 ± i ± j ± k)/2.
+(4)
+It consists of seven conjugacy classes of elements, represented by 1, −1, i and
+(±1 ± i + j + k)/2. It therefore [8] has seven complex irreducible representations,
+whose characters (giving the traces of the representing matrices) are the rows in the
+following table, where w denotes the quaternion (−1 + i + j + k)/2 and ω denotes
+the complex number (−1 + √−3)/2.
+±1
+i
+±w
+±w2
+1a
+1
+1
+1
+1
+1b
+1
+1
+ω
+¯ω
+1c
+1
+1
+¯ω
+ω
+3
+3
+−1
+0
+0
+2a
+±2
+0
+∓1
+∓1
+2b
+±2
+0
+∓ω
+∓¯ω
+2c
+±2
+0
+∓¯ω
+∓ω
+(5)
+Of these, 1a and 3 are real, 2a is pseudoreal, that is a one-dimensional quaternionic
+representation, and the rest are genuinely complex.
+The characters of the real representations are therefore as follows:
+±1
+i
+±w
+1
+1
+1
+1
+2
+2
+2
+−1
+3
+3
+−1
+0
+4a
+±4
+0
+∓2
+4b
+±4
+0
+±1
+(6)
+While the standard model always works with complex representations, real physical
+measurements are always expressed as real numbers, so that the real representations
+may ultimately provide a better model. For present purposes, we work mainly with
+the complex representations, since they are more familiar and also translate more
+easily into the standard model.
+2. Deriving the standard model
+2.1. Spinors and isospinors. The complex representations 2b and 2c are com-
+plex conjugates of each other, acted on by SL(2, C), and therefore correspond to
+left-handed and right-handed Weyl spinors in the standard model. The complex
+representation 2a, on the other hand, is acted on by SU(2), as the gauge group of
+the weak force, and therefore corresponds to (weak) isospin. There is no natural
+action of this gauge group on the left-handed Weyl spinor, but the finite group
+acts on both spinors and isospinors simultaneously, so can be used to transfer some
+approximate action of SU(2), or a subgroup thereof, from one to the other.
+
+4
+ROBERT ARNOTT WILSON
+For example, the tensor product
+1b ⊗ 2a = 2b
+(7)
+could be used to identify 2a with 2b as ‘the’ left-handed spinor. In particular, this
+has the effect that the copy of U(1) that acts on 1b/1c and 2 is identified with the
+copy of U(1) that acts as complex scalars on 2b/2c and 4b, thereby reducing to
+a single copy of U(1), as in the standard model. Alternatively, one can turn this
+around, and use only the representations 1b and 2b explicitly, leaving 2a implicit.
+This is in effect what the standard model does, using the complex representation
+1b to implement 1 − γ5 acting on the left-handed spinor 2b. In this way, only the
+first component of isospin appears explicitly in the model, and the ‘direction’ of
+this component, like the ‘direction’ of spin, is chosen arbitrarily, without regard to
+the geometry of the outside world.
+This procedure has the effect of incorporating the 120◦ rotation ω into the struc-
+ture of the spinor, in place of the discrete symmetry w. Compared to the standard
+multiplication by i, that is a 90◦ rotation, to get γ5 from −iγ5, there is a discrepancy
+of 30◦. It is plausible that this angle is a first approximation to the electro-weak
+mixing angle (Weinberg angle), that does not take into account the running with
+the energy scale. In order to include the running, it is necessary to include the real
+scalar 1a as well as the complex pseudoscalar 1b, so that a single angle is no longer
+sufficient to describe the full structure of the mixing.
+At the same time, the implicit identification of the real representation 1a with
+the imaginary part of the complex representation 1b destroys the discrete symme-
+try of order 3, which may explain why the standard model loses the generation
+symmetry at this point, and has to insert it again by hand later on. In this paper,
+I propose to reinstate the generation symmetry in such a way that it corresponds
+to permutations of the three ‘directions’ of isospin. This contrasts with the stan-
+dard model, in which the direction of isospin is not explicitly used, and all three
+generations use the same arbitrary direction.
+When we convert to the real representation, the Dirac spinor 2b + 2c reduces to
+a Majorana spinor 4b, while the isospinor 2a becomes 4a. It is no longer possible
+to separate 2b from 2c as left-handed and right-handed spinors, but the separation
+of isospinors 4a from spinors 4b is clear enough. On the other hand, if we lose the
+generation symmetry as above, then there is no distinction between the represen-
+tations 4a and 4b, so that the distinction between spin and isospin becomes less
+clear. It would seem therefore that the best way to relate the finite group algebra
+to the standard model would be to adopt a mixed strategy, working with the real
+group algebra, but the complex representations.
+The conclusion we must surely draw from this discussion is that although the
+standard model can be derived from this group algebra in the way it is done in [1],
+we lose the generation symmetry by doing so, and we lose the distinctions between
+the different representations.
+Indeed, we lose all the symmetry, since w, which
+does not commute with i, has been replaced by the scalar ω, which does. Hence the
+finite group itself plays no role in the standard model. This suggests that by keeping
+the finite group we may gain some insight into why there are three generations of
+fermions, and why the weak interaction is ‘chiral’. Moreover, since the finite group
+links the ‘direction’ of spin to the ‘direction’ of isospin, we may gain some insight
+into how the physical mechanisms of entanglement work.
+
+TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL
+5
+2.2. Tensor products. The Dirac algebra (complex Clifford algebra) used in the
+standard model is, as a representation of the Lorentz group, simply the square of
+the Dirac spinor. In order to identify it as a representation of the finite group, we
+must calculate:
+2a ⊗ 2a = 2b ⊗ 2c = 1a + 3,
+2b ⊗ 2b = 2a ⊗ 2c = 1c + 3,
+2c ⊗ 2c = 2a ⊗ 2b = 1b + 3.
+(8)
+If we regard 2b + 2c as the Dirac spinor, as is required by the interpretation of
+SL(2, C) as the Lorentz group, then the spacetime representation is the real 4-
+dimensional representation 1a + 3. But if we use the above identification of 2a
+with 2b, then spacetime becomes 1c + 3, with a complex time coordinate, rather
+like an anti-de Sitter spacetime.
+The Dirac equation can be obtained by interpreting 1a as time/energy, 3 as
+space/momentum, and the imaginary part of 1b/1c as mass. In discrete form, this
+equation comes from the tensor products
+(1a + 3) ⊗ (2b + 2c) = 2a2 + 2b3 + 2c3,
+(1b + 1c) ⊗ (2b + 2c) = 2a2 + 2b + 2c,
+(9)
+via a complicated process of projecting the right-handed sides onto a Dirac spinor.
+For a massive fermion at rest, this involves identifying the two spinors
+1a ⊗ (2b + 2c) = 2b + 2c,
+1b ⊗ (2b + 2c) = 2c + 2a.
+(10)
+Such a procedure can of course only be carried out if we ignore the finite symmetry
+group, and ignore the weak interaction.
+In particular, it should be noted that the action of SO(3, 1) on spacetime (or its
+dual) cannot be realised in the group algebra, since the latter does not contain a
+subgroup SO(3, 1). It is not possible to implement Lorentz transformations that
+mix space with time, but it is possible to implement time dilation by scalars on
+1a, and it is possible to implement space contraction in arbitrary directions by
+elements of GL(3, R).
+It is therefore possible to model the practical effects of
+Lorentz transformations to a large extent.
+If, on the other hand, we keep the finite symmetry group, then we need to
+consider all three of 1a, 1b and 1c as different versions of ‘time’, appropriate for
+different parts of the theory. The sum 1a + 1b + 1c is the regular representation
+of the cyclic group of order 3, and therefore is appropriate for a discrete triplet
+symmetry. This might, for example, be the generation symmetry, or perhaps it is
+more likely to be the broken symmetry of the intermediate vector bosons, Z0, W +
+and W −. The former suggests a possible interpretation of the three generations
+as a three-dimensional ‘time’, as is done explicitly by Chester et al. in [4], again
+in the context of E8. However, the group algebra model gives no support to the
+idea that there is a symmetry group SO(3, 3) of spacetime, even locally. What
+appears to happen is that the Euclidean metric on space disappears, and the group
+SO(3, 3) reduces to SL(3, R), which seems to indicate that the metric structure of
+spacetime breaks down at a sufficiently small scale. In Section 2.3 of this paper
+I use this group to propose instead a quite different implementation of the three
+generations.
+
+6
+ROBERT ARNOTT WILSON
+The corresponding calculations with the real representations are
+4a ⊗ 4a = (1 + 1 + 1 + 3) + (3 + 3 + 3 + 1)
+4a ⊗ 4b = 2 + 2 + 3 + 3 + 3 + 3
+4b ⊗ 4b = (1 + 2 + 3) + (3 + 3 + 3 + 1)
+(11)
+However, the natural interpretation of 4a is as a one-dimensional quaternionic
+representation, so that we should really calculate the quaternionic tensor product,
+which is a real four-dimensional representation:
+4a ⊗H 4a = 1 + 3.
+(12)
+Since 4b is naturally complex, the tensor product 4a⊗4b can only be real, as above,
+or complex 2a⊗ 2b, which requires breaking the symmetry of the quaternions, that
+is, breaking the symmetry of the weak SU(2). In other words, a complex model
+of electro-weak unification is forced to break the SU(2) symmetry, but a real or
+complex quaternionic model is not. In particular, a model of the latter type could
+in principle implement this symmetry-breaking as contingent, rather than absolute,
+and might therefore suggest a (physical, rather than mathematical) reason why this
+symmetry-breaking happens the way it does in the part of the universe that we can
+directly measure.
+2.3. Triplet symmetries. The only representation left to consider is 3, which
+supports a group SL(3, R) that we suppose must be related to the gauge group
+SU(3) of the strong force. Of its eight degrees of freedom, only three are compact,
+and can therefore be plausibly quantised. The other five cannot be quantised in any
+obvious way, but must nevertheless relate to properties of quantised (elementary)
+particles. The only plausible parameters of this type are masses, but there may
+be some freedom to choose which five masses are regarded as fundamental. The
+representation theory gives us
+3 ⊗ 3 = 1a + 1b + 1c + 3 + 3
+(13)
+which implies that there is a triplet of masses and two singlets. A possible choice
+may be three generations of electrons, together with a proton and a neutron, al-
+though there are certainly other possibilities.
+We should also note that the finite group links the SO(3) symmetry of space
+to an SO(3) subgroup of the SU(3) colour symmetry of quantum chromodynamics
+(QCD). It is not clear exactly how to interpret this physically, although it seems
+clear that the ‘colours’ must be related to some kind of ‘orientation’ in space. But
+this is all happening on such a small scale that it may not be possible to resolve
+such questions experimentally.
+Similarly, the triplet (flavour) symmetries of neutrinos can only plausibly be de-
+rived from the square of the isospinor 2a, that is 1a + 3, so must be implemented
+in 3, and not in 1a + 1b + 1c. On a macroscopic scale relevant for the study of
+neutrino oscillations, the representation 3 has interpretations as space, momentum,
+and various classical fields including electric, magnetic and gravitational. It is pos-
+sible, therefore, that the generation of a neutrino that is detected by a microscopic
+experiment using the finite group of symmetries of 3 can change when the ambi-
+ent spacetime coordinates change, due to rotations or other accelerations, or to a
+gravitational field.
+
+TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL
+7
+The simplest explanation of neutrino oscillations in this context is that 3 contains
+the momentum, measured with respect to the symmetry group SO(3) of the ambient
+space, and that 1a contains the energy, measured with respect to the speed of light.
+There is then no room for an intrinsic mass, but there is room for an effective change
+in mass due to bending of the path of the neutrino in a variable gravitational field,
+or due to the acceleration of one observer with respect to another. In contrast,
+the standard model effectively identifies 1a (here identified as the energy) with the
+imaginary part of 1b (the Dirac mass), which forces the neutrinos to have a non-
+zero (Dirac) mass. But general relativity does not require a Dirac mass in order
+for the neutrinos to be affected by the force of gravity, possibly even leading to a
+change of flavour in an appropriate quantum-gravitational context.
+2.4. The strong force. As complex tensor products of complex representations,
+we have
+1b ⊗ 3 = 1c ⊗ 3 = 3,
+(14)
+but there is a subtlety here caused by the fact that the factor 3 in the tensor
+products can be real, whereas the resulting 3 on the right-hand side cannot. Never-
+theless, a (real) basis can be chosen such that the latter splits as the sum of two real
+representations. If we extend to Lie groups acting on the two factors independently,
+then this splitting can still be achieved, giving rise to a group U(1)×SL(3, R) with
+compact part U(1) × SO(3), in which U(1) acts as scalars. But this is not what
+happens, either in the standard model, or in physical reality. There is a ‘mixing’
+between the U(1) factor and a copy of SO(2) inside SO(3) caused by the fact that
+an element of order 3 in the finite group acts on both simultaneously.
+What then happens in the standard model is that there is no clean relationship
+between the bases with respect to which the two groups U(1) and SO(3) are written.
+Moreover, since U(1) does not act as a group of scalars, these two groups no longer
+generate a direct product U(1)×SO(3), and instead generate SU(3). The theory of
+QCD is then built on this basis, and although it works well, it appears to be built
+on rather shaky foundations. If the group algebra is the fundamental structure
+underlying the standard model, then the fact that this algebra does not contain
+a copy of SU(3) is important. In this case, we must conclude that (at least) five
+of the eight ‘gluons’ of QCD are entirely fictitious concepts: not only are they
+unobservable, but they have no physical reality at all. Controversial as this may
+sound, it is not a new idea: it is already implicit (though not explicit) in [1], since
+there can be at most three gauge bosons arising from the compact part of SL(3, R).
+Since direct model-independent tests of properties of gluons are virtually impossible,
+this idea cannot be easily falsified or rejected.
+Nevertheless, the parameters that are required to change basis between the three
+isomorphic complex representations 1a ⊗C 3, 1b ⊗C 3 and 1c ⊗C 3 fill two complex
+3 × 3 matrices, represented in the standard model by the Cabibbo–Kobayashi–
+Maskawa (CKM) [21, 22] and Pontecorvo–Maki–Nakagata–Sakawa (PMNS) [23, 24]
+matrices. Each contains only four independent real parameters, because the only
+physical part is a map between two copies of U(1), and a map between two copies
+of SO(3). The rest of SU(3) does not contain physically meaningful parameters.
+
+8
+ROBERT ARNOTT WILSON
+3. Beyond the standard model
+3.1. Hidden variables. The group algebra described in this paper is just about
+big enough to describe the standard model, with nothing left over. By using the
+finite symmetries, we obtain three generations of fermions, which are characterised
+by three orthogonal components of a certain kind of isospin, where only one is
+used in the standard model. I have not chosen a basis for this isospin, which is
+not necessarily identical to the standard model weak isospin, and therefore have
+not chosen explicit quantum numbers for the different particles, but in principle
+this should not be too difficult. Similarly in the case of spin, we are able to use
+all three components in the model, rather than having to choose one. This allows
+an elementary particle to carry three spin quantum numbers rather than just one,
+although it is probable that only two of the three are independent. However, it is
+still the case that a measurement requires a macroscopic choice of this direction,
+in order to define the eigenstates, so that one can only measure spin in a specific
+direction, and not in two directions simultaneously.
+The effect of this combination of properties is that an elementary particle carries
+more information than it is possible to measure. This seemingly contradictory state
+of affairs is exactly what one needs to explain the puzzling aspects of measurements
+of properties of widely-separated entangled particles—both particles carry hidden
+variables that are discrete, and cannot be measured directly. The discreteness is
+crucial here, as it is what enables the model to evade the well-known problems
+with continuous hidden variable theories [25, 26, 27, 28, 29].
+What makes the
+experimental properties of quantum mechanics so puzzling, and counterintuitive, is
+that the hidden variables can only be detected by elementary particles, and not by
+macroscopic apparatus. Since the elementary particles have only a finite number of
+quantum states, they can only carry a finite amount of information, and can only
+distinguish a finite number of directions in space.
+This process naturally gives rise to the effect that is known in quantum mechan-
+ics as ‘collapse of the wave-function’. In the group algebra model, the wave-function
+itself is not associated to the individual particle, but to the macroscopic environ-
+ment, consisting of enough elementary particles to make it effectively continuous.
+The ‘collapse’ happens when the particle interacts with the environment, via an
+interaction with an individual particle in the environment. The Born probabilities
+arise from the fact that the experimenter has little or no control over the quantum
+properties of the environment.
+3.2. Mixing. The finite group implements some strange mixing between the direc-
+tion of spin and the direction of isospin. While the two directions certainly appear
+to be independent, they cannot apparently change independently, since every ele-
+ment of the finite group acts on both simultaneously. This fundamental property
+seems to underlie the observed chirality of the weak interaction. In the classic Wu
+experiment, a fixed direction of isospin is determined by the choice to observe the
+physical process of beta decay of a neutron, which breaks the generation symmetry,
+and a fixed direction of spin was chosen by the experimenters [30]. A third direction
+was observed, namely the direction of spin/momentum of the ejected antineutrino,
+and was found to be not independent of the pair of choices made. Thus what might
+appear at first sight to be three independent copies of 3 actually contains only two.
+Once two independent choices of direction have been made, the third is determined.
+
+TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL
+9
+A similar but even more mysterious mixing seems to underlie the phenomenon
+of neutrino oscillation.
+Here the experiments choose a particular generation of
+neutrino at both source and sink, and a particular direction of momentum between
+them. The natural expectation would be that the first two choices must be the same,
+but are independent of the third. But this is not what the experiments reveal. Yes,
+there are two independent choices, rather than three, but the relationship between
+them is more subtle than would naively be expected. There is a mixing between
+spin/momentum on the one hand and generation/flavour on the other. However,
+this is mixing is not apparent if one regards the momentum as fixed, so that a third
+copy of 3 is required to explain the observed effects.
+The group algebra does indeed contain a third copy of 3, which we can use if only
+we can identify what it is and how to use it. Since the experiments are large-scale,
+this copy of 3 must have a plausible macroscopic interpretation. Since neutrinos
+do not participate in electromagnetic interactions, the only available macroscopic
+interaction that is incontrovertibly known to science is gravitational. Assuming we
+do not wish to predict new forces at this point, we must then use the third copy of 3
+to choose the direction of the gravitational field. If this analysis is correct, then the
+probabilities of detecting neutrinos of particular flavours in particular experiments
+can be predicted from the geometry of the experiment relative to the gravitational
+field. In particular, if there is no correlation between the directions of the gravita-
+tional field at source and sink, as for example in solar neutrino experiments, then
+we should expect a 1/3 probability of detecting any specific flavour, as is indeed
+confirmed by experiment [31].
+3.3. Implications for quantum gravity. The essential part of any such exten-
+sion of the standard model to include gravity is a separation of the representations
+2a and 2b, which are mixed together in the standard model. In other words we
+must separate the neutrino representation 2a (without time, and therefore without
+intrinsic mass) from the matter (Dirac spinor) representation 2b + 2c (with time
+and intrinsic mass). The Newtonian gravitational field then appears as the sym-
+metric square of the neutrino field. In particular, the known fact that neutrinos
+interact very weakly with matter, combined with the squaring process, reduces the
+strength of the Newtonian gravity of an elementary particle to an undetectable level
+compared with the other forces.
+Putting a Dirac spinor in a gravitational field, we obtain the tensor products
+2b ⊗ 3 = 2a + 2b + 2c
+2c ⊗ 3 = 2a + 2b + 2c,
+(15)
+which not only causes a mixing between left-handed 2b and right-handed 2c, but
+also causes the Dirac particle to emit neutrinos from 2a.
+But it would appear
+from this discussion that gravitational effects only arise from the interactions of
+neutrinos with each other: since such interactions have no direct effect on matter,
+the obvious interpretation at a macroscopic level is that these undetectable neutrino
+interactions instead affect the shape of the underlying spacetime, as in the modern
+textbook interpretations of general relativity. On the other hand, a hypothetical
+future theory that models these neutrino interactions explicitly might in principle be
+able to explain gravity as an indirect effect of neutrino-neutrino interactions on the
+average effect of neutrino-matter interactions. If so, then ‘curvature of spacetime’
+will become a redundant concept, no longer required for an explanation of gravity.
+
+10
+ROBERT ARNOTT WILSON
+3.4. Comparison with general relativity. A theory of quantum gravity based
+on these ideas cannot be relativistic in the usual sense, because it is not acted on
+by the Lorentz group SL(2, C). Its gauge group is therefore not GL(4, R), as is
+generally assumed [32, 33], but only GL(3, R), or rather, the positive determinant
+component GL(3, R)+. However, the latter group can act on 4-dimensional space-
+time as a group of matrices of determinant 1, in which there are elements acting
+as time dilation on the (fixed) time coordinate, and simultaneously as length con-
+traction on an arbitrary space coordinate. These are not Lorentz transformations,
+but their effects on measurable quantities may be almost indistinguishable from the
+effects of Lorentz transformations.
+The quantisation of this gauge group forms the representation 1+2+3+3 of the
+binary tetrahedral group, while the symmetric rank 2 tensors of general relativity
+reduce to the form
+S2(1 + 3) = 1 + 3 + 1 + 2 + 3.
+(16)
+Removing the Ricci scalar from the Ricci tensor to leave the Einstein tensor shows
+that the gauge group GL(3, R) accurately quantises the Einstein tensor. But there
+is no SO(3, 1), so this gives us a non-relativistic version of general relativity, if that
+isn’t an oxymoron. In other words this quantum gravity is indistinguishable from
+general relativity in the non-relativistic limit. However, this quantisation procedure
+does not exhaust the group algebra, which contains an extra vector field 3 compared
+to the Einstein tensor.
+Since the entire bosonic part of the group algebra can be expected to have
+detectable effects on all scales, including galactic and cosmological scales, we must
+do three things to extend general relativity to a complete theory of gravity: first,
+remove the Ricci scalar; second, add a new vector field; and third, remove the
+requirement for invariance under SO(3, 1). Various models of gravity have been
+developed along these lines, of which the most successful to date are those that
+add a vector and one or two scalars to the symmetric tensor of general relativity
+[41, 42, 43]. But none of these removes the SO(3, 1) symmetry, and therefore, if
+my analysis is correct, they are all inconsistent with quantum mechanics.
+In the model proposed here, Lorentz invariance is only an approximate symme-
+try of the macroscopic world, valid in regions of the universe in which there is a
+reasonably well-defined concept of inertial frame. Presumably then, while an ap-
+proximately consistent definition of inertial frame can be made on a Solar System
+scale, provided that curvature of spacetime is invoked to compensate for the grav-
+itational fields of the objects within the Solar System, the same is not necessarily
+true on a galactic scale.
+It should be noted also that the separation of 2a from 2b+2c entails a separation
+of 1a from 1b+1c, and therefore a separation of a real scalar energy from a complex
+scalar mass plane. This entails adding an extra scalar field to the usual mass and
+energy fields. There are various possible ways to interpret this extra scalar, but its
+practical effect is to separate the concepts of inertial and gravitational mass. While
+the local experimental evidence is that these two concepts track each other closely,
+the non-local astronomical evidence is that they do not [34, 35, 36, 37, 38, 39, 40].
+The model proposed in this paper therefore provides a potential way to reconcile
+these two conflicting strands of evidence.
+
+TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL
+11
+3.5. Physical significance of a non-compact gauge group. In [1], the change
+from the compact gauge group SU(3) to the split group SL(3, R) is presented as a
+mere technicality, since the calculations required in the standard model can always
+be patched up by introducing factors of i from the group U(1). To a certain extent
+this is true, at least from a practical point of view. It is much the same process
+that is already incorporated into the standard model in the Dirac algebra, where
+the complexification allows one to switch between the Lorentz group in the form
+SL(2, C) = Spin(3, 1) and the compact form SU(2) × SU(2) = Spin(4).
+But in terms of the underlying conceptual structure, the difference between a
+Lorentzian (Minkowski) spacetime and a Euclidean (quantum) spacetime is im-
+portant. Similarly, the difference between SU(3) and SL(3, R) for describing the
+strong force is important. The split form is not compatible with the Yang–Mills
+paradigm of particle physics. The basic problem is that compactness is required for
+(second) quantisation to produce identifiable gauge bosons that can be detected in
+experiments. Non-compact degrees of freedom cannot be quantised into particles.
+On the other hand, the gauge bosons of the strong force (gluons) in the standard
+model cannot be experimentally detected as individual particles, which implies
+that there is no experimental (as opposed to theoretical) reason why it should be
+necessary to quantise all 8 degrees of freedom. It is certainly clear that at least three
+degrees of freedom must be quantised, in order for the three quarks in a proton to
+occupy orthogonal eigenstates, as required by the Pauli exclusion principle. We
+must conclude therefore that the strong theoretical arguments against a split gauge
+group SL(3, R) for the strong force are not in fact supported by experiment.
+It is useful at this point to compare with the non-compact Lorentz group in
+the form SO(3, 1), which is used in special relativity in order to define a concept
+of rest mass, abstracted from the measurable quantities of energy and momentum.
+Assuming an exact SO(3) symmetry of space, this defines a single mass value for any
+particular object. But if the symmetry of space is broken by a strongly directional
+gravitational field (for example), then one can define two different mass values, one
+in the direction of the gravitational field (usually called the gravitational mass), and
+the other perpendicular to it (usually called the inertial mass). There is nothing
+in any mathematical model to say that these two masses have to be the same,
+although the experimental evidence is that locally they are indistinguishable. The
+assumption that they are globally the same is not justified either by experiment or
+by theory, but is purely a phenomenological assumption.
+An analogous thing happens in the quantum form SL(2, C), if we consider the
+compact part SU(2) as the gauge group of the weak force, in order to define the
+intermediate vector bosons (Z0, W + and W −), and we use the non-compact part
+to define the masses of these three particles. In this case, the symmetry-breaking
+is provided by the electromagnetic force rather than the gravitational force, but
+the effect is the same, namely to produce two masses rather than one. The charge
+conjugation symmetry prevents a splitting into three distinct masses, so that we
+have only one dimensionless parameter to explain, namely the ratio of the masses
+of the W and Z bosons. While the standard model assumes that this mass ratio is
+a universal constant, the experimental evidence [44] currently provides a 7σ signal
+that this is not in fact the case. The group algebra model suggests that this problem
+might be resolved by providing greater clarity as to what precise mixture of inertial
+and gravitational masses is actually being measured by the experiments [45],
+
+12
+ROBERT ARNOTT WILSON
+Now let us consider the group SL(3, R). Here we have five non-compact degrees
+of freedom, reduced to three by the action of U(1). We therefore have three inde-
+pendent mass values available. The representation in which these mass values lie is
+the spin 2 representation of SO(3), which suggests that the underlying symmetry-
+breaking comes not from the gravitational field itself, but from the tidal effects of
+rotations within a gravitational field (or, equivalently, of a rotating gravitational
+field). I will not speculate here on which particle masses are most appropriate here,
+apart from noting that the measured values of such masses might change if the tidal
+effects of the rotation of the Earth relative to the the gravitational fields of the Sun
+and/or the Moon have a noticeable effect. Such effects are certainly small, but they
+may still be detectable. It is conceivable, for example, that the above-mentioned
+reported mass anomaly for the W/Z bosons might have its origin in such an effect,
+if one takes into account the experimental ‘mixing’ of the weak and strong forces.
+4. Conclusion
+4.1. Summary. In this paper I have taken the (coincidental) isomorphism between
+(1) the group used in the ‘octions’ model [1] in E8 as a modified version of the
+combined Lorentz group and gauge groups of the standard model, and
+(2) the group obtained from the determinant 1 subgroups of the Wedderburn
+components of the real group algebra of the binary tetrahedral group [12]
+as motivation to try to use the latter algebra instead of the former as a foundation
+for (a modified version of) the standard model, in an attempt to explain things
+that the standard model does not itself explain.
+I have shown that the apparently minor technical modifications proposed in
+[1] in fact entail profound conceptual changes compared to the standard model,
+while leaving almost all of the detailed predictions of the standard model intact.
+The points of difference concern only neutrinos and gluons, and therefore have
+almost no effect on any of the predictions of the standard model. They do however
+permit the modified model to predict (or postdict, or explain) some things that
+were not predicted in advance by the standard model, but have been detected by
+experiment. These necessarily involve subtle properties of neutrinos and/or gluons,
+and include neutrino oscillations, the chirality of the weak force, the existence of
+three generations of fermions, the mixing of the gauge groups, and possibly even
+the W/Z mass anomaly.
+Some of these predictions depend on an identification of part of the model as
+representing a quantised (Newtonian, i.e. non-relativistic) gravity. Because it is
+non-relativistic, it does not satisfy all of the properties that are predicted for quan-
+tum gravity by starting from the general theory of relativity. Nevertheless, it con-
+tains a quantised version of the Einstein tensor, and therefore contains corrections
+to Newtonian gravity, that are likely to agree with general relativity to first order
+locally, for example on the Solar System scale. The reason for making this asser-
+tion is that these corrections take account of accelerations, for example rotations,
+but not of relativistic velocities. At the same time, the model predicts a mixing of
+quantum gravity with the other forces, although I have not attempted in this paper
+to make any detailed predictions of measurable effects that could be attributed to
+such a cause.
+
+TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL
+13
+4.2. Speculations and further work. The basic principle of the mixing of grav-
+ity with the other forces is a mixing of neutrino generations with the direction of the
+gravitational field, which then leaks out via the weak force into a mixing of electron
+generations with gravity. This does not necessarily affect all three generations, but
+it does affect at least two of them. It therefore predicts that sufficiently precise
+experiments with muons in a varying gravitational field will eventually detect an
+anomaly caused by a mixing of electromagnetism with gravity.
+It is possible, in fact, that this has already happened: experimental measurement
+of the muon gyromagnetic ratio is in conflict with some (but not all) calculations
+of the standard model prediction. The magnitude of the discrepancy in the various
+calculated and measured values [46, 47, 48, 49] of g − 2 happens to be more or
+less equal to the sine of the angle by which the direction of the gravitational field
+changes over the experiment. This might be a coincidence, or it might not: with
+further work it may also turn out to be a prediction of the model.
+Similarly, one should expect to be able to detect mixing between gravity and the
+weak force by anomalies that involve strange quarks, investigated via experiments
+[50, 51, 52] with (neutral) kaons. The kind of effects that might be attributable
+to such a cause include switching between the long and short decay eigenstates,
+via interactions with (low energy, undetectable) neutrinos. Again, the magnitude
+of the effect, mixing two-pion and three-pion decays, that was detected [50] in
+1964 is consistent with the value of the sine of the angle by which the direction
+of the gravitational field changes over the size of the experiment.
+More recent
+experiments with rare one-pion decays [51, 52] exhibit further anomalies that might
+be attributable to the same cause. Extending to the third generation one should
+expect to see anomalies of the same kind exhibited by B-mesons [53].
+Evidence for mixing of the strong force with gravity is rather harder to find, but
+it is possible that an inconsistency between two different mass formulas [54, 55]
+for the mass of the tau particle might be attributable to the fact that one of these
+formulas uses the proton and neutron masses, and therefore involves the strong
+force, whereas the other one does not.
+The predictions in the two cases differ
+by a proportion of less than 10−4, and both are consistent with experiment, but
+not with each other.
+The formula in [54] relates only the three generations of
+electrons, so presumably depends only on neutrinos and quantum gravity, so that
+it represents a prediction of gravitational mass. The formula in [55], on the other
+hand, involves also the proton and neutron masses, so depends on electromagnetism,
+so that it represents a prediction of inertial mass. Since the model does not include
+an assumption that gravitational and inertial mass are equal, it is able, at least
+in principle, to accommodate both of these predictions simultaneously. Further
+work will be required to see whether it can in fact make either or both of these
+predictions.
+4.3. Evaluation. The model presented in this paper is not a Theory of Every-
+thing (TOE). It is not a Grand Unified Theory (GUT). It adds no new fields, no
+new particles, no new forces, no spin 2 gravitons, no supersymmetry, no strings,
+no new physics at all, to what is already known. All it does is explain how the
+Standard Model arises from something more fundamental. In doing so, it embeds
+the Standard Model in the gravitational background.
+This background is rela-
+tivistic only because the Standard Model is relativistic: the fundamental theory is
+non-relativistic.
+
+14
+ROBERT ARNOTT WILSON
+Perhaps it can be described as a Quantum Unification of All Non-relativistic
+Theories of the Universe and Matter (QUANTUM). It makes only one predic-
+tion, namely that, in order for fundamental quantum physics to be background-
+independent, as it obviously must be, the Standard Model must be background-
+dependent.
+Subtle changes in the gravitational background will therefore cause
+subtle changes in experimental measurements, particularly those that depend on
+measuring neutrino momenta. The Standard Model can be tweaked for each new
+anomaly, and surely will be, to fit the new background. But the anomalies will
+keep on coming, until such time as the background-dependence is recognised.
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+[54] Y. Koide (1983), A fermion-boson composite model of quarks and leptons, Physics Letters B
+120 (1–3), 161–165.
+[55] R. A. Wilson (2022), Remarks on the group-theoretical foundations of particle physics, Int.
+J. Geom. Meth. Mod. Phys.
+Queen Mary University of London
+Email address: r.a.wilson@qmul.ac.uk
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf,len=574
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='11727v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='gen-ph]  25 Jan 2023  TETRIONS: A DISCRETE APPROACH TO THE STANDARD  MODEL  ROBERT ARNOTT WILSON  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' I show how the symmetry-breaking of a recently proposed embed-  ding of the standard model of particle physics in E8 can be explained in terms  of the representation theory of the binary tetrahedral group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This finite group  provides a link between various types of spin and isospin that can be exploited  to ‘explain’ the chirality of the weak interaction, and the existence of three  generations of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Two apparently small technical differences between  the finite group model and the standard model turn out to have profound  consequences for the ways in which the weak and strong forces create mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The mathematical model  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In a recent paper [1], Manogue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' propose a new E8 model  of particle physics, that differs from other similar models [2, 3, 4] in one important  respect, namely that the gauge group of the strong force appears in its split rather  than compact real form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In order to reproduce the standard model, a complex  structure is then imposed on the representations, rather than the group itself, and  it is argued that this is sufficient for all practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The actual group that  is used is then  (SL(2, C) × U(1))/Z2 × U(1) × SU(2) × SL(3, R),  (1)  compared to the standard model:  SL(2, C) × (U(1) × SU(2) × SU(3))/Z6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  (2)  The technical differences in the scalars in the two cases are of no deep significance  for the underlying symmetries, as they merely select different representations of  the group for use in the physical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But the difference between SU(3) and  U(1)× SL(3, R) is, I shall argue in this paper, not merely technical, as proposed in  [1], but deeply significant, and has far-reaching consequences for the understanding  of mass and the quantisation of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  As is usual in grand unified theories, the key question of why the symmetries of  E8 break down to this group is not addressed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Regardless of the actual physical  mechanism of this symmetry-breaking, it is unsatisfactory from the theoretical point  of view that there is no mathematical reason for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' A Lie algebra [5, 6] does not  naturally break up in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' A Clifford algebra [7], often used in models of  physics, can break up as a sum of two matrix algebras, but not more, and when it  does break in this way, the two factors are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Hence neither of these types  of algebra can provide a convincing mathematical explanation for the breaking of  symmetry down to the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Date: First draft: 20th January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This version: 25th January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  1  2  ROBERT ARNOTT WILSON  The most obvious examples of algebras that do break up naturally as direct  sums of matrix algebras are the real and complex group algebras of finite groups  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In order to obtain the Lie group (1), in which real and complex matrix algebras  both appear, a real group algebra is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Moreover, the finite group must  have irreducible representations of complex dimensions 1 and 2, and another of  real dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Assuming that at least one of the 2-dimensional representations  is faithful, there are essentially just three possible groups [9, 10, 11], namely the  binary tetrahedral, binary octahedral and binary icosahedral groups, of orders 24,  48 and 120 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' These are precisely the three finite subgroups of SU(2)  than act irreducibly on the spin 1 (three-dimensional) representation, but they can  also be modified by arbitrary mixing with finite groups of complex scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The representation theory of these groups is studied in detail in [12, 13, 14, 15, 16]  with a view to using the group algebras in potential new models of fundamental  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The arbitrary scalars have minimal effect on the representation theory, so  they can be omitted without any real loss to the model, and may not contribute  anything tangible to physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In this paper I present an updated version of the  tetrahedral case [12], and explain how it can be related to the E8 model [1] and to  the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The binary tetrahedral group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is worth remarking at this point that the  use of the binary tetrahedral group as a finite version of the weak gauge group SU(2)  goes back to the original work of Yang and others [17] from the 1950s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' More recent  work [18, 19, 20] shows that this use of the finite group has real predictive power  for the measured values of mixing angles in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is therefore  of interest to try to turn this phenomenological approach into a deeper theoretical  basis for the properties of neutrinos and quarks, in order to explore where the  various masses and mixing angles ultimately come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Where the present paper  goes beyond these earlier papers is in proposing a theoretical basis in the binary  tetrahedral group, not only for the weak force, but also for the strong force and  electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  It turns out, and is well-known [12], that the real group algebra of the binary  tetrahedral group is isomorphic to the following sum of full matrix algebras  R1×1 + C1×1 + R3×3 + H1×1 + C2×2,  (3)  where H denotes the algebra of Hamiltonian quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' By taking the matrices  of real determinant 1 in each factor, we obtain exactly the group (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This remark-  able coincidence demands some explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But it is possible that it is purely a  coincidence, driven by the intuition of the authors of [1], rather than any real math-  ematical or physical connection between the group algebra and the Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' If  so, then the binary tetrahedral group might be capable of providing a much more  fundamental explanation for the standard model than any E8 model could ever do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  It is, of course, possible to derive the standard model of particle physics from the  group algebra by following the recipe in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' However, it should be clear that the  structure of E8 itself plays only a small part in this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It will therefore be  enlightening, if possible, to remove this complicating step, and derive the standard  model directly from the representation theory of the binary tetrahedral group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The finite group itself provides an extra ingredient that is not in any Lie group  model, and enables us to link the different gauge groups together in order to ‘ex-  plain’ the observed ‘mixing’ of the different fundamental forces at the quantum  TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL  3  level, but without introducing any new unobserved degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In partic-  ular, it turns out that the somewhat ad hoc formalism of electro-weak mixing in  the standard model can be explained by the fact that two different spinor represen-  tations of the finite group are treated as if they were the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Separating these  two representations again enables us to revise the formalism in such a way as to  incorporate the three generations of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Real and complex representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The binary tetrahedral group is most  easily described as the multiplicative group of the 24 quaternions  ±1, ±i, ±j, ±k, (±1 ± i ± j ± k)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  (4)  It consists of seven conjugacy classes of elements, represented by 1, −1, i and  (±1 ± i + j + k)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It therefore [8] has seven complex irreducible representations,  whose characters (giving the traces of the representing matrices) are the rows in the  following table, where w denotes the quaternion (−1 + i + j + k)/2 and ω denotes  the complex number (−1 + √−3)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  ±1  i  ±w  ±w2  1a  1  1  1  1  1b  1  1  ω  ¯ω  1c  1  1  ¯ω  ω  3  3  −1  0  0  2a  ±2  0  ∓1  ∓1  2b  ±2  0  ∓ω  ∓¯ω  2c  ±2  0  ∓¯ω  ∓ω  (5)  Of these, 1a and 3 are real, 2a is pseudoreal, that is a one-dimensional quaternionic  representation, and the rest are genuinely complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The characters of the real representations are therefore as follows:  ±1  i  ±w  1  1  1  1  2  2  2  −1  3  3  −1  0  4a  ±4  0  ∓2  4b  ±4  0  ±1  (6)  While the standard model always works with complex representations, real physical  measurements are always expressed as real numbers, so that the real representations  may ultimately provide a better model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' For present purposes, we work mainly with  the complex representations, since they are more familiar and also translate more  easily into the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Deriving the standard model  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Spinors and isospinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The complex representations 2b and 2c are com-  plex conjugates of each other, acted on by SL(2, C), and therefore correspond to  left-handed and right-handed Weyl spinors in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The complex  representation 2a, on the other hand, is acted on by SU(2), as the gauge group of  the weak force, and therefore corresponds to (weak) isospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' There is no natural  action of this gauge group on the left-handed Weyl spinor, but the finite group  acts on both spinors and isospinors simultaneously, so can be used to transfer some  approximate action of SU(2), or a subgroup thereof, from one to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  4  ROBERT ARNOTT WILSON  For example, the tensor product  1b ⊗ 2a = 2b  (7)  could be used to identify 2a with 2b as ‘the’ left-handed spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In particular, this  has the effect that the copy of U(1) that acts on 1b/1c and 2 is identified with the  copy of U(1) that acts as complex scalars on 2b/2c and 4b, thereby reducing to  a single copy of U(1), as in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Alternatively, one can turn this  around, and use only the representations 1b and 2b explicitly, leaving 2a implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  This is in effect what the standard model does, using the complex representation  1b to implement 1 − γ5 acting on the left-handed spinor 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In this way, only the  first component of isospin appears explicitly in the model, and the ‘direction’ of  this component, like the ‘direction’ of spin, is chosen arbitrarily, without regard to  the geometry of the outside world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  This procedure has the effect of incorporating the 120◦ rotation ω into the struc-  ture of the spinor, in place of the discrete symmetry w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Compared to the standard  multiplication by i, that is a 90◦ rotation, to get γ5 from −iγ5, there is a discrepancy  of 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is plausible that this angle is a first approximation to the electro-weak  mixing angle (Weinberg angle), that does not take into account the running with  the energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In order to include the running, it is necessary to include the real  scalar 1a as well as the complex pseudoscalar 1b, so that a single angle is no longer  sufficient to describe the full structure of the mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  At the same time, the implicit identification of the real representation 1a with  the imaginary part of the complex representation 1b destroys the discrete symme-  try of order 3, which may explain why the standard model loses the generation  symmetry at this point, and has to insert it again by hand later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In this paper,  I propose to reinstate the generation symmetry in such a way that it corresponds  to permutations of the three ‘directions’ of isospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This contrasts with the stan-  dard model, in which the direction of isospin is not explicitly used, and all three  generations use the same arbitrary direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  When we convert to the real representation, the Dirac spinor 2b + 2c reduces to  a Majorana spinor 4b, while the isospinor 2a becomes 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is no longer possible  to separate 2b from 2c as left-handed and right-handed spinors, but the separation  of isospinors 4a from spinors 4b is clear enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' On the other hand, if we lose the  generation symmetry as above, then there is no distinction between the represen-  tations 4a and 4b, so that the distinction between spin and isospin becomes less  clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It would seem therefore that the best way to relate the finite group algebra  to the standard model would be to adopt a mixed strategy, working with the real  group algebra, but the complex representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The conclusion we must surely draw from this discussion is that although the  standard model can be derived from this group algebra in the way it is done in [1],  we lose the generation symmetry by doing so, and we lose the distinctions between  the different representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Indeed, we lose all the symmetry, since w, which  does not commute with i, has been replaced by the scalar ω, which does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Hence the  finite group itself plays no role in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This suggests that by keeping  the finite group we may gain some insight into why there are three generations of  fermions, and why the weak interaction is ‘chiral’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Moreover, since the finite group  links the ‘direction’ of spin to the ‘direction’ of isospin, we may gain some insight  into how the physical mechanisms of entanglement work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Tensor products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The Dirac algebra (complex Clifford algebra) used in the  standard model is, as a representation of the Lorentz group, simply the square of  the Dirac spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In order to identify it as a representation of the finite group, we  must calculate:  2a ⊗ 2a = 2b ⊗ 2c = 1a + 3,  2b ⊗ 2b = 2a ⊗ 2c = 1c + 3,  2c ⊗ 2c = 2a ⊗ 2b = 1b + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  (8)  If we regard 2b + 2c as the Dirac spinor, as is required by the interpretation of  SL(2, C) as the Lorentz group, then the spacetime representation is the real 4-  dimensional representation 1a + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But if we use the above identification of 2a  with 2b, then spacetime becomes 1c + 3, with a complex time coordinate, rather  like an anti-de Sitter spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The Dirac equation can be obtained by interpreting 1a as time/energy, 3 as  space/momentum, and the imaginary part of 1b/1c as mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In discrete form, this  equation comes from the tensor products  (1a + 3) ⊗ (2b + 2c) = 2a2 + 2b3 + 2c3,  (1b + 1c) ⊗ (2b + 2c) = 2a2 + 2b + 2c,  (9)  via a complicated process of projecting the right-handed sides onto a Dirac spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  For a massive fermion at rest, this involves identifying the two spinors  1a ⊗ (2b + 2c) = 2b + 2c,  1b ⊗ (2b + 2c) = 2c + 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  (10)  Such a procedure can of course only be carried out if we ignore the finite symmetry  group, and ignore the weak interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  In particular, it should be noted that the action of SO(3, 1) on spacetime (or its  dual) cannot be realised in the group algebra, since the latter does not contain a  subgroup SO(3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is not possible to implement Lorentz transformations that  mix space with time, but it is possible to implement time dilation by scalars on  1a, and it is possible to implement space contraction in arbitrary directions by  elements of GL(3, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  It is therefore possible to model the practical effects of  Lorentz transformations to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  If, on the other hand, we keep the finite symmetry group, then we need to  consider all three of 1a, 1b and 1c as different versions of ‘time’, appropriate for  different parts of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The sum 1a + 1b + 1c is the regular representation  of the cyclic group of order 3, and therefore is appropriate for a discrete triplet  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This might, for example, be the generation symmetry, or perhaps it is  more likely to be the broken symmetry of the intermediate vector bosons, Z0, W +  and W −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The former suggests a possible interpretation of the three generations  as a three-dimensional ‘time’, as is done explicitly by Chester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' in [4], again  in the context of E8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' However, the group algebra model gives no support to the  idea that there is a symmetry group SO(3, 3) of spacetime, even locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' What  appears to happen is that the Euclidean metric on space disappears, and the group  SO(3, 3) reduces to SL(3, R), which seems to indicate that the metric structure of  spacetime breaks down at a sufficiently small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='3 of this paper  I use this group to propose instead a quite different implementation of the three  generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  6  ROBERT ARNOTT WILSON  The corresponding calculations with the real representations are  4a ⊗ 4a = (1 + 1 + 1 + 3) + (3 + 3 + 3 + 1)  4a ⊗ 4b = 2 + 2 + 3 + 3 + 3 + 3  4b ⊗ 4b = (1 + 2 + 3) + (3 + 3 + 3 + 1)  (11)  However, the natural interpretation of 4a is as a one-dimensional quaternionic  representation, so that we should really calculate the quaternionic tensor product,  which is a real four-dimensional representation:  4a ⊗H 4a = 1 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  (12)  Since 4b is naturally complex, the tensor product 4a⊗4b can only be real, as above,  or complex 2a⊗ 2b, which requires breaking the symmetry of the quaternions, that  is, breaking the symmetry of the weak SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In other words, a complex model  of electro-weak unification is forced to break the SU(2) symmetry, but a real or  complex quaternionic model is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In particular, a model of the latter type could  in principle implement this symmetry-breaking as contingent, rather than absolute,  and might therefore suggest a (physical, rather than mathematical) reason why this  symmetry-breaking happens the way it does in the part of the universe that we can  directly measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Triplet symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The only representation left to consider is 3, which  supports a group SL(3, R) that we suppose must be related to the gauge group  SU(3) of the strong force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Of its eight degrees of freedom, only three are compact,  and can therefore be plausibly quantised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The other five cannot be quantised in any  obvious way, but must nevertheless relate to properties of quantised (elementary)  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The only plausible parameters of this type are masses, but there may  be some freedom to choose which five masses are regarded as fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The  representation theory gives us  3 ⊗ 3 = 1a + 1b + 1c + 3 + 3  (13)  which implies that there is a triplet of masses and two singlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' A possible choice  may be three generations of electrons, together with a proton and a neutron, al-  though there are certainly other possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  We should also note that the finite group links the SO(3) symmetry of space  to an SO(3) subgroup of the SU(3) colour symmetry of quantum chromodynamics  (QCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is not clear exactly how to interpret this physically, although it seems  clear that the ‘colours’ must be related to some kind of ‘orientation’ in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But  this is all happening on such a small scale that it may not be possible to resolve  such questions experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Similarly, the triplet (flavour) symmetries of neutrinos can only plausibly be de-  rived from the square of the isospinor 2a, that is 1a + 3, so must be implemented  in 3, and not in 1a + 1b + 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' On a macroscopic scale relevant for the study of  neutrino oscillations, the representation 3 has interpretations as space, momentum,  and various classical fields including electric, magnetic and gravitational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is pos-  sible, therefore, that the generation of a neutrino that is detected by a microscopic  experiment using the finite group of symmetries of 3 can change when the ambi-  ent spacetime coordinates change, due to rotations or other accelerations, or to a  gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL  7  The simplest explanation of neutrino oscillations in this context is that 3 contains  the momentum, measured with respect to the symmetry group SO(3) of the ambient  space, and that 1a contains the energy, measured with respect to the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  There is then no room for an intrinsic mass, but there is room for an effective change  in mass due to bending of the path of the neutrino in a variable gravitational field,  or due to the acceleration of one observer with respect to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In contrast,  the standard model effectively identifies 1a (here identified as the energy) with the  imaginary part of 1b (the Dirac mass), which forces the neutrinos to have a non-  zero (Dirac) mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But general relativity does not require a Dirac mass in order  for the neutrinos to be affected by the force of gravity, possibly even leading to a  change of flavour in an appropriate quantum-gravitational context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The strong force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' As complex tensor products of complex representations,  we have  1b ⊗ 3 = 1c ⊗ 3 = 3,  (14)  but there is a subtlety here caused by the fact that the factor 3 in the tensor  products can be real, whereas the resulting 3 on the right-hand side cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Never-  theless, a (real) basis can be chosen such that the latter splits as the sum of two real  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' If we extend to Lie groups acting on the two factors independently,  then this splitting can still be achieved, giving rise to a group U(1)×SL(3, R) with  compact part U(1) × SO(3), in which U(1) acts as scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But this is not what  happens, either in the standard model, or in physical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' There is a ‘mixing’  between the U(1) factor and a copy of SO(2) inside SO(3) caused by the fact that  an element of order 3 in the finite group acts on both simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  What then happens in the standard model is that there is no clean relationship  between the bases with respect to which the two groups U(1) and SO(3) are written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Moreover, since U(1) does not act as a group of scalars, these two groups no longer  generate a direct product U(1)×SO(3), and instead generate SU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The theory of  QCD is then built on this basis, and although it works well, it appears to be built  on rather shaky foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' If the group algebra is the fundamental structure  underlying the standard model, then the fact that this algebra does not contain  a copy of SU(3) is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In this case, we must conclude that (at least) five  of the eight ‘gluons’ of QCD are entirely fictitious concepts: not only are they  unobservable, but they have no physical reality at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Controversial as this may  sound, it is not a new idea: it is already implicit (though not explicit) in [1], since  there can be at most three gauge bosons arising from the compact part of SL(3, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Since direct model-independent tests of properties of gluons are virtually impossible,  this idea cannot be easily falsified or rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Nevertheless, the parameters that are required to change basis between the three  isomorphic complex representations 1a ⊗C 3, 1b ⊗C 3 and 1c ⊗C 3 fill two complex  3 × 3 matrices, represented in the standard model by the Cabibbo–Kobayashi–  Maskawa (CKM) [21, 22] and Pontecorvo–Maki–Nakagata–Sakawa (PMNS) [23, 24]  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Each contains only four independent real parameters, because the only  physical part is a map between two copies of U(1), and a map between two copies  of SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The rest of SU(3) does not contain physically meaningful parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  8  ROBERT ARNOTT WILSON  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Beyond the standard model  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The group algebra described in this paper is just about  big enough to describe the standard model, with nothing left over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' By using the  finite symmetries, we obtain three generations of fermions, which are characterised  by three orthogonal components of a certain kind of isospin, where only one is  used in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' I have not chosen a basis for this isospin, which is  not necessarily identical to the standard model weak isospin, and therefore have  not chosen explicit quantum numbers for the different particles, but in principle  this should not be too difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Similarly in the case of spin, we are able to use  all three components in the model, rather than having to choose one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This allows  an elementary particle to carry three spin quantum numbers rather than just one,  although it is probable that only two of the three are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' However, it is  still the case that a measurement requires a macroscopic choice of this direction,  in order to define the eigenstates, so that one can only measure spin in a specific  direction, and not in two directions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The effect of this combination of properties is that an elementary particle carries  more information than it is possible to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This seemingly contradictory state  of affairs is exactly what one needs to explain the puzzling aspects of measurements  of properties of widely-separated entangled particles—both particles carry hidden  variables that are discrete, and cannot be measured directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The discreteness is  crucial here, as it is what enables the model to evade the well-known problems  with continuous hidden variable theories [25, 26, 27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  What makes the  experimental properties of quantum mechanics so puzzling, and counterintuitive, is  that the hidden variables can only be detected by elementary particles, and not by  macroscopic apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Since the elementary particles have only a finite number of  quantum states, they can only carry a finite amount of information, and can only  distinguish a finite number of directions in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  This process naturally gives rise to the effect that is known in quantum mechan-  ics as ‘collapse of the wave-function’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In the group algebra model, the wave-function  itself is not associated to the individual particle, but to the macroscopic environ-  ment, consisting of enough elementary particles to make it effectively continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The ‘collapse’ happens when the particle interacts with the environment, via an  interaction with an individual particle in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The Born probabilities  arise from the fact that the experimenter has little or no control over the quantum  properties of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The finite group implements some strange mixing between the direc-  tion of spin and the direction of isospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' While the two directions certainly appear  to be independent, they cannot apparently change independently, since every ele-  ment of the finite group acts on both simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This fundamental property  seems to underlie the observed chirality of the weak interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In the classic Wu  experiment, a fixed direction of isospin is determined by the choice to observe the  physical process of beta decay of a neutron, which breaks the generation symmetry,  and a fixed direction of spin was chosen by the experimenters [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' A third direction  was observed, namely the direction of spin/momentum of the ejected antineutrino,  and was found to be not independent of the pair of choices made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Thus what might  appear at first sight to be three independent copies of 3 actually contains only two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Once two independent choices of direction have been made, the third is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL  9  A similar but even more mysterious mixing seems to underlie the phenomenon  of neutrino oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Here the experiments choose a particular generation of  neutrino at both source and sink, and a particular direction of momentum between  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The natural expectation would be that the first two choices must be the same,  but are independent of the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But this is not what the experiments reveal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Yes,  there are two independent choices, rather than three, but the relationship between  them is more subtle than would naively be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' There is a mixing between  spin/momentum on the one hand and generation/flavour on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' However,  this is mixing is not apparent if one regards the momentum as fixed, so that a third  copy of 3 is required to explain the observed effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The group algebra does indeed contain a third copy of 3, which we can use if only  we can identify what it is and how to use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Since the experiments are large-scale,  this copy of 3 must have a plausible macroscopic interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Since neutrinos  do not participate in electromagnetic interactions, the only available macroscopic  interaction that is incontrovertibly known to science is gravitational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Assuming we  do not wish to predict new forces at this point, we must then use the third copy of 3  to choose the direction of the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' If this analysis is correct, then the  probabilities of detecting neutrinos of particular flavours in particular experiments  can be predicted from the geometry of the experiment relative to the gravitational  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In particular, if there is no correlation between the directions of the gravita-  tional field at source and sink, as for example in solar neutrino experiments, then  we should expect a 1/3 probability of detecting any specific flavour, as is indeed  confirmed by experiment [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Implications for quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The essential part of any such exten-  sion of the standard model to include gravity is a separation of the representations  2a and 2b, which are mixed together in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In other words we  must separate the neutrino representation 2a (without time, and therefore without  intrinsic mass) from the matter (Dirac spinor) representation 2b + 2c (with time  and intrinsic mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The Newtonian gravitational field then appears as the sym-  metric square of the neutrino field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In particular, the known fact that neutrinos  interact very weakly with matter, combined with the squaring process, reduces the  strength of the Newtonian gravity of an elementary particle to an undetectable level  compared with the other forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Putting a Dirac spinor in a gravitational field, we obtain the tensor products  2b ⊗ 3 = 2a + 2b + 2c  2c ⊗ 3 = 2a + 2b + 2c,  (15)  which not only causes a mixing between left-handed 2b and right-handed 2c, but  also causes the Dirac particle to emit neutrinos from 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  But it would appear  from this discussion that gravitational effects only arise from the interactions of  neutrinos with each other: since such interactions have no direct effect on matter,  the obvious interpretation at a macroscopic level is that these undetectable neutrino  interactions instead affect the shape of the underlying spacetime, as in the modern  textbook interpretations of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' On the other hand, a hypothetical  future theory that models these neutrino interactions explicitly might in principle be  able to explain gravity as an indirect effect of neutrino-neutrino interactions on the  average effect of neutrino-matter interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' If so, then ‘curvature of spacetime’  will become a redundant concept, no longer required for an explanation of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  10  ROBERT ARNOTT WILSON  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Comparison with general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' A theory of quantum gravity based  on these ideas cannot be relativistic in the usual sense, because it is not acted on  by the Lorentz group SL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Its gauge group is therefore not GL(4, R), as is  generally assumed [32, 33], but only GL(3, R), or rather, the positive determinant  component GL(3, R)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' However, the latter group can act on 4-dimensional space-  time as a group of matrices of determinant 1, in which there are elements acting  as time dilation on the (fixed) time coordinate, and simultaneously as length con-  traction on an arbitrary space coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' These are not Lorentz transformations,  but their effects on measurable quantities may be almost indistinguishable from the  effects of Lorentz transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The quantisation of this gauge group forms the representation 1+2+3+3 of the  binary tetrahedral group, while the symmetric rank 2 tensors of general relativity  reduce to the form  S2(1 + 3) = 1 + 3 + 1 + 2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  (16)  Removing the Ricci scalar from the Ricci tensor to leave the Einstein tensor shows  that the gauge group GL(3, R) accurately quantises the Einstein tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But there  is no SO(3, 1), so this gives us a non-relativistic version of general relativity, if that  isn’t an oxymoron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In other words this quantum gravity is indistinguishable from  general relativity in the non-relativistic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' However, this quantisation procedure  does not exhaust the group algebra, which contains an extra vector field 3 compared  to the Einstein tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Since the entire bosonic part of the group algebra can be expected to have  detectable effects on all scales, including galactic and cosmological scales, we must  do three things to extend general relativity to a complete theory of gravity: first,  remove the Ricci scalar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' second, add a new vector field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' and third, remove the  requirement for invariance under SO(3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Various models of gravity have been  developed along these lines, of which the most successful to date are those that  add a vector and one or two scalars to the symmetric tensor of general relativity  [41, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But none of these removes the SO(3, 1) symmetry, and therefore, if  my analysis is correct, they are all inconsistent with quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  In the model proposed here, Lorentz invariance is only an approximate symme-  try of the macroscopic world, valid in regions of the universe in which there is a  reasonably well-defined concept of inertial frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Presumably then, while an ap-  proximately consistent definition of inertial frame can be made on a Solar System  scale, provided that curvature of spacetime is invoked to compensate for the grav-  itational fields of the objects within the Solar System, the same is not necessarily  true on a galactic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  It should be noted also that the separation of 2a from 2b+2c entails a separation  of 1a from 1b+1c, and therefore a separation of a real scalar energy from a complex  scalar mass plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This entails adding an extra scalar field to the usual mass and  energy fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' There are various possible ways to interpret this extra scalar, but its  practical effect is to separate the concepts of inertial and gravitational mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' While  the local experimental evidence is that these two concepts track each other closely,  the non-local astronomical evidence is that they do not [34, 35, 36, 37, 38, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The model proposed in this paper therefore provides a potential way to reconcile  these two conflicting strands of evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Physical significance of a non-compact gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In [1], the change  from the compact gauge group SU(3) to the split group SL(3, R) is presented as a  mere technicality, since the calculations required in the standard model can always  be patched up by introducing factors of i from the group U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' To a certain extent  this is true, at least from a practical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is much the same process  that is already incorporated into the standard model in the Dirac algebra, where  the complexification allows one to switch between the Lorentz group in the form  SL(2, C) = Spin(3, 1) and the compact form SU(2) × SU(2) = Spin(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  But in terms of the underlying conceptual structure, the difference between a  Lorentzian (Minkowski) spacetime and a Euclidean (quantum) spacetime is im-  portant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Similarly, the difference between SU(3) and SL(3, R) for describing the  strong force is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The split form is not compatible with the Yang–Mills  paradigm of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The basic problem is that compactness is required for  (second) quantisation to produce identifiable gauge bosons that can be detected in  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Non-compact degrees of freedom cannot be quantised into particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  On the other hand, the gauge bosons of the strong force (gluons) in the standard  model cannot be experimentally detected as individual particles, which implies  that there is no experimental (as opposed to theoretical) reason why it should be  necessary to quantise all 8 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is certainly clear that at least three  degrees of freedom must be quantised, in order for the three quarks in a proton to  occupy orthogonal eigenstates, as required by the Pauli exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' We  must conclude therefore that the strong theoretical arguments against a split gauge  group SL(3, R) for the strong force are not in fact supported by experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  It is useful at this point to compare with the non-compact Lorentz group in  the form SO(3, 1), which is used in special relativity in order to define a concept  of rest mass, abstracted from the measurable quantities of energy and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Assuming an exact SO(3) symmetry of space, this defines a single mass value for any  particular object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But if the symmetry of space is broken by a strongly directional  gravitational field (for example), then one can define two different mass values, one  in the direction of the gravitational field (usually called the gravitational mass), and  the other perpendicular to it (usually called the inertial mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' There is nothing  in any mathematical model to say that these two masses have to be the same,  although the experimental evidence is that locally they are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The  assumption that they are globally the same is not justified either by experiment or  by theory, but is purely a phenomenological assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  An analogous thing happens in the quantum form SL(2, C), if we consider the  compact part SU(2) as the gauge group of the weak force, in order to define the  intermediate vector bosons (Z0, W + and W −), and we use the non-compact part  to define the masses of these three particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In this case, the symmetry-breaking  is provided by the electromagnetic force rather than the gravitational force, but  the effect is the same, namely to produce two masses rather than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The charge  conjugation symmetry prevents a splitting into three distinct masses, so that we  have only one dimensionless parameter to explain, namely the ratio of the masses  of the W and Z bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' While the standard model assumes that this mass ratio is  a universal constant, the experimental evidence [44] currently provides a 7σ signal  that this is not in fact the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The group algebra model suggests that this problem  might be resolved by providing greater clarity as to what precise mixture of inertial  and gravitational masses is actually being measured by the experiments [45],  12  ROBERT ARNOTT WILSON  Now let us consider the group SL(3, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Here we have five non-compact degrees  of freedom, reduced to three by the action of U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' We therefore have three inde-  pendent mass values available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The representation in which these mass values lie is  the spin 2 representation of SO(3), which suggests that the underlying symmetry-  breaking comes not from the gravitational field itself, but from the tidal effects of  rotations within a gravitational field (or, equivalently, of a rotating gravitational  field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' I will not speculate here on which particle masses are most appropriate here,  apart from noting that the measured values of such masses might change if the tidal  effects of the rotation of the Earth relative to the the gravitational fields of the Sun  and/or the Moon have a noticeable effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Such effects are certainly small, but they  may still be detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is conceivable, for example, that the above-mentioned  reported mass anomaly for the W/Z bosons might have its origin in such an effect,  if one takes into account the experimental ‘mixing’ of the weak and strong forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Conclusion  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In this paper I have taken the (coincidental) isomorphism between  (1) the group used in the ‘octions’ model [1] in E8 as a modified version of the  combined Lorentz group and gauge groups of the standard model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' and  (2) the group obtained from the determinant 1 subgroups of the Wedderburn  components of the real group algebra of the binary tetrahedral group [12]  as motivation to try to use the latter algebra instead of the former as a foundation  for (a modified version of) the standard model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' in an attempt to explain things  that the standard model does not itself explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  I have shown that the apparently minor technical modifications proposed in  [1] in fact entail profound conceptual changes compared to the standard model,  while leaving almost all of the detailed predictions of the standard model intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The points of difference concern only neutrinos and gluons, and therefore have  almost no effect on any of the predictions of the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' They do however  permit the modified model to predict (or postdict, or explain) some things that  were not predicted in advance by the standard model, but have been detected by  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' These necessarily involve subtle properties of neutrinos and/or gluons,  and include neutrino oscillations, the chirality of the weak force, the existence of  three generations of fermions, the mixing of the gauge groups, and possibly even  the W/Z mass anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Some of these predictions depend on an identification of part of the model as  representing a quantised (Newtonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' non-relativistic) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Because it is  non-relativistic, it does not satisfy all of the properties that are predicted for quan-  tum gravity by starting from the general theory of relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Nevertheless, it con-  tains a quantised version of the Einstein tensor, and therefore contains corrections  to Newtonian gravity, that are likely to agree with general relativity to first order  locally, for example on the Solar System scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The reason for making this asser-  tion is that these corrections take account of accelerations, for example rotations,  but not of relativistic velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' At the same time, the model predicts a mixing of  quantum gravity with the other forces, although I have not attempted in this paper  to make any detailed predictions of measurable effects that could be attributed to  such a cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  TETRIONS: A DISCRETE APPROACH TO THE STANDARD MODEL  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Speculations and further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The basic principle of the mixing of grav-  ity with the other forces is a mixing of neutrino generations with the direction of the  gravitational field, which then leaks out via the weak force into a mixing of electron  generations with gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This does not necessarily affect all three generations, but  it does affect at least two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It therefore predicts that sufficiently precise  experiments with muons in a varying gravitational field will eventually detect an  anomaly caused by a mixing of electromagnetism with gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  It is possible, in fact, that this has already happened: experimental measurement  of the muon gyromagnetic ratio is in conflict with some (but not all) calculations  of the standard model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The magnitude of the discrepancy in the various  calculated and measured values [46, 47, 48, 49] of g − 2 happens to be more or  less equal to the sine of the angle by which the direction of the gravitational field  changes over the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' This might be a coincidence, or it might not: with  further work it may also turn out to be a prediction of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Similarly, one should expect to be able to detect mixing between gravity and the  weak force by anomalies that involve strange quarks, investigated via experiments  [50, 51, 52] with (neutral) kaons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The kind of effects that might be attributable  to such a cause include switching between the long and short decay eigenstates,  via interactions with (low energy, undetectable) neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Again, the magnitude  of the effect, mixing two-pion and three-pion decays, that was detected [50] in  1964 is consistent with the value of the sine of the angle by which the direction  of the gravitational field changes over the size of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  More recent  experiments with rare one-pion decays [51, 52] exhibit further anomalies that might  be attributable to the same cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Extending to the third generation one should  expect to see anomalies of the same kind exhibited by B-mesons [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Evidence for mixing of the strong force with gravity is rather harder to find, but  it is possible that an inconsistency between two different mass formulas [54, 55]  for the mass of the tau particle might be attributable to the fact that one of these  formulas uses the proton and neutron masses, and therefore involves the strong  force, whereas the other one does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The predictions in the two cases differ  by a proportion of less than 10−4, and both are consistent with experiment, but  not with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  The formula in [54] relates only the three generations of  electrons, so presumably depends only on neutrinos and quantum gravity, so that  it represents a prediction of gravitational mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The formula in [55], on the other  hand, involves also the proton and neutron masses, so depends on electromagnetism,  so that it represents a prediction of inertial mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Since the model does not include  an assumption that gravitational and inertial mass are equal, it is able, at least  in principle, to accommodate both of these predictions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Further  work will be required to see whether it can in fact make either or both of these  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The model presented in this paper is not a Theory of Every-  thing (TOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It is not a Grand Unified Theory (GUT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It adds no new fields, no  new particles, no new forces, no spin 2 gravitons, no supersymmetry, no strings,  no new physics at all, to what is already known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' All it does is explain how the  Standard Model arises from something more fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' In doing so, it embeds  the Standard Model in the gravitational background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  This background is rela-  tivistic only because the Standard Model is relativistic: the fundamental theory is  non-relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  14  ROBERT ARNOTT WILSON  Perhaps it can be described as a Quantum Unification of All Non-relativistic  Theories of the Universe and Matter (QUANTUM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' It makes only one predic-  tion, namely that, in order for fundamental quantum physics to be background-  independent, as it obviously must be, the Standard Model must be background-  dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content='  Subtle changes in the gravitational background will therefore cause  subtle changes in experimental measurements, particularly those that depend on  measuring neutrino momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' The Standard Model can be tweaked for each new  anomaly, and surely will be, to fit the new background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
+page_content=' But the anomalies will  keep on coming, until such time as the background-dependence is recognised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFKT4oBgHgl3EQfGy3D/content/2301.11727v1.pdf'}
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+arXiv:2301.01872v1  [math.GT]  5 Jan 2023
+Smallest nonabelian quotients of surface braid groups
+Cindy Tan
+Abstract.
+We give a sharp lower bound on the size of nonabelian quotients of the surface
+braid group 퐵푛(Σ푔) and classify all quotients that attain the lower bound: Depending on 푛 and
+푔, a quotient of minimum order is either a symmetric group or a 2-step nilpotent group.
+1
+Introduction
+The Artin braid group 퐵푛 arises as the fundamental group of UConf푛(D), the configuration space of 푛
+distinct unordered points on the open disk D. One can generalize this construction to define for an oriented,
+closed genus 푔 surface Σ푔 the surface braid groups
+퐵푛(Σ푔) = 휋1(UConf푛(Σ푔)).
+It was shown by Kolay [4] that for 푛 = 3 or 푛 ≥ 5, the smallest noncyclic finite quotient of 퐵푛 is the
+symmetric group 푆푛, in the sense that 푆푛 has minimum order amongst noncyclic quotients of 퐵푛 and 푆푛 is
+the unique noncyclic quotient of 퐵푛 of minimum order.
+In this paper we show that in contrast to the regular braid groups, for a fixed genus 푔 surface Σ푔 there exist
+infinitely many 푛 for which 푆푛 is not the smallest nonabelian quotient of 퐵푛(Σ푔). The new minimal
+quotients that arise are 2-step nilpotent groups which will be defined in Construction 2.6.
+Theorem 1.1 (Smallest nonabelian quotients of 퐵푛(Σ푔)). Let 푛 ≥ 5 and 푔 ≥ 1. Suppose that 퐺 is a finite
+nonabelian quotient of 퐵푛(Σ푔).
+(a) If 푔 + 푛 − 1 is even then |퐺| ≥ min(푛!, 22푔+2). Furthermore,
+(i) If 푛! < 22푔+2 and |퐺| = 푛! then 퐺 � 푆푛.
+(ii) If 푛! > 22푔+2 and |퐺| = 22푔+2 then either 퐺 � I(22, 푔) or 퐺 � II(22, 푔) (and both quotients
+arise in every such case).
+(b) If 푔 + 푛 − 1 is odd then |퐺| ≥ min(푛!, 푝2푔+1), where 푝 is the smallest prime divisor of 푔 + 푛 − 1.
+Furthermore,
+(i) If 푛! < 푝2푔+1 and |퐺| = 푛! then 퐺 � 푆푛.
+(ii) If 푛! > 푝2푔+1 and |퐺| = 푝2푔+1 then either 퐺 � I(푝, 푔) or 퐺 � II(푝, 푔) (and both quotients arise
+in every such case).
+The groups I(푝 푗, 푔) and II(푝 푗, 푔) are nonisomorphic 2-step nilpotent groups defined in Construction 2.6.
+Note that Theorem 1.1 implies the following qualitative result.
+Corollary 1.2.
+(a) Fix 푔 ≥ 1. There exist infinitely many 푛 for which the smallest quotient of 퐵푛(Σ푔) is 푆푛 and also
+infinitely many 푛 for which the smallest quotient is not 푆푛.
+(b) Fix 푛 ≥ 5. For all sufficiently large 푔, the smallest nonabelian quotient of 퐵푛(Σ푔) is 푆푛. Also, there
+exists a (small) 푔 for which this not true.
+1
+
+Remarks 1.3 (Smaller cases).
+(a) If 푛 = 1, 2, 3, 4 and 푔 ≥ 1 (with the exception of (푛, 푔) = (1, 1) where 퐵푛(Σ푔) = 휋1(푇2) = Z2 is
+abelian) then the symmetric group 푆3 is the smallest nonabelian quotient of 퐵푛(Σ푔).
+(b) If 푔 = 0 then 퐵푛(Σ푔) is the spherical braid group 퐵푛(푆2) which is an intermediate quotient of the
+map 퐵푛 → 푆푛 [3]. It follows from the result of Kolay [4] that the smallest quotient of 퐵푛(푆2) is 푆푛
+for 푛 ≥ 5 and 푆3 for 푛 = 3, 4. For 푛 = 1 and 2 we note that 퐵푛(푆2) is abelian.
+Question 1.4 (Geometric interpretation). The quotient map 퐵푛(Σ푔) → 푆푛 can be interpreted geometrically
+as mapping a surface braid to the permutation it performs on basepoints. We pose the question: What is the
+geometric interpretation of the new nilpotent minimal quotients, or more generally the nilpotent quotients
+of 퐵푛(Σ푔) enumerated in Lemma 3.4?
+Proof method.
+By mapping a braid to its permutation on points, 푆푛 is a finite quotient of 퐵푛(Σ푔). For
+푔 ≥ 1, there is an embedding 퐵푛 ↩→ 퐵푛(Σ푔) [2]. It follows from Kolay that if 퐵푛 → 퐵푛(Σ푔) → 퐺 has
+noncyclic image, then |퐺| ≥ 푛! with the bound attained only by 퐺 � 푆푛.
+It remains to consider nonabelian quotients of 퐵푛(Σ푔) where 퐵푛 has cyclic image, which we will refer to as
+braid-reduced quotients. For certain choices of 푝 and 푗 the nilpotent groups I(푝 푗, 푔) and II(푝 푗, 푔) defined
+in Construction 2.6 are nonabelian braid-reduced quotients of 퐵푛(Σ푔), and in fact can be constructed to have
+minimum order amongst braid-reduced quotients; this comprises the primary contribution of this paper:
+Theorem 1.5 (Smallest nonabelian braid-reduced quotients of 퐵푛(Σ푔)). Let 푛 ≥ 3 and 푔 ≥ 1. Suppose 퐺 is
+a finite nonabelian quotient of 퐵푛(Σ푔) where 퐵푛 has cyclic image in 퐺. Then 퐺 is 2-step nilpotent and
+(a) If 푔 + 푛 − 1 is even then |퐺| ≥ 22푔+2. In the case of equality, either 퐺 � I(22, 푔) or 퐺 � II(22, 푔) (and
+both quotients arise in every such case).
+(b) If 푔 + 푛 − 1 is odd then |퐺| ≥ 푝2푔+1 where 푝 is the smallest prime divisor of 푔 + 푛 − 1. In the case of
+equality, either 퐺 � I(푝, 푔) or 퐺 � II(푝, 푔) (and both quotients arise in every such case).
+To prove Theorem 1.5 we utilize a presentation of 퐵푛(Σ푔) (Theorem 3.1) due to Bellingeri [1] and assume
+that 퐵푛 has cyclic image to reduce the relations to conclude that a braid-reduced quotient 퐺 must be
+nilpotent. If we further assume that 퐺 is a nonabelian braid-reduced quotient of minimum order then 퐺
+belongs to a class of nilpotent groups called JN2 groups (Definition 2.1) which were classified by Newman
+in 1960 [5]. Section 2 provides a self-contained exposition of the classification of JN2 groups and we prove
+Theorem 1.5 in Section 3.
+Acknowledgements.
+I am grateful to my advisor Benson Farb for continued support throughout this
+project and for detailed comments on many revisions of this paper, as well as for suggesting this problem in
+the first place. I thank Peter Huxford for useful discussions about braid groups and small 푝-groups, and for
+many helpful suggestions during the editing process. I also thank Dan Margalit for taking the time to read
+and comment on an earlier draft.
+2
+Just 2-step nilpotent groups
+In this section we introduce and classify JN2 groups, a class of nilpotent groups which includes all minimal
+nonabelian quotients of 퐵푛(Σ푔) in which 퐵푛 has cyclic image.
+Definition 2.1. A group 퐺 is just 2-step nilpotent (JN2) if 퐺 is 2-step nilpotent (in particular, nonabelian)
+and every proper quotient of 퐺 is abelian.
+2
+
+Finite JN2 groups admit a complete and explicit classification due to Newman [5]: Any finite JN2 group
+can be assigned a unique class (푝 푗, 푚) where 푝 is a prime and 푗 and 푚 are positive integers; up to
+isomorphism, there are precisely two JN2 groups of a given class (푝 푗, 푚). We will state and prove this
+classification theorem in Theorem 2.7, following the general ideas of [5].
+All JN2 groups will hereafter be assumed to be finite. The following proposition will allow us to define the
+class (푝 푗, 푚) of a JN2 group.
+Proposition 2.2 (Characterization of JN2 groups [5, Theorem 1]). A finite group 퐺 is JN2 if and only if
+there exists a prime 푝 such that
+(a) 퐺′ ≔ [퐺, 퐺] is cyclic of order 푝,
+(b) the center 푍퐺 is cyclic of order a power of 푝, and
+(c) 퐺/푍퐺 is elementary abelian of exponent 푝.
+Proof.
+⇒ Let 퐺 be a finite JN2 group. For every nontrivial normal subgroup 푁 ⊴ 퐺, we have that 퐺′ ≤ 푁 since
+any proper quotient of 퐺 is abelian. Since 퐺 is 2-step nilpotent, 퐺′ ≤ 푍퐺. Consequently,
+(a) 퐺′ is abelian and admits no proper nontrivial subgroups so 퐺′ � Z/푝Z for some prime 푝.
+(b) 푍퐺 cannot be properly decomposed as a direct sum: Any nontrivial subgroup of 푍퐺 contains 퐺′
+so no two nontrivial subgroups intersect trivially. Since 푍퐺 is finite abelian, it must be cyclic of
+prime power order. The prime must be 푝 because 퐺′ ≤ 푍퐺.
+(c) 퐺/푍퐺 is abelian because 퐺′ ≤ 푍퐺. For 푥, 푦 ∈ 퐺, we have that [푥 푝, 푦] = [푥, 푦] 푝 by using the
+identity
+[푥푧, 푦] = 푧[푥, 푦]푧−1[푧, 푦]
+and noting that [푥, 푦] is central because 퐺′ ≤ 푍퐺. But 퐺′ has order 푝, so in fact [푥 푝, 푦] = 1.
+Thus 푥 푝 ∈ 푍퐺 for all 푥 ∈ 퐺, which is to say that 퐺/푍퐺 has exponent 푝.
+⇐ Suppose 퐺 is a finite group satisfying (a), (b), and (c). Then 퐺′ ≠ {1} by (a) and 퐺′ ≤ 푍퐺 by (c) so 퐺
+is 2-step nilpotent.
+If 푁 ⊴ 퐺 is a normal subgroup with 퐺′ ≰ 푁 then 푁 ∩ 퐺′ = {1} by (a). Since 푁 is normal,
+[푁, 퐺] ≤ 푁 ∩ 퐺′ = {1} so 푁 ≤ 푍퐺. But 퐺′ ≤ 푍퐺, and (a) and (b) imply that any nontrivial subgroup
+of 푍퐺 intersects 퐺′ nontrivially. Thus 푁 = {1}. We conclude that every proper quotient of 퐺 is
+abelian.
+□
+An immediate corollary of Proposition 2.2(c) is that 푉 ≔ 퐺/푍퐺 has the structure of an F푝-vector space.
+Note that vector addition in 푉 is written multiplicatively and scalar multiplication of an element
+푥 mod 푍퐺 ∈ 푉 by a scalar 푟 ∈ F푝 is written as
+푟 · (푥 mod 푍퐺) = 푥푟 mod 푍퐺.
+Fix a generator 푧 of 푍퐺. This fixes a generator 푧푝 푗−1 of 퐺′ and hence an identification of 퐺′ with F푝.
+Define a pairing
+푉 × 푉 → 퐺′ = F푝
+(푥 mod 푍퐺, 푦 mod 푍퐺) ↦→ [푥, 푦]
+This pairing is a well-defined, bilinear, nondegenerate, alternating form which makes 푉 into a symplectic
+vector space. In particular, dim푉 is even.
+3
+
+Thus associated to each JN2 group 퐺 is a class (푝 푗, 푚) where |푍퐺| = 푝 푗 and dim푉 = 2푚, so 퐺 fits into the
+short exact sequence
+1
+Z/푝 푗Z
+퐺
+(Z/푝Z)2푚
+0.
+The symplectic structure on central factor groups 푉 = 퐺/푍퐺 is key to the classification theorem because
+symplectic automorphisms on central factor groups can be used to construct isomorphisms between certain
+JN2 groups of the same class. The following lemma extracts from a JN2 group a normalized symplectic
+basis on its associated vector space 푉.
+Lemma 2.3. Let 퐺 be JN2 of class (푝 푗, 푚) where 푝 푗 ≠ 2, with a fixed generator 푧 of 푍퐺. Then there exists
+a symplectic basis B = {푎푖 mod 푍퐺, 푏푖 mod 푍퐺}푚
+푖=1 of 푉 = 퐺/푍퐺 such that the representatives 푎푖, 푏푖 ∈ 퐺
+satisfy either
+(I) 푎푝
+푖 = 푏푝
+푖 = 1 for all 푖, or
+(II) 푎푝
+1 = 푏푝
+1 = 푧 and 푎푝
+푖 = 푏푝
+푖 = 1 for 2 ≤ 푖 ≤ 푚.
+We will say that B is type I or II accordingly.
+Remark 2.4 (Nomenclature). For the reader familiar with existing terminology from [5], a “type I (II)
+basis” as named in our Lemma 2.3 corresponds to a “canonic normal basis with zero (one) pairs of type II”
+in the vocabulary of Newman.
+Proof. Note that 푥 푝 ∈ 푍퐺 for all 푥 ∈ 퐺 because 퐺/푍퐺 has exponent 푝. Let (푍퐺) 푝 = {푢푝 : 푢 ∈ 푍퐺} and
+identify 푍퐺/(푍퐺) 푝 with F푝 by mapping 푧 mod (푍퐺) 푝 ↦→ 1. Define a map
+휈 : 푉 → 푍퐺/(푍퐺) 푝 = F푝
+푥 mod 푍퐺 ↦→ 푥 푝 mod (푍퐺) 푝
+Viewing 푉 = 퐺/푍퐺 as a vector space written multiplicatively, 휈 commutes with scalar multiplication and
+휈((푥 mod 푍퐺)(푦 mod 푍퐺)) = (푥푦) 푝 mod (푍퐺) 푝 = [푦, 푥]
+푝( 푝−1)
+2
+푥 푝푦 푝 mod (푍퐺) 푝
+for 푥, 푦 ∈ 퐺 so 휈 is a linear functional as long as [푦, 푥]
+푝( 푝−1)
+2
+= 1 mod (푍퐺) 푝. This holds if 푝 푗 ≠ 2: If 푝 is
+odd then 푝 | 푝(푝−1)
+2
+so [푦, 푥]
+푝( 푝−1)
+2
+= 1 because 퐺′ has order 푝. If 푗 ≥ 2 then 퐺′ ≰ 푍퐺 so 퐺′ ≤ (푍퐺) 푝.
+If 휈 is the trivial linear functional on 푉, take B to be any symplectic basis of 푉. Otherwise, there exists a
+symplectic basis B of 푉 such that 휈 written with respect to B is the row vector
+휈 =
+�1
+1
+0
+· · ·
+0�
+because symplectic automorphisms act transitively on nontrivial vectors.
+In other words, for each basis vector 푥 푗 mod 푍퐺 ∈ B,
+푥 푝
+푗 mod (푍퐺) 푝 = 휈(푥 푗 mod 푍퐺) = 푧휈푗 mod (푍퐺) 푝
+so there exists 푢 푗 ∈ 푍퐺 such that 푥 푝
+푗 = 푧휈푗푢푝
+푗 . Then 푥 푗푢−1
+푗
+≡ 푥 푗 mod 푍퐺 and (푥 푗푢−1
+푗 ) 푝 = 푧휈푗. Thus
+푥 푗푢−1
+푗
+∈ 퐺 are representatives of the basis B satisfying (I) if 휈 is trivial and (II) otherwise.
+□
+We will now construct two standard non-isomorphic JN2 groups for each given class (푝 푗, 푚). The proof of
+the classification theorem will exhibit an isomorphism from any arbitrary JN2 group to a standard one. The
+primary method of constructing larger JN2 groups from smaller ones is taking a central product.
+4
+
+Definition 2.5 (Central product). Let 퐺 and 퐻 be groups for which there exists an isomorphism
+휑 : 푍퐺 → 푍퐻. Define the central product of 퐺 and 퐻 (with respect to 휑) to be
+퐺 ⊙ 퐻 = (퐺 × 퐻)/푁
+where 푁 =
+�
+(푔, 휑(푔)−1) : 푔 ∈ 푍퐺
+�
+, namely identifying 푍퐺 × 1 with 1 × 푍퐻 by the isomorphism 휑. By
+퐺 ⊙푛 we mean the central product of 푛 copies of 퐺 with the identity isomorphism on 푍퐺.
+Note that if 퐺, 퐻 are JN2 of class (푝 푗, 푚1) and (푝 푗, 푚2) then 퐺 ⊙ 퐻 is JN2 of class (푝 푗, 푚1 + 푚2) by
+Proposition 2.2 since
+1. (퐺 ⊙ 퐻)′ = 퐺′ × 퐻′/푁 � 퐺′ � 퐻′ � Z/푝Z,
+2. 푍(퐺 ⊙ 퐻) � 푍퐺 � 푍퐻, and
+3. (퐺 ⊙ 퐻)/푍(퐺 ⊙ 퐻) � (퐺/푍퐺) × (퐻/푍퐺).
+Construction 2.6 (Standard JN2 groups). Define the groups
+푀(푝 푗) =
+�
+푧, 푎, 푏 : [푧, 푎] = [푧, 푏] = 1; [푎, 푏] = 푧푝 푗−1; 푧푝 푗 = 푎푝 = 푏푝 = 1
+�
+푁(푝 푗) =
+�
+푧, 푎, 푏 : [푧, 푎] = [푧, 푏] = 1; [푎, 푏] = 푧푝 푗−1; 푧푝 푗 = 1; 푎푝 = 푏푝 = 푧
+�
+I(푝 푗, 푚) = 푀(푝 푗) ⊙푚
+II(푝 푗, 푚) = 푁(푝 푗) ⊙ 푀(푝 푗) ⊙(푚−1)
+Observe the following:
+1. 푀(푝 푗) and 푁(푝 푗) are JN2 (by Proposition 2.2) of class (푝 푗, 1) with each center generated by 푧 and
+{푎, 푏} as a symplectic basis of 푉.
+2. I(푝 푗, 푚) and II(푝 푗, 푚) are JN2 of class (푝 푗, 푚) by the remarks following Definition 2.5.
+3. I(푝 푗, 푚) and II(푝 푗, 푚) are not isomorphic when 푝 푗 ≠ 2: The group 푁(푝 푗) has an element of order
+푝 푗+1 (for example, 푎 or 푏) and therefore so does II(푝 푗, 푚). On the contrary, the group 푀(푝 푗), and
+consequently also I(푝 푗, 푚) = 푀(푝 푗) ⊙푚, has exponent at most 푝 푗: The linear functional 휈 (as in the
+proof of Lemma 2.3) is trivial on the symplectic basis {푎, 푏} so 푀(푝 푗) 푝 ≤ (푍퐺) 푝, hence
+푀(푝 푗) 푝 푗 ≤ (푍퐺) 푝 푗 = 1.
+Note: If 푝 푗 = 2, then I(푝 푗, 푚) and II(푝 푗, 푚) are still non-isomorphic: 푀(2) is the dihedral group 퐷8
+and 푁(2) is the quaternion group 푄8, which contain two and six elements of order 4 respectively and
+both have centers of order 2. In particular no elements of order 4 are central. The larger groups
+I(푝 푗, 푚) and II(푝 푗, 푚) can then be distinguished by counting the number of elements of order 4
+because only central elements are identified in the central product. We will not require this case.
+We are now ready to state and prove the classification theorem of JN2 groups.
+Theorem 2.7 (Classification of finite JN2 groups [5, Theorem 5, Theorem 7(c), Lemma 8(i)]). Let 퐺 be
+JN2 of class (푝 푗, 푚). Suppose that 푝 푗 ≠ 2. Then 퐺 is isomorphic to either I(푝 푗, 푚) or II(푝 푗, 푚).
+Proof. Let 푧 be a generator of 푍퐺 and let B be the symplectic basis given by Lemma 2.3. In the notation
+of Lemma 2.3, let 퐻푖 = ⟨푧, 푎푖, 푏푖⟩. If B is type I then 퐻푖 = 푀(푝 푗) for all 푖. If B is type II then
+퐻1 = 푁(푝 푗) and 퐻푖 = 푀(푝 푗) for 푖 ≥ 2.
+The subgroups 퐻푖 commute pairwise, together generate 퐺, and intersect precisely in their centres ⟨푧⟩, so
+퐺 � �푚
+푖=1 퐻푖. Hence 퐺 is isomorphic to I(푝 푗, 푚) or II(푝 푗, 푚) according to the type of the basis B.
+□
+5
+
+Remarks 2.8.
+(a) (Generalizations) For brevity, we have excluded the case of 푝 푗 = 2 and specialized to finite groups.
+With additional work, the 푝 푗 = 2 case and some infinite JN2 groups (those with a countable
+symplectic basis) also admit a classification as central products of elementary JN2 groups, see [5].
+(b) (Special cases) Note that 푀(푝) and 푁(푝) are the only two groups of order 푝3. The group
+푀(푝) = I(푝, 1) is isomorphic to the Heisenberg group over F푝. A generalization of the finite
+Heisenberg groups are the extraspecial groups, which are defined to be 푝-groups 퐺 with 푍퐺 order 푝
+and 퐺/푍퐺 nontrivial elementary abelian. In particular, extraspecial groups are JN2 and it follows
+from Theorem 2.7 that there are precisely two distinct extraspecial groups of order 푝1+2푚 for each
+choice of a prime 푝 and positive integer 푚, and that this exhausts all extraspecial groups.
+3
+Minimal nonabelian quotients of 퐵푛(Σ푔)
+In this section we provide the proof of Theorem 1.5.
+By a braid-reduced quotient, we mean a finite quotient of 퐵푛(Σ푔) with 퐵푛 having cyclic image. The
+content of Theorem 1.5 is to establish the minimum order of nonabelian braid-reduced quotients given 푔
+and 푛 and to classify nonabelian braid-reduced quotients of minimum order.
+The strategy of the proof will be to utilize an explicit presentation of the surface braid groups (Theorem 3.1)
+to characterize braid-reduced quotients by the relations that they must satisfy (Lemma 3.3). We will then
+show that many JN2 groups are nonabelian braid-reduced quotients (Lemma 3.4) and finally prove that all
+nonabelian braid-reduced quotients of minimum order are JN2 groups that were considered in Lemma 3.4.
+The following presentation of 퐵푛(Σ푔) is due to Bellingeri [1].
+Theorem 3.1 (Presentation of 퐵푛(Σ푔) [1, Theorem 1.2]). For 푔 ≥ 1 and 푛 ≥ 2, the surface braid group
+퐵푛(Σ푔) admits the presentation:
+• generators: 휎1, . . . , 휎푛−1, 푎1, . . . , 푎푔, 푏1, . . . , 푏푔.
+• relations:
+braid relations:
+[휎푖, 휎푗] = 1
+(1 ≤ 푖, 푗 ≤ 푛 − 1 and |푖 − 푗| ≥ 2)
+휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1
+(1 ≤ 푖 ≤ 푛 − 2)
+mixed relations:
+(R1)
+[푎푟, 휎푖] = [푏푟, 휎푖] = 1
+(1 ≤ 푟 ≤ 푔 and 푖 ≠ 1)
+(R2)
+[푎푟, 휎−1
+1 푎푟휎−1
+1 ] = [푏푟, 휎−1
+1 푏푟휎−1
+1 ] = 1
+(1 ≤ 푟 ≤ 푔)
+(R3)
+[푎푠, 휎1푎푟휎−1
+1 ] = [푏푠, 휎1푏푟휎−1
+1 ] = 1
+(1 ≤ 푠 < 푟 ≤ 푔)
+[푏푠, 휎1푎푟휎−1
+1 ] = [푎푠, 휎1푏푟휎−1
+1 ] = 1
+(1 ≤ 푠 < 푟 ≤ 푔)
+(R4)
+[푎푟, 휎−1
+1 푏푟휎−1
+1 ] = 휎2
+1
+(1 ≤ 푟 ≤ 푔)
+(TR)
+[푎1, 푏−1
+1 ] · · · [푎푔, 푏−1
+푔 ] = 휎1휎2 · · · 휎2
+푛−1 · · · 휎2휎1
+Remark 3.2 (Geometric interpretation of the presentation). The embeddings 퐵푛 ↩→ 퐵푛(Σ푔) identify the
+Artin braid generators with the Bellingeri generators 휎푖. The remaining generators 푎푟, 푏푟 can be
+understood loosely to be the standard generators of 휋1(Σ푔).
+More precisely, let {푝1, . . . , 푝푛} ∈ UConf푛(Σ푔) denote the basepoint of 퐵푛(Σ푔) and let 퐷 ⊂ Σ푔 be an open
+6
+
+disk with 푝1 ∈ 휕퐷, with 푝2, . . . , 푝푛 in the interior of 퐷. There is an inclusion
+휋1(Σ푔 − 퐷, 푝1) ↩→ 퐵푛(Σ푔)
+which takes a loop 훾 in Σ푔 − 퐷 to the braid on Σ푔 with first strand 훾 and all other strands trivial. The group
+휋1(Σ푔 − 퐷, 푝1) is free on 2푔 generators and surjects onto 휋1(Σ푔, 푝1) which has a standard presentation.
+The surface braid group generators 푎푟, 푏푟 ∈ 퐵푛(Σ푔) can then be understood as a choice of a free generating
+set of 휋1(Σ푔 − 퐷, 푝1) which lifts the standard generating set of 휋1(Σ푔, 푝1). It should be emphasized that
+the lifts are not canonical and that the presentation depends on the choices; the curious reader may refer to
+[1] for illustrations of the loops which produce this particular presentation.
+Lemma 3.3 (Characterization of braid-reduced quotients). Let 푛 ≥ 3 and 푔 ≥ 1. A finite group 퐺 is a
+braid-reduced quotient of 퐵푛(Σ푔) if and only if 퐺 admits a generating set {휎, 푎1, 푏1, . . . , 푎푔, 푏푔} satisfying
+the relations
+(R1′)
+[푎푟, 휎] = [푏푟, 휎] = 1
+(1 ≤ 푟 ≤ 푔)
+(R3′)
+[푎푠, 푎푟] = [푏푠, 푏푟] = [푏푠, 푎푟] = [푎푠, 푏푟] = 1
+(1 ≤ 푠 < 푟 ≤ 푔)
+(R4′)
+[푎푟, 푏푟] = 휎2
+(1 ≤ 푟 ≤ 푔)
+(TR′)
+휎2(푔+푛−1) = 1
+Proof. A finite quotient of 퐵푛(Σ푔) is presented by Theorem 3.1 with additional relations. The condition
+that 퐵푛 has cyclic image in a quotient is equivalent to adding the relations
+휎푖 = 휎1,
+1 ≤ 푖 ≤ 푛.
+If we add these relations and write 휎 = 휎1, the relation (R2) is made redundant and (R1), (R3), and (R4)
+respectively reduce to the relations (R1′), (R3′), and (R4′) as in the statement of the lemma. The final
+relation (TR) reduces to
+[푎1, 푏−1
+1 ] · · · [푎푔, 푏−1
+푔 ] = 휎2(푛−1)
+which is equivalent to (TR′) because from (R4′) we can write 푎푟 = 푏−1
+푟 휎−2푎푟 푏푟 so that
+[푎푟, 푏−1
+푟 ] = 푎푟 푏−1
+푟 푎−1
+푟 푏푟
+(R4′)
+=
+(푏−1
+푟 휎−2푎푟 푏푟)푏−1
+푟 푎−1
+푟 푏푟 = 푏−1
+푟 휎−2푏푟
+(R1′)
+= 휎−2.
+□
+The following lemma proves that many JN2 groups are braid-reduced quotients.
+Lemma 3.4. Let 푛 ≥ 3 and 푔 ≥ 1. Let 푝 be a prime dividing 푔 + 푛 − 1.
+(a) If 푝 = 2, then I(22, 푔) and II(2푗, 푔) for all 푗 ≥ 2 are nonabelian braid-reduced quotients of 퐵푛(Σ푔).
+(b) If 푝 is odd, then I(푝, 푔) and II(푝 푗, 푔) for all 푗 ≥ 1 are nonabelian braid-reduced quotients of 퐵푛(Σ푔).
+Proof. Let 푝 be a prime dividing 푔 + 푛 − 1. By Lemma 3.3 we need to exhibit a generating set {휎, 푎푟, 푏푟 }
+of each group satisfying relations (R1′), (R3′), (R4′), and (TR′).
+In any of the JN2 groups in the statement of the theorem, fix a generator 푧 of the center and choose
+푎1, 푏1, . . . , 푎푔, 푏푔 to be the representatives of a symplectic basis of 푉 given by Lemma 2.3. By
+Theorem 2.7 this basis will be type I for I(22, 푔) and I(푝, 푔), and type II for II(2푗, 푔) and II(푝 푗, 푔). Note
+that with the given symplectic form, the condition that a basis is symplectic is simply that all basis elements
+commute except symplectic pairs [푎푟, 푏푟] = 푧푝 푗−1. In particular, (R3′) is satisfied.
+We will now choose 휎 for each group and verify that {휎, 푎푟, 푏푟 } generate the group and satisfy (R4′).
+1. I(22, 푔) is generated by 휎 = 푧 and the 푎푟, 푏푟. These generators satisfy (R4′) because
+[푎푟, 푏푟] = 푧2 = 휎2.
+7
+
+2. II(2푗, 푔), for a given 푗 ≥ 2, is generated by the 푎푟, 푏푟 alone because 푎푝
+1 = 푧. If we choose 휎 = 푧2 푗−2
+then (R4′) is satisfied because [푎푟, 푏푟] = 푧2 푗−1 = 휎2.
+3. I(푝, 푔) for odd prime 푝 is generated by 휎 = 푧(푝 푗+1)/2 and the 푎푟, 푏푟. Then (R4′) is satisfied because
+[푎푟, 푏푟] = 푧 = 휎2.
+4. II(푝 푗, 푔), for given odd prime 푝 and 푗 ≥ 1, is generated by 푎푟, 푏푟 alone because 푎푝
+1 = 푧. Then set
+휎 = 푧(푝 푗+푝 푗−1)/2 so that (R4′) is satisfied because [푎푟, 푏푟] = 푧푝 푗−1 = 휎2.
+In all cases 휎 was chosen to be central, hence (R1′) is satisfied.
+It remains to check that (TR′) holds, namely that |휎| divides 2(푔 + 푛 − 1). Recall that we are assuming that
+푝 | (푔 + 푛 − 1). In cases 1 and 2, we have 푝 = 2 and |휎| = 4 = 2푝 | 2(푔 + 푛 − 1). In cases 3 and 4, we have
+|휎| = 푝 | (푔 + 푛 − 1).
+□
+We are now prepared to prove our main result, Theorem 1.5.
+Proof of Theorem 1.5. Let 퐺 be a nonabelian braid-reduced quotient of minimum order and let
+{휎, 푎1, 푏1, . . . , 푎푔, 푏푔} denote the generating set of 퐺 as given by Lemma 3.3. By (R1′) and (R3′), all pairs
+of these generators commute except for pairs 푎푟, 푏푟 so 퐺′ =
+�
+휎2�
+by (R4′). Then 퐺′ is central and
+nontrivial, which is to say that 퐺 is 2-step nilpotent. By minimality, 퐺 has no proper nonabelian quotients.
+Therefore 퐺 is JN2 of some class (푝 푗, 푚).
+We make three claims:
+1. 푝 푗 ≠ 2,
+2. 푚 = 푔, and
+3. 푝 | (푔 + 푛 − 1).
+These claims will complete the proof: Since |퐺| = 푝2푚+푗, it follows from claims 1 and 2 that |퐺| ≥ 푝2푔+1 if
+푝 is odd and |퐺| ≥ 22푔+2 if 푝 = 2. Claims 1 and 3 along with the minimality of 퐺 together imply that 퐺 is
+one of (in particular, the smallest of) the quotients constructed in Lemma 3.4. Explicitly: If 푔 + 푛 − 1 is
+even then 푝 푗 = 22. Otherwise, 푝 푗 = 푝 where 푝 is the smallest prime dividing 푔 + 푛 − 1. Finally, 퐺 must be
+isomorphic to either I(푝 푗, 푔) or II(푝 푗, 푔) by Theorem 2.7.
+Proof of claims: Let 푑 = |휎|. By (R1′), 휎 is central so 푑 | 푝 푗. But 푝 = |퐺′| = |휎2| so 푑 | 2푝. Thus either 푝
+is odd and 푝 = 푑, or 푝 = 2 and 푑 = 4.
+1. If 푝 is odd then 푝 푗 ≠ 2. If 푝 = 2 then 푝 푗 ≥ 푑 = 4 so 푝 푗 ≠ 2.
+2. We will show that dim푉 = 2푔 by proving that
+B = {푎푟 mod 푍퐺, 푏푟 mod 푍퐺}푔
+푟=1
+is a basis of 푉. Every element 푥 ∈ 퐺 can be written uniquely in the form
+푥 = 휎푘푎푖1
+1 · · · 푎푖푔
+푔 푏 푗1
+1 · · · 푏 푗푔
+푔
+using commuting relations (R1′), (R3′), (R4′) so B is a generating set. To prove that B is linearly
+independent, let
+푦 = 푎푖1
+1 · · · 푎푖푔
+푔 푏 푗1
+1 · · · 푏 푗푔
+푔 ∈ 퐺
+and suppose that 푦 = 0 mod 푍퐺, which is to suppose that an arbitrary linear combination of elements
+of B is trivial in 푉. Then 푦 is central so
+[푦, 푏1] = [푎푖1
+1 , 푏1] = 휎−2푖1 = 1
+8
+
+which implies that 푑 | 2푖1 and thus 푖1 = 0 mod 푝: If 푝 is odd then 푑 = 푝 so 푝 | 푖1. If 푝 = 2 then
+푑 = 4 | 2푖1 so 푝 = 2 | 푖1.
+Similarly 푖푟 = 푗푟 = 0 mod 푝 for all 푟, which is to say that all coefficients of the linear combination are
+trivial over the base field F푝. This proves the linear independence of B.
+3. The relation (TR′) imposes the relation 푑 | 2(푔 + 푛 − 1). Either 푑 = 푝 is odd or 푑 = 4 and 푝 = 2; in
+both cases (TR′) implies that 푝 | (푔 + 푛 − 1).
+□
+Remark 3.5. In his paper [1], Bellingeri also gives a presentation of the braid group of the genus 푔 surface
+with 푝 punctures (equivalently for the purposes of braid groups, 푝 boundary components). The above
+methods can be used nearly verbatim to prove that the smallest nonabelian quotient of 퐵푛(Σ푔,푝) is the
+smaller of 푆푛 or I(22, 푔) and II(22, 푔).
+References
+[1] P. Bellingeri. On presentations of surface braid groups. Journal of Algebra, 274(2):543–563, Apr. 2004.
+[2] J. S. Birman. On braid groups. Communications on Pure and Applied Mathematics, 22(1):41–72, Jan. 1969.
+[3] E. Fadell and J. Van Buskirk. On the braid groups of 퐸2 and 푆2. Bulletin of the American Mathematical Society,
+67(2):211–213, 1961.
+[4] S. Kolay. Smallest non-cyclic quotients of braid and mapping class groups, Oct. 2021. arXiv: 2110.02162.
+[5] M. F. Newman. On a Class of Nilpotent Groups. Proceedings of the London Mathematical Society,
+s3-10(1):365–375, 1960.
+9
+
diff --git a/r9AzT4oBgHgl3EQf6P5j/content/tmp_files/load_file.txt b/r9AzT4oBgHgl3EQf6P5j/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..500e3f6c589a3d97fbca828ae17ea06fc99f93df
--- /dev/null
+++ b/r9AzT4oBgHgl3EQf6P5j/content/tmp_files/load_file.txt
@@ -0,0 +1,377 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf,len=376
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='01872v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='GT]  5 Jan 2023  Smallest nonabelian quotients of surface braid groups  Cindy Tan  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  We give a sharp lower bound on the size of nonabelian quotients of the surface  braid group 퐵푛(Σ푔) and classify all quotients that attain the lower bound: Depending on 푛 and  푔, a quotient of minimum order is either a symmetric group or a 2-step nilpotent group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  1  Introduction  The Artin braid group 퐵푛 arises as the fundamental group of UConf푛(D), the configuration space of 푛  distinct unordered points on the open disk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' One can generalize this construction to define for an oriented,  closed genus 푔 surface Σ푔 the surface braid groups  퐵푛(Σ푔) = 휋1(UConf푛(Σ푔)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  It was shown by Kolay [4] that for 푛 = 3 or 푛 ≥ 5, the smallest noncyclic finite quotient of 퐵푛 is the  symmetric group 푆푛, in the sense that 푆푛 has minimum order amongst noncyclic quotients of 퐵푛 and 푆푛 is  the unique noncyclic quotient of 퐵푛 of minimum order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  In this paper we show that in contrast to the regular braid groups, for a fixed genus 푔 surface Σ푔 there exist  infinitely many 푛 for which 푆푛 is not the smallest nonabelian quotient of 퐵푛(Σ푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The new minimal  quotients that arise are 2-step nilpotent groups which will be defined in Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='1 (Smallest nonabelian quotients of 퐵푛(Σ푔)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 푛 ≥ 5 and 푔 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Suppose that 퐺 is a finite  nonabelian quotient of 퐵푛(Σ푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (a) If 푔 + 푛 − 1 is even then |퐺| ≥ min(푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=', 22푔+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Furthermore,  (i) If 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' < 22푔+2 and |퐺| = 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' then 퐺 � 푆푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (ii) If 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' > 22푔+2 and |퐺| = 22푔+2 then either 퐺 � I(22, 푔) or 퐺 � II(22, 푔) (and both quotients  arise in every such case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (b) If 푔 + 푛 − 1 is odd then |퐺| ≥ min(푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=', 푝2푔+1), where 푝 is the smallest prime divisor of 푔 + 푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Furthermore,  (i) If 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' < 푝2푔+1 and |퐺| = 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' then 퐺 � 푆푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (ii) If 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' > 푝2푔+1 and |퐺| = 푝2푔+1 then either 퐺 � I(푝, 푔) or 퐺 � II(푝, 푔) (and both quotients arise  in every such case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  The groups I(푝 푗, 푔) and II(푝 푗, 푔) are nonisomorphic 2-step nilpotent groups defined in Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='1 implies the following qualitative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (a) Fix 푔 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' There exist infinitely many 푛 for which the smallest quotient of 퐵푛(Σ푔) is 푆푛 and also  infinitely many 푛 for which the smallest quotient is not 푆푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (b) Fix 푛 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' For all sufficiently large 푔, the smallest nonabelian quotient of 퐵푛(Σ푔) is 푆푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Also, there  exists a (small) 푔 for which this not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  1  Remarks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3 (Smaller cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (a) If 푛 = 1, 2, 3, 4 and 푔 ≥ 1 (with the exception of (푛, 푔) = (1, 1) where 퐵푛(Σ푔) = 휋1(푇2) = Z2 is  abelian) then the symmetric group 푆3 is the smallest nonabelian quotient of 퐵푛(Σ푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (b) If 푔 = 0 then 퐵푛(Σ푔) is the spherical braid group 퐵푛(푆2) which is an intermediate quotient of the  map 퐵푛 → 푆푛 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' It follows from the result of Kolay [4] that the smallest quotient of 퐵푛(푆2) is 푆푛  for 푛 ≥ 5 and 푆3 for 푛 = 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' For 푛 = 1 and 2 we note that 퐵푛(푆2) is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='4 (Geometric interpretation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The quotient map 퐵푛(Σ푔) → 푆푛 can be interpreted geometrically  as mapping a surface braid to the permutation it performs on basepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' We pose the question: What is the  geometric interpretation of the new nilpotent minimal quotients, or more generally the nilpotent quotients  of 퐵푛(Σ푔) enumerated in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Proof method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  By mapping a braid to its permutation on points, 푆푛 is a finite quotient of 퐵푛(Σ푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' For  푔 ≥ 1, there is an embedding 퐵푛 ↩→ 퐵푛(Σ푔) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' It follows from Kolay that if 퐵푛 → 퐵푛(Σ푔) → 퐺 has  noncyclic image, then |퐺| ≥ 푛!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' with the bound attained only by 퐺 � 푆푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  It remains to consider nonabelian quotients of 퐵푛(Σ푔) where 퐵푛 has cyclic image, which we will refer to as  braid-reduced quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' For certain choices of 푝 and 푗 the nilpotent groups I(푝 푗, 푔) and II(푝 푗, 푔) defined  in Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='6 are nonabelian braid-reduced quotients of 퐵푛(Σ푔), and in fact can be constructed to have  minimum order amongst braid-reduced quotients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' this comprises the primary contribution of this paper:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5 (Smallest nonabelian braid-reduced quotients of 퐵푛(Σ푔)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 푛 ≥ 3 and 푔 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Suppose 퐺 is  a finite nonabelian quotient of 퐵푛(Σ푔) where 퐵푛 has cyclic image in 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Then 퐺 is 2-step nilpotent and  (a) If 푔 + 푛 − 1 is even then |퐺| ≥ 22푔+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In the case of equality, either 퐺 � I(22, 푔) or 퐺 � II(22, 푔) (and  both quotients arise in every such case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (b) If 푔 + 푛 − 1 is odd then |퐺| ≥ 푝2푔+1 where 푝 is the smallest prime divisor of 푔 + 푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In the case of  equality, either 퐺 � I(푝, 푔) or 퐺 � II(푝, 푔) (and both quotients arise in every such case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5 we utilize a presentation of 퐵푛(Σ푔) (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='1) due to Bellingeri [1] and assume  that 퐵푛 has cyclic image to reduce the relations to conclude that a braid-reduced quotient 퐺 must be  nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' If we further assume that 퐺 is a nonabelian braid-reduced quotient of minimum order then 퐺  belongs to a class of nilpotent groups called JN2 groups (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='1) which were classified by Newman  in 1960 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Section 2 provides a self-contained exposition of the classification of JN2 groups and we prove  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  I am grateful to my advisor Benson Farb for continued support throughout this  project and for detailed comments on many revisions of this paper, as well as for suggesting this problem in  the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' I thank Peter Huxford for useful discussions about braid groups and small 푝-groups, and for  many helpful suggestions during the editing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' I also thank Dan Margalit for taking the time to read  and comment on an earlier draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  2  Just 2-step nilpotent groups  In this section we introduce and classify JN2 groups, a class of nilpotent groups which includes all minimal  nonabelian quotients of 퐵푛(Σ푔) in which 퐵푛 has cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' A group 퐺 is just 2-step nilpotent (JN2) if 퐺 is 2-step nilpotent (in particular, nonabelian)  and every proper quotient of 퐺 is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  2  Finite JN2 groups admit a complete and explicit classification due to Newman [5]: Any finite JN2 group  can be assigned a unique class (푝 푗, 푚) where 푝 is a prime and 푗 and 푚 are positive integers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' up to  isomorphism, there are precisely two JN2 groups of a given class (푝 푗, 푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' We will state and prove this  classification theorem in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='7, following the general ideas of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  All JN2 groups will hereafter be assumed to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The following proposition will allow us to define the  class (푝 푗, 푚) of a JN2 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='2 (Characterization of JN2 groups [5, Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' A finite group 퐺 is JN2 if and only if  there exists a prime 푝 such that  (a) 퐺′ ≔ [퐺, 퐺] is cyclic of order 푝,  (b) the center 푍퐺 is cyclic of order a power of 푝, and  (c) 퐺/푍퐺 is elementary abelian of exponent 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  ⇒ Let 퐺 be a finite JN2 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' For every nontrivial normal subgroup 푁 ⊴ 퐺, we have that 퐺′ ≤ 푁 since  any proper quotient of 퐺 is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Since 퐺 is 2-step nilpotent, 퐺′ ≤ 푍퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Consequently,  (a) 퐺′ is abelian and admits no proper nontrivial subgroups so 퐺′ � Z/푝Z for some prime 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (b) 푍퐺 cannot be properly decomposed as a direct sum: Any nontrivial subgroup of 푍퐺 contains 퐺′  so no two nontrivial subgroups intersect trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Since 푍퐺 is finite abelian, it must be cyclic of  prime power order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The prime must be 푝 because 퐺′ ≤ 푍퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (c) 퐺/푍퐺 is abelian because 퐺′ ≤ 푍퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' For 푥, 푦 ∈ 퐺, we have that [푥 푝, 푦] = [푥, 푦] 푝 by using the  identity  [푥푧, 푦] = 푧[푥, 푦]푧−1[푧, 푦]  and noting that [푥, 푦] is central because 퐺′ ≤ 푍퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' But 퐺′ has order 푝, so in fact [푥 푝, 푦] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Thus 푥 푝 ∈ 푍퐺 for all 푥 ∈ 퐺, which is to say that 퐺/푍퐺 has exponent 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  ⇐ Suppose 퐺 is a finite group satisfying (a), (b), and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Then 퐺′ ≠ {1} by (a) and 퐺′ ≤ 푍퐺 by (c) so 퐺  is 2-step nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  If 푁 ⊴ 퐺 is a normal subgroup with 퐺′ ≰ 푁 then 푁 ∩ 퐺′ = {1} by (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Since 푁 is normal,  [푁, 퐺] ≤ 푁 ∩ 퐺′ = {1} so 푁 ≤ 푍퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' But 퐺′ ≤ 푍퐺, and (a) and (b) imply that any nontrivial subgroup  of 푍퐺 intersects 퐺′ nontrivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Thus 푁 = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' We conclude that every proper quotient of 퐺 is  abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  □  An immediate corollary of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='2(c) is that 푉 ≔ 퐺/푍퐺 has the structure of an F푝-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Note that vector addition in 푉 is written multiplicatively and scalar multiplication of an element  푥 mod 푍퐺 ∈ 푉 by a scalar 푟 ∈ F푝 is written as  푟 · (푥 mod 푍퐺) = 푥푟 mod 푍퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Fix a generator 푧 of 푍퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' This fixes a generator 푧푝 푗−1 of 퐺′ and hence an identification of 퐺′ with F푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Define a pairing  푉 × 푉 → 퐺′ = F푝  (푥 mod 푍퐺, 푦 mod 푍퐺) ↦→ [푥, 푦]  This pairing is a well-defined, bilinear, nondegenerate, alternating form which makes 푉 into a symplectic  vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In particular, dim푉 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  3  Thus associated to each JN2 group 퐺 is a class (푝 푗, 푚) where |푍퐺| = 푝 푗 and dim푉 = 2푚, so 퐺 fits into the  short exact sequence  1  Z/푝 푗Z  퐺  (Z/푝Z)2푚  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  The symplectic structure on central factor groups 푉 = 퐺/푍퐺 is key to the classification theorem because  symplectic automorphisms on central factor groups can be used to construct isomorphisms between certain  JN2 groups of the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The following lemma extracts from a JN2 group a normalized symplectic  basis on its associated vector space 푉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 퐺 be JN2 of class (푝 푗, 푚) where 푝 푗 ≠ 2, with a fixed generator 푧 of 푍퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Then there exists  a symplectic basis B = {푎푖 mod 푍퐺, 푏푖 mod 푍퐺}푚  푖=1 of 푉 = 퐺/푍퐺 such that the representatives 푎푖, 푏푖 ∈ 퐺  satisfy either  (I) 푎푝  푖 = 푏푝  푖 = 1 for all 푖, or  (II) 푎푝  1 = 푏푝  1 = 푧 and 푎푝  푖 = 푏푝  푖 = 1 for 2 ≤ 푖 ≤ 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  We will say that B is type I or II accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='4 (Nomenclature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' For the reader familiar with existing terminology from [5], a “type I (II)  basis” as named in our Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3 corresponds to a “canonic normal basis with zero (one) pairs of type II”  in the vocabulary of Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Note that 푥 푝 ∈ 푍퐺 for all 푥 ∈ 퐺 because 퐺/푍퐺 has exponent 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let (푍퐺) 푝 = {푢푝 : 푢 ∈ 푍퐺} and  identify 푍퐺/(푍퐺) 푝 with F푝 by mapping 푧 mod (푍퐺) 푝 ↦→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Define a map  휈 : 푉 → 푍퐺/(푍퐺) 푝 = F푝  푥 mod 푍퐺 ↦→ 푥 푝 mod (푍퐺) 푝  Viewing 푉 = 퐺/푍퐺 as a vector space written multiplicatively, 휈 commutes with scalar multiplication and  휈((푥 mod 푍퐺)(푦 mod 푍퐺)) = (푥푦) 푝 mod (푍퐺) 푝 = [푦, 푥]  푝( 푝−1)  2  푥 푝푦 푝 mod (푍퐺) 푝  for 푥, 푦 ∈ 퐺 so 휈 is a linear functional as long as [푦, 푥]  푝( 푝−1)  2  = 1 mod (푍퐺) 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' This holds if 푝 푗 ≠ 2: If 푝 is  odd then 푝 | 푝(푝−1)  2  so [푦, 푥]  푝( 푝−1)  2  = 1 because 퐺′ has order 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' If 푗 ≥ 2 then 퐺′ ≰ 푍퐺 so 퐺′ ≤ (푍퐺) 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  If 휈 is the trivial linear functional on 푉, take B to be any symplectic basis of 푉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Otherwise, there exists a  symplectic basis B of 푉 such that 휈 written with respect to B is the row vector  휈 =  �1  1  0  · ·  0�  because symplectic automorphisms act transitively on nontrivial vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  In other words, for each basis vector 푥 푗 mod 푍퐺 ∈ B,  푥 푝  푗 mod (푍퐺) 푝 = 휈(푥 푗 mod 푍퐺) = 푧휈푗 mod (푍퐺) 푝  so there exists 푢 푗 ∈ 푍퐺 such that 푥 푝  푗 = 푧휈푗푢푝  푗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Then 푥 푗푢−1  푗  ≡ 푥 푗 mod 푍퐺 and (푥 푗푢−1  푗 ) 푝 = 푧휈푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Thus  푥 푗푢−1  푗  ∈ 퐺 are representatives of the basis B satisfying (I) if 휈 is trivial and (II) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  □  We will now construct two standard non-isomorphic JN2 groups for each given class (푝 푗, 푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The proof of  the classification theorem will exhibit an isomorphism from any arbitrary JN2 group to a standard one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The  primary method of constructing larger JN2 groups from smaller ones is taking a central product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  4  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5 (Central product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 퐺 and 퐻 be groups for which there exists an isomorphism  휑 : 푍퐺 → 푍퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Define the central product of 퐺 and 퐻 (with respect to 휑) to be  퐺 ⊙ 퐻 = (퐺 × 퐻)/푁  where 푁 =  �  (푔, 휑(푔)−1) : 푔 ∈ 푍퐺  �  , namely identifying 푍퐺 × 1 with 1 × 푍퐻 by the isomorphism 휑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' By  퐺 ⊙푛 we mean the central product of 푛 copies of 퐺 with the identity isomorphism on 푍퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Note that if 퐺, 퐻 are JN2 of class (푝 푗, 푚1) and (푝 푗, 푚2) then 퐺 ⊙ 퐻 is JN2 of class (푝 푗, 푚1 + 푚2) by  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='2 since  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' (퐺 ⊙ 퐻)′ = 퐺′ × 퐻′/푁 � 퐺′ � 퐻′ � Z/푝Z,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푍(퐺 ⊙ 퐻) � 푍퐺 � 푍퐻, and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' (퐺 ⊙ 퐻)/푍(퐺 ⊙ 퐻) � (퐺/푍퐺) × (퐻/푍퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='6 (Standard JN2 groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Define the groups  푀(푝 푗) =  �  푧, 푎, 푏 : [푧, 푎] = [푧, 푏] = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' [푎, 푏] = 푧푝 푗−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푧푝 푗 = 푎푝 = 푏푝 = 1  �  푁(푝 푗) =  �  푧, 푎, 푏 : [푧, 푎] = [푧, 푏] = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' [푎, 푏] = 푧푝 푗−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푧푝 푗 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푎푝 = 푏푝 = 푧  �  I(푝 푗, 푚) = 푀(푝 푗) ⊙푚  II(푝 푗, 푚) = 푁(푝 푗) ⊙ 푀(푝 푗) ⊙(푚−1)  Observe the following:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푀(푝 푗) and 푁(푝 푗) are JN2 (by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='2) of class (푝 푗, 1) with each center generated by 푧 and  {푎, 푏} as a symplectic basis of 푉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' I(푝 푗, 푚) and II(푝 푗, 푚) are JN2 of class (푝 푗, 푚) by the remarks following Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' I(푝 푗, 푚) and II(푝 푗, 푚) are not isomorphic when 푝 푗 ≠ 2: The group 푁(푝 푗) has an element of order  푝 푗+1 (for example, 푎 or 푏) and therefore so does II(푝 푗, 푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' On the contrary, the group 푀(푝 푗), and  consequently also I(푝 푗, 푚) = 푀(푝 푗) ⊙푚, has exponent at most 푝 푗: The linear functional 휈 (as in the  proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3) is trivial on the symplectic basis {푎, 푏} so 푀(푝 푗) 푝 ≤ (푍퐺) 푝, hence  푀(푝 푗) 푝 푗 ≤ (푍퐺) 푝 푗 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Note: If 푝 푗 = 2, then I(푝 푗, 푚) and II(푝 푗, 푚) are still non-isomorphic: 푀(2) is the dihedral group 퐷8  and 푁(2) is the quaternion group 푄8, which contain two and six elements of order 4 respectively and  both have centers of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In particular no elements of order 4 are central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The larger groups  I(푝 푗, 푚) and II(푝 푗, 푚) can then be distinguished by counting the number of elements of order 4  because only central elements are identified in the central product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' We will not require this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  We are now ready to state and prove the classification theorem of JN2 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='7 (Classification of finite JN2 groups [5, Theorem 5, Theorem 7(c), Lemma 8(i)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 퐺 be  JN2 of class (푝 푗, 푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Suppose that 푝 푗 ≠ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Then 퐺 is isomorphic to either I(푝 푗, 푚) or II(푝 푗, 푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 푧 be a generator of 푍퐺 and let B be the symplectic basis given by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In the notation  of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3, let 퐻푖 = ⟨푧, 푎푖, 푏푖⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' If B is type I then 퐻푖 = 푀(푝 푗) for all 푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' If B is type II then  퐻1 = 푁(푝 푗) and 퐻푖 = 푀(푝 푗) for 푖 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  The subgroups 퐻푖 commute pairwise, together generate 퐺, and intersect precisely in their centres ⟨푧⟩, so  퐺 � �푚  푖=1 퐻푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Hence 퐺 is isomorphic to I(푝 푗, 푚) or II(푝 푗, 푚) according to the type of the basis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  □  5  Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (a) (Generalizations) For brevity, we have excluded the case of 푝 푗 = 2 and specialized to finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  With additional work, the 푝 푗 = 2 case and some infinite JN2 groups (those with a countable  symplectic basis) also admit a classification as central products of elementary JN2 groups, see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (b) (Special cases) Note that 푀(푝) and 푁(푝) are the only two groups of order 푝3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The group  푀(푝) = I(푝, 1) is isomorphic to the Heisenberg group over F푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' A generalization of the finite  Heisenberg groups are the extraspecial groups, which are defined to be 푝-groups 퐺 with 푍퐺 order 푝  and 퐺/푍퐺 nontrivial elementary abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In particular, extraspecial groups are JN2 and it follows  from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='7 that there are precisely two distinct extraspecial groups of order 푝1+2푚 for each  choice of a prime 푝 and positive integer 푚, and that this exhausts all extraspecial groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  3  Minimal nonabelian quotients of 퐵푛(Σ푔)  In this section we provide the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  By a braid-reduced quotient, we mean a finite quotient of 퐵푛(Σ푔) with 퐵푛 having cyclic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The  content of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5 is to establish the minimum order of nonabelian braid-reduced quotients given 푔  and 푛 and to classify nonabelian braid-reduced quotients of minimum order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  The strategy of the proof will be to utilize an explicit presentation of the surface braid groups (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='1)  to characterize braid-reduced quotients by the relations that they must satisfy (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' We will then  show that many JN2 groups are nonabelian braid-reduced quotients (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='4) and finally prove that all  nonabelian braid-reduced quotients of minimum order are JN2 groups that were considered in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  The following presentation of 퐵푛(Σ푔) is due to Bellingeri [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='1 (Presentation of 퐵푛(Σ푔) [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' For 푔 ≥ 1 and 푛 ≥ 2, the surface braid group  퐵푛(Σ푔) admits the presentation:  generators: 휎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' , 휎푛−1, 푎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' , 푎푔, 푏1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' , 푏푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  relations:  braid relations:  [휎푖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎푗] = 1  (1 ≤ 푖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푗 ≤ 푛 − 1 and |푖 − 푗| ≥ 2)  휎푖휎푖+1휎푖 = 휎푖+1휎푖휎푖+1  (1 ≤ 푖 ≤ 푛 − 2)  mixed relations:  (R1)  [푎푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎푖] = [푏푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎푖] = 1  (1 ≤ 푟 ≤ 푔 and 푖 ≠ 1)  (R2)  [푎푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎−1  1 푎푟휎−1  1 ] = [푏푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎−1  1 푏푟휎−1  1 ] = 1  (1 ≤ 푟 ≤ 푔)  (R3)  [푎푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎1푎푟휎−1  1 ] = [푏푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎1푏푟휎−1  1 ] = 1  (1 ≤ 푠 < 푟 ≤ 푔)  [푏푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎1푎푟휎−1  1 ] = [푎푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎1푏푟휎−1  1 ] = 1  (1 ≤ 푠 < 푟 ≤ 푔)  (R4)  [푎푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 휎−1  1 푏푟휎−1  1 ] = 휎2  1  (1 ≤ 푟 ≤ 푔)  (TR)  [푎1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푏−1  1 ] · · · [푎푔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푏−1  푔 ] = 휎1휎2 · · · 휎2  푛−1 · · · 휎2휎1  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='2 (Geometric interpretation of the presentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The embeddings 퐵푛 ↩→ 퐵푛(Σ푔) identify the  Artin braid generators with the Bellingeri generators 휎푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The remaining generators 푎푟, 푏푟 can be  understood loosely to be the standard generators of 휋1(Σ푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  More precisely, let {푝1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' , 푝푛} ∈ UConf푛(Σ푔) denote the basepoint of 퐵푛(Σ푔) and let 퐷 ⊂ Σ푔 be an open  6  disk with 푝1 ∈ 휕퐷, with 푝2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' , 푝푛 in the interior of 퐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' There is an inclusion  휋1(Σ푔 − 퐷, 푝1) ↩→ 퐵푛(Σ푔)  which takes a loop 훾 in Σ푔 − 퐷 to the braid on Σ푔 with first strand 훾 and all other strands trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The group  휋1(Σ푔 − 퐷, 푝1) is free on 2푔 generators and surjects onto 휋1(Σ푔, 푝1) which has a standard presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  The surface braid group generators 푎푟, 푏푟 ∈ 퐵푛(Σ푔) can then be understood as a choice of a free generating  set of 휋1(Σ푔 − 퐷, 푝1) which lifts the standard generating set of 휋1(Σ푔, 푝1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' It should be emphasized that  the lifts are not canonical and that the presentation depends on the choices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' the curious reader may refer to  [1] for illustrations of the loops which produce this particular presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3 (Characterization of braid-reduced quotients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 푛 ≥ 3 and 푔 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' A finite group 퐺 is a  braid-reduced quotient of 퐵푛(Σ푔) if and only if 퐺 admits a generating set {휎, 푎1, 푏1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' , 푎푔, 푏푔} satisfying  the relations  (R1′)  [푎푟, 휎] = [푏푟, 휎] = 1  (1 ≤ 푟 ≤ 푔)  (R3′)  [푎푠, 푎푟] = [푏푠, 푏푟] = [푏푠, 푎푟] = [푎푠, 푏푟] = 1  (1 ≤ 푠 < 푟 ≤ 푔)  (R4′)  [푎푟, 푏푟] = 휎2  (1 ≤ 푟 ≤ 푔)  (TR′)  휎2(푔+푛−1) = 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' A finite quotient of 퐵푛(Σ푔) is presented by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='1 with additional relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The condition  that 퐵푛 has cyclic image in a quotient is equivalent to adding the relations  휎푖 = 휎1,  1 ≤ 푖 ≤ 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  If we add these relations and write 휎 = 휎1, the relation (R2) is made redundant and (R1), (R3), and (R4)  respectively reduce to the relations (R1′), (R3′), and (R4′) as in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The final  relation (TR) reduces to  [푎1, 푏−1  1 ] · · · [푎푔, 푏−1  푔 ] = 휎2(푛−1)  which is equivalent to (TR′) because from (R4′) we can write 푎푟 = 푏−1  푟 휎−2푎푟 푏푟 so that  [푎푟, 푏−1  푟 ] = 푎푟 푏−1  푟 푎−1  푟 푏푟  (R4′)  =  (푏−1  푟 휎−2푎푟 푏푟)푏−1  푟 푎−1  푟 푏푟 = 푏−1  푟 휎−2푏푟  (R1′)  = 휎−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  □  The following lemma proves that many JN2 groups are braid-reduced quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 푛 ≥ 3 and 푔 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 푝 be a prime dividing 푔 + 푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (a) If 푝 = 2, then I(22, 푔) and II(2푗, 푔) for all 푗 ≥ 2 are nonabelian braid-reduced quotients of 퐵푛(Σ푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  (b) If 푝 is odd, then I(푝, 푔) and II(푝 푗, 푔) for all 푗 ≥ 1 are nonabelian braid-reduced quotients of 퐵푛(Σ푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 푝 be a prime dividing 푔 + 푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3 we need to exhibit a generating set {휎, 푎푟, 푏푟 }  of each group satisfying relations (R1′), (R3′), (R4′), and (TR′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  In any of the JN2 groups in the statement of the theorem, fix a generator 푧 of the center and choose  푎1, 푏1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' , 푎푔, 푏푔 to be the representatives of a symplectic basis of 푉 given by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' By  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='7 this basis will be type I for I(22, 푔) and I(푝, 푔), and type II for II(2푗, 푔) and II(푝 푗, 푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Note  that with the given symplectic form, the condition that a basis is symplectic is simply that all basis elements  commute except symplectic pairs [푎푟, 푏푟] = 푧푝 푗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In particular, (R3′) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  We will now choose 휎 for each group and verify that {휎, 푎푟, 푏푟 } generate the group and satisfy (R4′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' I(22, 푔) is generated by 휎 = 푧 and the 푎푟, 푏푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' These generators satisfy (R4′) because  [푎푟, 푏푟] = 푧2 = 휎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' II(2푗, 푔), for a given 푗 ≥ 2, is generated by the 푎푟, 푏푟 alone because 푎푝  1 = 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' If we choose 휎 = 푧2 푗−2  then (R4′) is satisfied because [푎푟, 푏푟] = 푧2 푗−1 = 휎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' I(푝, 푔) for odd prime 푝 is generated by 휎 = 푧(푝 푗+1)/2 and the 푎푟, 푏푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Then (R4′) is satisfied because  [푎푟, 푏푟] = 푧 = 휎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' II(푝 푗, 푔), for given odd prime 푝 and 푗 ≥ 1, is generated by 푎푟, 푏푟 alone because 푎푝  1 = 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Then set  휎 = 푧(푝 푗+푝 푗−1)/2 so that (R4′) is satisfied because [푎푟, 푏푟] = 푧푝 푗−1 = 휎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  In all cases 휎 was chosen to be central, hence (R1′) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  It remains to check that (TR′) holds, namely that |휎| divides 2(푔 + 푛 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Recall that we are assuming that  푝 | (푔 + 푛 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In cases 1 and 2, we have 푝 = 2 and |휎| = 4 = 2푝 | 2(푔 + 푛 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In cases 3 and 4, we have  |휎| = 푝 | (푔 + 푛 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  □  We are now prepared to prove our main result, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Let 퐺 be a nonabelian braid-reduced quotient of minimum order and let  {휎, 푎1, 푏1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' , 푎푔, 푏푔} denote the generating set of 퐺 as given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' By (R1′) and (R3′), all pairs  of these generators commute except for pairs 푎푟, 푏푟 so 퐺′ =  �  휎2�  by (R4′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Then 퐺′ is central and  nontrivial, which is to say that 퐺 is 2-step nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' By minimality, 퐺 has no proper nonabelian quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Therefore 퐺 is JN2 of some class (푝 푗, 푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  We make three claims:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푝 푗 ≠ 2,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푚 = 푔, and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 푝 | (푔 + 푛 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  These claims will complete the proof: Since |퐺| = 푝2푚+푗, it follows from claims 1 and 2 that |퐺| ≥ 푝2푔+1 if  푝 is odd and |퐺| ≥ 22푔+2 if 푝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Claims 1 and 3 along with the minimality of 퐺 together imply that 퐺 is  one of (in particular, the smallest of) the quotients constructed in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Explicitly: If 푔 + 푛 − 1 is  even then 푝 푗 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Otherwise, 푝 푗 = 푝 where 푝 is the smallest prime dividing 푔 + 푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Finally, 퐺 must be  isomorphic to either I(푝 푗, 푔) or II(푝 푗, 푔) by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Proof of claims: Let 푑 = |휎|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' By (R1′), 휎 is central so 푑 | 푝 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' But 푝 = |퐺′| = |휎2| so 푑 | 2푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Thus either 푝  is odd and 푝 = 푑, or 푝 = 2 and 푑 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' If 푝 is odd then 푝 푗 ≠ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' If 푝 = 2 then 푝 푗 ≥ 푑 = 4 so 푝 푗 ≠ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' We will show that dim푉 = 2푔 by proving that  B = {푎푟 mod 푍퐺, 푏푟 mod 푍퐺}푔  푟=1  is a basis of 푉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Every element 푥 ∈ 퐺 can be written uniquely in the form  푥 = 휎푘푎푖1  1 · · · 푎푖푔  푔 푏 푗1  1 · · · 푏 푗푔  푔  using commuting relations (R1′), (R3′), (R4′) so B is a generating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' To prove that B is linearly  independent, let  푦 = 푎푖1  1 · · · 푎푖푔  푔 푏 푗1  1 · · · 푏 푗푔  푔 ∈ 퐺  and suppose that 푦 = 0 mod 푍퐺, which is to suppose that an arbitrary linear combination of elements  of B is trivial in 푉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Then 푦 is central so  [푦, 푏1] = [푎푖1  1 , 푏1] = 휎−2푖1 = 1  8  which implies that 푑 | 2푖1 and thus 푖1 = 0 mod 푝: If 푝 is odd then 푑 = 푝 so 푝 | 푖1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' If 푝 = 2 then  푑 = 4 | 2푖1 so 푝 = 2 | 푖1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  Similarly 푖푟 = 푗푟 = 0 mod 푝 for all 푟, which is to say that all coefficients of the linear combination are  trivial over the base field F푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' This proves the linear independence of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The relation (TR′) imposes the relation 푑 | 2(푔 + 푛 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Either 푑 = 푝 is odd or 푑 = 4 and 푝 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' in  both cases (TR′) implies that 푝 | (푔 + 푛 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' In his paper [1], Bellingeri also gives a presentation of the braid group of the genus 푔 surface  with 푝 punctures (equivalently for the purposes of braid groups, 푝 boundary components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' The above  methods can be used nearly verbatim to prove that the smallest nonabelian quotient of 퐵푛(Σ푔,푝) is the  smaller of 푆푛 or I(22, 푔) and II(22, 푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  References  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Bellingeri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' On presentations of surface braid groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Journal of Algebra, 274(2):543–563, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Birman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' On braid groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Communications on Pure and Applied Mathematics, 22(1):41–72, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Fadell and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Van Buskirk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' On the braid groups of 퐸2 and 푆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Bulletin of the American Mathematical Society,  67(2):211–213, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Kolay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Smallest non-cyclic quotients of braid and mapping class groups, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' arXiv: 2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='02162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' On a Class of Nilpotent Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content=' Proceedings of the London Mathematical Society,  s3-10(1):365–375, 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQf6P5j/content/2301.01872v1.pdf'}
diff --git a/r9E4T4oBgHgl3EQfVQyb/content/tmp_files/2301.05023v1.pdf.txt b/r9E4T4oBgHgl3EQfVQyb/content/tmp_files/2301.05023v1.pdf.txt
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+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+1 
+Self-assembly of soot nanoparticles on the surface of 
+resistively heated carbon microtubes in near-
+hexagonal arrays of micropyramids.  
+Valeriy A. Luchnikov,1* Yukie Saito,2* Luc Delmotte,1 Joseph Dentzer,1 Emmanuel Denys,1 
+Vincent Malesys,1 Ludovic Josien,1 Laurent Simon,1 Simon Gree.1 
+1Université de Haute-Alsace, CNRS, IS2M, UMR 7361, Mulhouse F-68057, France  
+2Graduate School of Agricultural and Life Sciences, The University of Tokyo, Tokyo, 113-
+8657, Japan 
+Corresponding authors: Valeriy A. Luchnikov, email: valeriy.luchnikov@uha.fr and Yukie 
+Saito, email:   aysaito@g.ecc.u-tokyo.ac.jp 
+ABSTRACT Almost regular hexagonal arrays of a few micrometers tall and wide 
+micropyramids consisting of soot nano-particles are formed on the surface of graphitized hollow 
+filaments, which are resistively heated to  1800°C−2400°C in an Ar atmosphere containing 
+trace amounts of oxygen (~300 p.p.m.). At the higher temperatures (T>2300°C, approximately) 
+the soot particles are represented mainly by  multi-shell carbon nano-onions. The height and the 
+width of the pyramids is strongly dependent on the temperature of the resistive heating, 
+diminishing from 5-10m at T1800°C to  1m at 2300-2400°C.  Quasi-hexagonal arrays of 
+the micropyramids are organized in the convex “craters” on the surface of the microtubes, which 
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+2 
+grow with the time of the thermal treatment. The pyramids are pointing always normally to the 
+surface of the craters, except at the boundaries between the craters, where the normal direction is 
+not well defined.  The pyramids are soft and can be easy destroyed by touching them, but can be 
+hardened by heating them in the oxygen-free atmosphere. The pyramids are observed only on the 
+exterior surface of the microtubes, but not on their inner surface. This suggests that the 
+thermophoretic force generated by a strong temperature gradient near the external surface of the 
+tubes may be the cause of the micropyramids formation. Electrostatic charging of the soot 
+nanoparticles due to thermionic emission may also be relevant to this phenomenon. The 
+micropyramids can function as field emission point sources, as demonstrated with the use of a 
+micro-nanoprobing station, mounted in a scanning electron microscope.  
+KEYWORDS: Self-assembly; carbon nano-onions; resistive heating; thermophoretic force; 
+micropyramids; hexagonal array.  
+ 
+Carbon is known for its capacity to form plenty of architectures at the nanoscale as well as the 
+microscale. Apart of the most well known carbon single and multishell tubes, fullerens, and 
+graphene, there have been discovered numerous interesting entities such as carbon nano-horns 
+and nano-cones,1-3 conical micro-crystals,4 carbon whiskers,5,6 graphite micro-pyramides,7 
+graphene flowers or vertical graphene.8-12   Electrical, optical, and morphological peculiarities of 
+these architectures make them perspective for advanced applications such as field electron 
+emission sources,9,10  supercapacitors,13 biosensors,11ultra-black materials,14  superhydrophobic15  
+and bactericidal16 coatings.  The methods of production of these structures include argon plasma 
+etching of graphite substrates,7,14 reactive sputtering using methane gas,9 thermal chemical vapor 
+deposition on carbon fibers,10  reduction of graphene oxide nanosheets,11 combustion flame 
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+3 
+deposition,8 microwave plasma chemical vapor deposition, followed by bias-assisted reactive ion 
+etching,16 wood charcoal heat treatment above 2000°C.6 
+Here we report soot micropyramids, which we have found on the surface of resistively heated 
+amorphous carbon microtubes in Ar atmosphere, to which vanishingly small amount of air was 
+admixed. The pyramids are always pointing along the local normal direction to the tube’s 
+surface, suggesting that the soot nanoparticles, formed in the oxygen-deficient atmosphere, 
+organize into the pyramids under the action of some force, directed outward the surface. 
+Moreover, the pyramids are assembled in almost hexagonal arrays, resembling the spontaneously 
+formed hexagonal patterns, which appear at certain deformable interfaces under the action of 
+normally applied forces, and which have been extensively studied in the past. Examples include 
+the Rosenweig instability of a ferromagnetic fluid under a magnetic field,17 the gravity-driven 
+Rayleigh−Taylor instability of suspended liquids,18  elastic solids,19  and electrostatically induced 
+structure formations at the polymer−air interface.20  The patterns arise due to the competition of 
+external fields (gravity, magnetic, or electrostatic), which amplify the interface deformation with 
+surface tension and elastic energy, which have stabilizing effects. Linear stability analyses 
+establish the dispersion relation between the amplitude growth rates of the normal modes of 
+deformation and the wave numbers of the modes. The modes with the highest growth rates 
+dominate and determine the characteristic wavelength of the pattern. Hexagonal patterns arise 
+due to the resonance of the dominant modes, whose wave vectors form an equilateral triangle in 
+the reciprocal space, as shown by nonlinear stability analysis.18,21   
+A natural candidate for the role of such a normally directed force in our system of soot particles 
+on the surface of a resistively heated microtube is the thermophoretic force,    , which results 
+from the difference of the average momentum transferred to the particle by molecules arriving  
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+4 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 1. The most characteristic morphology features of the carbon microtubes resistively heated  in the 
+oxygen-deficient atmosphere. (a) Craters on the surface of a microtube. (b,c) An array of pyramids at different 
+magnification. (d) Almost hexagonal array of the pyramids. (e) The interface between pyramid’s bottoms and the 
+unchanged wall of a tube. The white arrow : inner surface of the tube. (f) Pyramids with thin filaments on the top. 
+(g)  A magnified view of a crater. Note that the pyramids are pointing normally to the walls of the craters (h) A 
+boundary between two craters. (i) Irregular multi-apex pyramids at the bondary.  
+from the forward and the backward directions, with respect to the temperature gradient’s 
+direction.22  More rapid molecules, which arrive from the region of higher temperature, transfer 
+larger momentum on average than the molecules that arrive from the colder regions. This results 
+in a drag force on the particle, which drives it from hotter regions to colder ones. Thermoelectric 
+(a) 
+(b) 
+(c) 
+(d) 
+(e) 
+(f) 
+(g) 
+(h) 
+(i) 
+
+100umLm20um1 μm50μm1 μumumManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+5 
+phenomena, such as electrostatic charging of the particles due to thermionic loss of electrons at 
+high temperatures may be also essential for the process of the micropyramids formation. 
+Results and discussion.  
+Figure 1 presents the most representative features of the micro-scale morphology  observed on 
+the surface of a tube heated to                and at an oxygen concentration  
+           for 5 hours. The surface of the tube, which was initially smooth, became covered by 
+micro-craters (Figure 1a) whose concave bottoms are roofed by the arrays of the features which 
+we call micro-pyramids, because their shape resembles this geometric form (Figure 1b,c). No 
+similar structure was found on the surface of the control tubes heated without air admixing (see 
+Figure S1 of the Supporting Information, SI). The pyramids have sharp apices with 
+submicrometric curvature radii, and the apices of some pyramids are split. The surface of the 
+pyramids is not smooth, rather, it has porous aspect (see Figure 1c and Figure S2 of SI for a 
+high-resolution image). The pyramids are often prolonged by a thread whose thickness does not 
+exceed 100 nm   (Figure 1d).  A fragment of  an extremity of an intentionally broken tube is 
+shown in Figure 1e. The image reveals the interface between the loose material of the pyramids 
+and the monolithic structure of the tube wall, which was not yet etched by the oxygen-containing 
+atmosphere. The pyramids appear only on the outer surface of the tube but not on their interior 
+walls, shown by the white arrow.   
+ Seen from the top, the pyramid arrays have nearly hexagonal coordination in the centers of the 
+craters (Figure 1f). The pyramids are always oriented normally to the concave craters walls 
+(Figure 1g). At the boundaries between the craters, where the normal direction is not defined, the 
+pyramids have irregular shapes characterized by split apices (Figures 1g,1i). 
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+6 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2 The inner structure of the pyramids. (a) Pyramids after the application of Scotch tape to the surface of a 
+tube (                        . Some pyramids are partially destroyed, and some are completely 
+removed. Note that the soot pyramids cover the pyramidal elevations of the unchanged material of the tubes. (b) 
+High-resolution SEM of the interior of a partially destroyed pyramid. (c,d) HRTEM of the material of the 
+miccropyramids, formed at          and         , respectively. A CNO particle is shown on the figure 
+(c) by the white arrow. (e, f) the Raman spectra of the material of the pyramids, formed at          and 
+        , respectively. 
+ To collect the material of the pyramids for Raman spectroscopy characterization, a tube was 
+touched by a narrow strip of Scotch tape. Some of the pyramids were destroyed partially by this 
+operation (Figure 2a), which allowed the examination of their inner structure. High-resolution 
+SEM reveals that the pyramids consist of nanoparticles whose diameters do not exceed 30nm 
+(Figure 2b).  The fact that the pyramids can be easily disrupted by touching signifies that the 
+particles are not connected by covalent bonds; rather, they are held together by van der Waals 
+forces. A high-resolution TEM reveals that particles are the closed-shell ones, similar to the 
+particles produced by the carbon arc method.23 Some particles can be identified as multi-shell 
+carbon nano-onions (CNO) (Figure 2c).   The graphitization degree and the fraction of CNOs 
+(a) 
+(b) 
+(c) 
+(d) 
+(e) 
+(f) 
+
+Lim100nm10nm20nm600
+D
+T=2000°C
+1。/1,=0.69
+400
+G
+G
+'n'
+a
+200
+2D
+0
+0
+1000
+2000
+3000
+4000
+ (cm-1)2000
+D
+T=2380°C
+1500
+G
+L-/_=0.77
+1000
+e
+2D
+500
+0
+0
+1000
+2000
+3000
+4000
+y(cmManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+7 
+increase at higher temperatures. For a tube heated to T≈2380°C (Figure 2d) the quasi-totality of 
+the particles, collected on the surface of the tubes, can be classified as CNO.  The Raman spectra 
+of the particles  (Figure 2e,f) have well-resolved G (graphitic) and D (disordered) bands,24  
+meaning that the particles contain a relatively low concentration of defects of the sp2-
+coordination of carbon atoms. The ratio of the intensities of the G and the D peaks,     
+ 
+, grows 
+with temperature, implying the improving the sp2-coordination of the CNO particles.  
+The mechanism of formation of the soot particles on the surface of the resistively heated 
+graphitized microtubes is not yet clear.  The physico-chemical conditions at the surface of the 
+microtubes are different from the conditions of the well-understood processes of soot production 
+as  hydrocarbon flames,25,26   spark discharges27 and laser ablation.28  In the flames, the process 
+starts by the gas-phase chemical reactions leading to the formation of polycyclic aromatic 
+hydrocarbons (PAH), followed by the particles nucleation via the PAH dimerization, particle 
+growth by the surface reactions, and particles coagulation. It is unlikely that the carbonized 
+microtubes can be the source of  PAH precursors, because at temperatures above 1000°K 
+carbon films are almost completely dehydrogenated.29 In the cases of spark discharge and laser 
+ablation, soot nuclei are formed due to evaporation of a small amount of carbon electrode or 
+carbon target material, and condensation of the supersaturated vapor in the neutral gas flow. The 
+soot nuclei coagulate, forming the soot particles, which aggregate eventually. This mechanism is 
+also unlikely be relevant to the formation of the soot particles in our system, because the 
+temperatures of the resistive heating of the microtubes is well below of graphite sublimation 
+point (around 4000°K at the ambient pressure).30  
+To the best of our knowledge, soot formation on the surface of graphitized materials, exposed to 
+the temperatures close to 2000°C and oxygen concentrations of a few hundreds of p.p.m.  was 
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+8 
+not reported to the date, despite a great deal of literature on oxidation of graphite materials, 
+especially for the nuclear applications.31 Typically, temperature at which graphite oxidation was 
+studied did not exceed 1000°K. At these temperatures, the basal graphite plane is inert to 
+molecular oxygen.  The graphite oxidation proceeds from the zig-zag and armchair active sites of 
+the graphite crystallites, attacked by molecular oxygen, with the CO and CO2 desorbtion.  
+Oxidation of multilayer graphene flakes suspended on amorphous carbon grid in air and heated 
+by laser to temperatures close to 2000°K resulted in the layer-by-layer thinning of the flakes, 
+assuming that flakes are eroded in a highly anisotropic manner. Yet, no formation of carbon 
+nanoparticles, such as soot, was reported.32  
+In order to gain a deeper insight into the conditions of the soot particles formation in our system, 
+we have made a control experiment, in which a 25μm-thick pyrolytic graphite film was 
+resistively heated to 2000°C in the Ar atmosphere at oxygen concentration 600 p.p.m. The 
+sample was eroded in course of 20min of the treatment (Figure S3), but no soot and pyramids 
+formation was detected. In particular, the graphite stripe remained shiny, while the carbonized 
+chitosan tubes became violet black upon etching.  This means that the mechanism of the soot 
+formation in our system requires a relatively high number of defects of the sp2-coordination of 
+the carbon material.   
+ 
+ 
+1800°C 
+1900°C 
+2000°C 
+2100°C 
+2200°C 
+  (m) 
+        
+        
+        
+        
+        
+Table 1. Average distance between the pyramide centers as the function of resistive heating temperature, for 
+                  
+The size of the pyramids and their number per unit surface depend strongly on temperature of the 
+resistive heating (Table 1). Within the temperature interval 1800°C – 2200°C, in which the  
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+9 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 3 Voronoi analysis of the pyramids arrangement. Color coding of the Voronoi cells coordination: brown – 4-
+edged cells; blue – 5-edged cells; green- 6-edged cells; pink – 7-edged cells, and orange – 8-edged cells. (a) 
+Superposition of the SEM image of a crater bottom for a tube heated at         , and                 during 
+4 hours,  and the Voronoi diagram, calculated by the pyramids vortices. (b) Coloring of the Voronoi diagram cells 
+according to the number of the cells edges. (c) Distribution of the cells by the number of their edges. (d-f) The same 
+as (a-c), respectively, for a tube heated at          ,               , during 2 hours (see figure S5, SI).  
+micropyramids arrays can be unambiguously identified, the average distance between the 
+pyramids centers decreases by ~6 times as temperature grows. The pyramids also became more 
+« slim », and their apices are more often continued by a thin filament. 
+The crater-like features imply that the pyramids do not appear simultaneously on the surface of 
+the tubes. Indeed, we have found that initially small, isolated craters, having an almost perfectly 
+circular shape and containing a few pyramids, appear on the surface of the tubes a few dozens of 
+minutes after the beginning of the resistive heating (Figure S4a,b, SI). During the experiment, the 
+craters became wider and deeper, and the number of pyramids in them increased. New craters 
+(a) 
+(f) 
+(c) 
+(b) 
+(d) 
+(e) 
+
+Co,~300p.p.m
+0.8
+Fraction
+0.6
+0.4
+0.2
+0
+4
+5
+6
+8
+NumberofVoronoicelledges-02
+~600p.p.m
+0.8
+E 0.4
+0.2
+0
+4
+8
+NumberofVonoicell edgesO
+O
+O
+OManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+10 
+appeared and grew. Eventually, all the surfaces became covered by the craters, which touched 
+each other (Figure S4c,d).  
+The characteristic time of the pyramids formation can be strongly reduced by increasing the 
+oxygen concentration in the vessel atmosphere. Thus, the almost regular arrays of the pyramids 
+were observed on the tubes heated at         ,                after 120 min of resistive 
+heating. A few craters were observed on the surface of the tubes, but the majority of the 
+pyramids were formed outside them (Figure S5, SI). Apparently, the rate of the pyramids 
+formation affects their mutual coordination. This fact follows from the analysis of the Voronoi 
+diagrams of the SEM images of the pyramids arrays produced at at           and at the 
+relative oxygen concentrations                 and               (Figure 3). By 
+definition, the Voronoi cell of a vortex is defined by the ensemble of the points which are closer 
+to a given vortex than to any other vortex of the system. The edges of the polygons correspond 
+roughly to the faces of the pyramids, and the vortices of the polygons correspond to the edges of 
+the pyramids. In our analysis, the vortices were defined as the extremities of the micropyramids. 
+At the lower oxygen concentration, the absolute majority (~80%) of the Voronoi cells have 6 
+edges, while at the higher oxygen concentration, the fraction of the 6-coordinated cells drops to 
+around 50%. Further increase of oxygen concentration to                 led to rapid 
+consumption of the microtubes (typically, within 10-20 min) and irregular shapes of the 
+pyramids, as well as poor degree of their hexagonal arrangement.  
+The hypothetical mechanism of the pyramids growing and the propagation of the pyramids 
+arrays is shown on the Figure 4. We suppose that a pyramid’s formation starts from a local defect  
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+11 
+ on the surface of a tube, such as a local 
+microscale elevation (Figure 4a). Soot particles, 
+generated at the bottom of the defect, roll up 
+along its walls in the vertex direction, increasing 
+the defect’s height and transforming it into a 
+pyramide. Simultaneously, local cavities are 
+formed at the bottom of the pyramide, because of 
+the particles excavation from the tube. The particles start to roll along the walls of the cavities, 
+forming new pyramids at their borders (Figure 4b). This process is repeated, leading to formation 
+of the pyramid’s arrays (Figure 4c). Similar process may start from a defect in form of a micro-
+cavity on the surface of a tube. The growing of the pyramids seems to be limited by their 
+sputtering from the pyramid’s vortices. The formation of the concave “craters” signifies that the 
+pyramids accelerate somehow the transformation of the graphitized material in soot, because the 
+older pyramids have propagated deeper in the walls of the tubes. This is probably because the 
+migration of the soot particles onto the pyramid’s walls unmasks the tube’s surface and makes it 
+more accessible to the molecules of oxygen present in the gas mixture.    
+The height of the pyramids in the largest craters is much smaller than the craters’ depth (see 
+Figure 1g). This means that the soot nanoparticles, which constitute the pyramids, are constantly 
+removed from the pyramids and replaced by new particles, which are produced by the interaction 
+of oxygen with the tube’s walls. To clarify the fate of the eliminated particles, we placed a 
+tungsten wire close to a microtube. In 1 hour of resistive heating at         , the so-called 
+graphene flowers8,9 formed on the surface of the wire (Figure S6). Thus, the carbon material did 
+not burn  
+ 
+Figure 4. Hypothetical mechanism of pyramids 
+growing and pyramids arrays propagation.  
+
+(a)
+(b)
+CManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+12 
+ 
+ 
+ 
+Figure 5 Normal view (a) and side view (b) of the micropyramids produced at              
+           and  stabilized by heating at          without air admixing during 30min. The sintering of 
+soot particles rigidifies the pyramids, so that they are not smashed or disrupted when touched (cf. Figure 2a); 
+rather, they are wholly unrooted (c). 
+completely; rather, it sputtered into the space around the tubes, where it can be sedimented on a 
+support.  
+The pyramids can be stabilized by sintering soot particles in an oxygen-free atmosphere. This 
+operation makes them mechanically stable (Figure 5), so that the pyramids are wholly unrooted 
+rather than smashed, when they are touched. The sintering procedure increases also the height to 
+base aspect ratio of the pyramids, and cleans up the space between them from the soot particles, 
+so that the pyramids are transformed into needle-like structures, well separated from each other.  
+We suppose that the effect of the « cleaning up » of the space between the pyramids is due to the 
+fact that, in the absence of oxygen, soot particles are not produced anymore in course of the 
+erosion of the material of the tube. But they are not sintered immediately, and therefore migrate 
+to the top of the pyramids, increasing their height to width aspect ratio. The particles which are 
+inside the pyramids (and not on their surface) are less mobile, and for them the probability of 
+sintering is higher, therefore these particles constitute the rigid « skeleton » of the pyramids, 
+making them more robust mechanically.   
+(a) 
+(b) 
+(c) 
+
+3um2 μmManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+13 
+The ensemble of the experimental facts presented above allows us to suppose the existence of a 
+force normally directed to the interface between the soot layer and the material of the tube, 
+which is not yet transformed into soot by its interaction with oxygen. A natural candidate for the 
+role of such a force is thermophoretic force,    , which results from the difference of the average 
+momentum transferred to the particle by molecules arriving from the forward and the backward 
+directions, respecting the temperature gradient’s direction. More rapid molecules, which arrive 
+from the region of higher temperature, transfer larger momentum on average than the molecules 
+that arrive from the colder regions. This results in a drag force on the particle, which drives it 
+from hotter regions to colder ones. The fact that the pyramids do not appear inside the tubes, 
+where the temperature is presumably uniform, seems to support the hypothesis that the 
+temperature gradient giving rise to thermophoretic force is the factor that causes the formation of 
+pyramids. The intensity of     can be estimated from the following considerations. At   
+      , the mean free path of argon atoms is                       
+ 
+ 
+      , where 
+   is the Boltzmann constant,   is the pressure of the gas, and            is the diameter of 
+the argon atoms. From the HRTEM images (Figure 2c,d), one can evaluate the characteristic 
+diameter of the soot particles as       . Since the Knudsen number   
+ 
+    is large, the 
+thermophoretic 
+force 
+can 
+be 
+calculated 
+in 
+the 
+free 
+molecular 
+limit22 
+ 
+as 
+     
+    
+  
+   
+        
+ 
+   where                  is argon gas thermal conductivity at 
+the given temperature,33      is the mass of argon atoms. The temperature gradient at the 
+filament’s surface is usually evaluated with the assumption that there exists a layer of almost 
+immovable gas around the filament (the so-called stagnant layer of Langmuir).34  Coherent anti-
+Stokes Raman scattering measurements35  and computer simulations36 estimate this gradient to 
+              for tungsten filaments heated to         at the normal pressure of the 
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+14 
+nitrogen surrounding the filament. Assuming that, for argon, the temperature gradient is on the 
+same order of magnitude, one can estimate the thermophoretic force acting on a particle of the 
+characteristic size        as                 . It is several orders of magnitude 
+smaller than the typical adhesion force of soot particles measured by atomic force microscopy to 
+be in the range of a few nN.37  Therefore, it is hardly possible that the thermophoretic force alone 
+can detach the soot particles from each other. Conversely, it is likely that, at elevated 
+temperatures, the average kinetic energy of the particles,                  
+ , takes over 
+the van der Waals adhesion energy, allowing the rearrangement of the mutual position of the 
+particles. The thermophoretic force field might then bias these rearrangements so they have the 
+effect of migrating the particles outward from the tube’s surface. At the tips of the pyramids, the 
+particles have a relatively small number of neighbors and can be “evaporated” into the hot gas 
+surrounding the tube. This scenario is in line with current theories about the formation of 
+nanoparticle aggregates, which assume that the coagulation efficiency of the particles depends on 
+both their size and temperature.38,39  At sufficiently high temperatures, smaller soot particles are 
+likely to bounce off rather than adhere to each other because of the relatively high kinetic energy 
+compared with the interaction energy of the particles (the so-called thermal rebound effect). Hou 
+et al.38 simulated the coagulation of soot particles in frames of the Hamaker model, in which the 
+effective interaction of two spherical atomic clusters is the sum of all pairwise Lennard−Jones 
+interactions of the atoms belonging to opposite clusters. As follows from their model, the van der 
+Waals interaction cannot hold together two spherical soot particles of diameters       when 
+the temperature exceeds         . These values are uncertain, since the soot particles are 
+mostly aspherical at this temperature. Moreover, in close packing, each particle has more than 
+one neighbor; therefore, it needs more kinetic energy to escape from them. Also, the formation of 
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+15 
+covalent bonds between some particles is not excluded. Nevertheless, it seems plausible that at 
+high temperatures, which are reached in the resistively heated tubes, the soot particles may move 
+with respect to each other, at least on the surface of the pyramids, where they have a smaller 
+average number of contacts. It may, then, be possible to make an analogy between the ensemble 
+of soot particles at high temperatures and a layer of suspended viscous fluid whose interface is 
+deformed by gravity.18 This analogy is supported by the existence of thin filaments by which the 
+extremities of some pyramids are prolonged. Such filaments are unlikely to be formed by the 
+random aggregation of soot particles. Rather, they look as the result of the viscous flow in a 
+force field. Another argument for the liquid-like behavior of the soot layer on the surface of 
+resistively heated microtubes is the circular growth of pyramid clusters (Figure S4). The already 
+formed pyramids seem to induce the formation of subsequent generations of pyramids in their 
+neighborhood. Similar behavior was observed for the arrays of pending droplets in the layers of 
+suspended viscous fluid film,18 in which the concentric rings of hexagonally coordinated droplet 
+arrays grow around the initial perturbations.  
+Thermoelectric phenomena may be also relevant to the pyramids formation. A grounded carbon 
+spherical nanoparticle of the diameter        loses one electron every         at   
+       (       )  and every         at          (       ) , as calculated by the 
+Richardson−Dushman equation for the thermoelectric current,             
+ 
+, where 
+      is the surface of the particle,                   is the Richardson’s constant, 
+and               is the work function for carbon.40,41 Filippov et al. considered the 
+possibility of an electrostatic breakdown (Coulomb explosion) of soot agglomerates heated by 
+laser pulses to high temperatures.40  If the analogy between the soot layer at high temperature 
+and a layer of viscous liquid having a finite surface tension is correct, then charging particles on 
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+16 
+the surface of the layer may lead to the development of electrohydrodynamic instability.42  
+However, unlike the particles in the isolated soot clusters, the particles on the surface of the tubes 
+constantly restore their electroneutrality due to the electron flux from the grounded tubes. 
+Unfortunately, in the absence of literature and data on the conductivity and effective permittivity 
+of the soot layer at high temperatures, it is impossible to make reliable estimates of the surface 
+charge.  
+Theoretically, it might be possible to discriminate, which of the two factor is more relevant to the 
+pyramids formation, by heating the carbonized tubes in an oven, thus eliminating the temperature  
+gradient at the tubes surface. But, the maximal temperature, at which the alumina ovens can 
+operate in continuous regime does not exceed 1600° - 1700°C, which is essentially below of the 
+lowest temperature, at which we succeeded to observe the pyramids. The graphitic ovens can 
+reach much higher temperatures (up to 3000°C), but they are incompatible with the oxygen-
+containing atmosphere.  
+The arrays of the carbon micropyramids may be explored as the field emitter arrays.9,10,43   As a 
+first proof of concept we present here preliminary results to test the capacity of field emission 
+properties of individual pyramids. We used a micro-nanoprobing station mounted in a scanning 
+electron microscope, see Figure 6a. The nanoprobes are standard tungsten tips without any 
+coating (such as for example gold to improve the workfunction of the tip probe). The 
+measurements are taken at 10-6 mbar at the ambient temperature.   An example of current-voltage 
+(I-V) characteristic of a pyramid, which is separated from the tip by a 200nm wide gap, is shown 
+in Figure 6b. The observed emission curves are very similar to those observed in a previous work 
+devoted to the emission from carbon nanotube films and approximated by the classical Fowler–
+Nordheim (FN) model.43   The first few sweeps show a noisy curves (blue, red, green lines).  
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+17 
+ 
+ 
+ 
+ 
+Figure 6. The measurement of the tunneling current from the micropyramids. (a) The set-up: four micro-nano-
+manipulator IMINA® mounted around a cryo stage (600K-77K), Kammrath® (b) SEM image of the nanoprobe in 
+front of a pyramid  (d=200nm) and the emitted current  vs applied potential. The current increases with a threshold 
+bias of -75V.  (c) SEM image and  anomalous (non-FN) emitted current versus bias curve for two pyramids. (d) 
+The I-V curve for a pyramide annealed at 2300°C.   
+ 
+After several sweeps the I-V characteristics became regular (black lines). This is usually 
+attributed to a “conditioning” of the emitter, such as oxygen desorption.43   The threshold of 
+electron emission is -75V and the measured emitted current reach 60nA for a bias of -100V.  It 
+should be said that the emissive characteristics of the pyramids are highly variable and may 
+demonstrate completely different I-V curves. The Figure 6c shows two type of I-V curve 
+(a) 
+(b) 
+(c) 
+(d) 
+
+70
+60
+50
+Current (nA)
+40
+30
+20
+10
+d=200nm
+20
+0
+-20
+-40
+-60
+-80
+-100
+Voltage (V)0.06
+d=200nm
+0.05
+0.04
+Current (nA)
+0.03
+0.02
+0.01
+0.0
+-0.01
+0
+-50
+-100
+-150
+-200
+Voltage (V)60-
+50-
+Current (nA)
+40
+30
+20-
+10-
+d=800nm
+0
+0
+-50
+-100
+-150
+-200
+Voltage (V)Tip
+1μmTipTip
+5umManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+18 
+recorded for two other pyramids. The shape of the curve is very different than the expected FN 
+model. The current increases slowly up to a sharp increase at different bias voltage with a 
+saturation at 0.5nA which is a value of one order of magnitude lower than for the  pyramid 
+shown on the Figure 6b. Possibly, the morphology of the pyramids changes during the I-V 
+measurement, however a plateau of emissive current is reached.  The FN-like emission curves 
+were detected for the micropyramids annealed at 2300°C (Figure 6d). 
+From these measurements follows that the carbon micropyramids do can be considered as the 
+field emission sources, yet their emissive characteristics are very variable. It is not surprising 
+since, according to the FN theory, the tunneling currents are highly dependent on the shape of the 
+extremities of the emitters (mathematically, it is expressed by the local field enhancement 
+factor), and the extremities of the pyramids are all different.  We need much more studies to 
+understand the behavior of individual pyramids, to probe the emissivity with lower tip radius 
+curvature (these experiments were done with 500nm but tip with 100 and 50 nm) to see the effect 
+of the pyramids shapes and morphology, as well as the coating on the tungsten tip.    
+Conclusion and outlook. 
+Hexagonal arrays of micropyramids composed of soot nanoparticles are observed on the surface 
+of graphitized microtubes resistively heated to                in the Ar atmosphere 
+containing a small (~300-600p.p.m.) fraction of oxygen. The few micrometer tall and wide 
+micropyramids are arranged in quasi-regular hexagonal arrays on the surface of the microtubes. 
+The pyramids are pointing normally to the walls of the craters, except the borders between the 
+craters, where the normal direction is not defined. The pyramids are formed only on the external 
+walls of the microtubes. HRTEM reveals that the increasing fraction soot particles is 
+morphologically identical to the carbon nano-onions, when resistive heating temperature grows.    
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+19 
+The pyramids are soft, and can be easily destroyed by touching them, but they can be rigidified 
+by the resistive heating without oxygen admixing in the Ar stream passing through the system.  
+The mechanism of the pyramids formation and their self-assembly in the quasi-hexagonal arrays 
+is not yet clear. The fact that the pyramids are always directing parallel to the local normals to 
+the tube surface indicates on the existence of some factor which favors the drift of the 
+nanoparticles in these directions. The thermophoretic force, arising as the consequence of a 
+strong temperature gradient in the vicinity of the tube, is a natural candidate on the role of such a 
+factor. Under this assumption, one can draw an analogy with the gravity-driven Rayleigh−Taylor 
+instability of a thin suspended liquid layer. It is also possible that the electrostatic charging of the 
+soot particle layer due to thermoelectric emission contributes to the pyramids formation and their 
+assembly in the hexagonal arrays.   
+Regular mutual arrangement and the sharp extremities of the micropyramids allow to consider 
+them as perspective micro-structures for advanced applications. In our study, we have 
+demonstrated that the individual carbon micropyramids emit the tunneling current upon 
+application of local electrical field. This indicates on the principal possibility to create field 
+emitter arrays45 on the base of the micropyramids ensembles.  To achieve this goal, a reliable 
+approach should be found for the formation of the pyramids arrays on the flat substrates and on 
+sufficiently large surfaces. The pyramids ensembles on the flat substrates may be then 
+investigated also for the creation of superhydrophobic15 and bactericidal16  coatings, as well as 
+the ultra-black materials.14  
+Methods. 
+The experimental setup for the generation of the soot micropyramid arrays is shown 
+schematically in Figure 7. An electro-conductive hollow-core microtube derived from a  chitosan 
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+20 
+self-rolled microtube, made according to the procedure given in Ref. [44], was placed inside 
+quartz vessel 1, where it was suspended on crocodile-like electrodes. An electrical current was 
+generated in the tube by a DC power supply 2. The vessel was filled with an air/argon mixture at 
+normal pressure             with the use of valves 3 and 4, guided by a multichannel gas 
+flow controller 5.  
+ 
+ 
+Figure 7. The experimental setup to produce arrays of self-assembled soot micropyramids.  
+ 
+After passing through the system, the gas mixture was rejected in atmosphere 6. The relative 
+concentration,    , of oxygen was varied between 300 and 3,000 p.p.m. The temperature of the 
+microtube was measured by Modline 5 Raytec optical pyrometer 7 conjugated with a computer 
+8. Typically, the formation of well-developed pyramid patterns requires several dozens of 
+minutes              , but at a relatively high O2 concentration, this time period was 
+shorter. After a joule heating experiment, the filament was gently dismounted and examined by 
+scanning electron microscopy (SEM) and transmission electron microscopy (TEM). Soot 
+particles transferred onto Scotch tape were examined by Raman spectroscopy (Horiba Labram 
+300,  =632nm). The field emission was measured with the use of a nanoprobing station mounted 
+
+2
+8
+Air
+5
+1
+Ar
+4Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+21 
+in a scanning electron microscope FEG-MEB XL30. The station consisted of four micro-
+nanomanipulators IMINA®  mounted around a Kammrath® cryostage (600K-77K). Panasonic 
+pyrolytic graphite sheets were purchased at Farnell Electronics.  
+Acknowledgements   
+We thank L. Vidal, S. Knopf, B. Rety, and A. Beda (all-IS2M CNRS) for their assistance with 
+electron microscopy imaging and the carbonization of chitosan microtubes.  The measurements 
+of the tunneling currents were done with the financial support by the CNRS, the Région Grand 
+Est and FEDER (EU) through the NanoteraHertz project, and the French National Research 
+Agency (ANR) through the MIXES project (Grant ANR-19-CE09-0028).  
+Supporting Information. 
+The Supporting Information Available :  
+Figure S1, a control experiment on resistive heating of a tube without air admixing.  
+Figure S2, high resolution SEM images of the micropyramids.  
+Figure S3, a control experiment, etching of a pyrolytic graphite film.  
+Figure S4, the grow of “craters” on the surface of the resistively heated microtubes.  
+Figure S5, the morphology of a microtube, resistively heated at          and oxygen relative 
+concentration              .  
+Figure S6, graphene flowers grown on the surface of a tungsten wire suspended in the vicinity of 
+a resistively heated tube.  
+
+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+22 
+References 
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+ 
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+Against Oxidation with Molecular Oxygen by Synthetic Conditions.  J. Aerosol Sci. 2020, 143, 43-
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+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
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+Adhesive force, Adhesion Energy and Hamaker Constant of Soot Particles Generated From a 
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+Carbonaceous Nanoparticles.  J. Aerosol Sci. 2019,  140, 18-29.  
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+Manuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+25 
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+Under Normal Electric Fields.  Phys. Rev. Fluids. 2021,  6, 103703 (17pp).  
+[43]  Di Bartolomeo A.;  Scarfato, A.;  Giubileo, F.; Bobba, F.;  Biasiucci, M.;  Cucolo, A. M.; Santucci, 
+S.;  Passacantando, M.  A Local Field Emission Study of Partially Aligned Carbon-Nanotubes by 
+Atomic Force Microscope Probe. Carbon. 2007,  45, 2957-2971.  
+[44]  Beda, A.;  Yamada, H.;  Egunov, A.;  Matei Ghimbeu, C.;  Malval, J.-P.; Saito, Y.;  Luchnikov, V.  
+Carbon Microtubes Derived from Self-Rolled Chitosan Acetate Films and Graphitized by Joule 
+Heating. J. Mater. Sci. 2019,  54, 11345–11356.  
+[45]  Guibileo, F.;  Di Bartolomeo, A.;  Iemmo, L.;  Luongo, G.;  Urban, F. Field Emission from Carbon 
+Nanostructures.  Appl. Sci. 2018,  8, 526 (21pp).  
+ 
+ 
+ 
+TOC figure 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+200um
+5Lm
+Self-assemblyofsootnanoparticlesintoregulararraysofmicropyramidswasobserved
+onthesurfaceofcarbonmicrotubesresistivelyheated inoxygen-deficientatmosphereManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+26 
+Self-assembly of soot nanoparticles on the surface of resistively heated carbon microtubes 
+in near-hexagonal arrays of micropyramids - Supporting Information. 
+ 
+Valeriy A. Luchnikov,1* Yukie Saito,2* Luc Delmotte,1 Joseph Dentzer,1 Emmanuel Denys,1 
+Vincent Malesys,1 Ludovic Josien,1 Laurent Simon,1 Simon Gree.1  
+ 
+1Université de Haute-Alsace, CNRS, IS2M, UMR 7361, Mulhouse F-68057, France  
+2Graduate School of Agricultural and Life Sciences, The University of Tokyo, Tokyo, 113-
+8657, Japan 
+ 
+*Corresponding authors: Valeriy A. Luchnikov, email: valeriy.luchnikov@uha.fr and Yukie 
+Saito, email: aysaito@g.ecc.u-tokyo.ac.jp     
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure S1. Control experiment on resistive heating of a tube without air admixing, at         . 
+(a,b) Surface of the tube, at different magnifications. (c) An impurity escaping the microtube 
+surface. (d) Cross-section view of a surface of a joule-heated microtube, showing the traces of 
+impurities which have migrated on the surface and which are responsible for the surface roughness. 
+(a) 
+(b) 
+(c) 
+(d) 
+
+500um20 um20ur5umManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+27 
+ 
+ 
+ 
+Figure S2. Micropyramids, obtained at          (high resolution SEM).    
+ 
+
+5um1 μmManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+28 
+ 
+ 
+ 
+Figure S3. The control experiment – etching of a pyrolytic graphite film at 2000°C and oxygen relative 
+concentration 600p.p.m. during 20 min. No soot formation is detected.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure S4. Grow of the “craters” on the surface of the microtubes resistively heated at          and 
+oxygen relative concentration                 (a,b): a tube surface after 80 min of heating. (c,d) the same 
+(a) 
+(b) 
+(c) 
+(d) 
+(a) 
+(b) 
+
+500um50μm500um50um1 mm100umManuscript accepted to ACS Nano 
+https://pubs.acs.org/doi/10.1021/acsnano.2c04395 
+ 
+ 
+29 
+tube surface, after 320 minutes of heating. 
+ 
+ 
+ 
+ 
+ 
+Figure S5. The morphology of a microtube, resistively heated at          and oxygen relative 
+concentration                 (after 120 minutes of heating). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure S6. (a) Graphene flowers grown on the surface of a tungsten wire suspended in a carbon tube resistively 
+heated to 2300°C. (b) Soot particles between the “petals” of the GF.  
+ 
+ 
+(a) 
+(b) 
+(b) 
+(a) 
+
+100nm200μm10μm1 μm
\ No newline at end of file
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+page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='jp ABSTRACT Almost regular hexagonal arrays of a few micrometers tall and wide micropyramids consisting of soot nano-particles are formed on the surface of graphitized hollow filaments, which are resistively heated to  1800°C−2400°C in an Ar atmosphere containing trace amounts of oxygen (~300 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' At the higher temperatures (T>2300°C, approximately) the soot particles are represented mainly by  multi-shell carbon nano-onions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The height and the width of the pyramids is strongly dependent on the temperature of the resistive heating, diminishing from 5-10\uf06dm at T\uf0bb1800°C to  \uf0bb1\uf020\uf06dm at 2300-2400°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Quasi-hexagonal arrays of the micropyramids are organized in the convex “craters” on the surface of the microtubes, which  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  2 grow with the time of the thermal treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids are pointing always normally to the surface of the craters, except at the boundaries between the craters, where the normal direction is not well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids are soft and can be easy destroyed by touching them, but can be hardened by heating them in the oxygen-free atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids are observed only on the exterior surface of the microtubes, but not on their inner surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This suggests that the thermophoretic force generated by a strong temperature gradient near the external surface of the tubes may be the cause of the micropyramids formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Electrostatic charging of the soot nanoparticles due to thermionic emission may also be relevant to this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The micropyramids can function as field emission point sources, as demonstrated with the use of a micro-nanoprobing station, mounted in a scanning electron microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' KEYWORDS: Self-assembly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' carbon nano-onions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' resistive heating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' thermophoretic force;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' micropyramids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' hexagonal array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Carbon is known for its capacity to form plenty of architectures at the nanoscale as well as the microscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Apart of the most well known carbon single and multishell tubes, fullerens, and graphene, there have been discovered numerous interesting entities such as carbon nano-horns and nano-cones,1-3 conical micro-crystals,4 carbon whiskers,5,6 graphite micro-pyramides,7 graphene flowers or vertical graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='8-12   Electrical, optical, and morphological peculiarities of these architectures make them perspective for advanced applications such as field electron emission sources,9,10  supercapacitors,13 biosensors,11ultra-black materials,14  superhydrophobic15 and bactericidal16 coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The methods of production of these structures include argon plasma etching of graphite substrates,7,14 reactive sputtering using methane gas,9 thermal chemical vapor deposition on carbon fibers,10  reduction of graphene oxide nanosheets,11 combustion flame  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  3 deposition,8 microwave plasma chemical vapor deposition, followed by bias-assisted reactive ion etching,16 wood charcoal heat treatment above 2000°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='6 Here we report soot micropyramids, which we have found on the surface of resistively heated amorphous carbon microtubes in Ar atmosphere, to which vanishingly small amount of air was admixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids are always pointing along the local normal direction to the tube’s surface, suggesting that the soot nanoparticles, formed in the oxygen-deficient atmosphere, organize into the pyramids under the action of some force, directed outward the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Moreover, the pyramids are assembled in almost hexagonal arrays, resembling the spontaneously formed hexagonal patterns, which appear at certain deformable interfaces under the action of normally applied forces, and which have been extensively studied in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Examples include the Rosenweig instability of a ferromagnetic fluid under a magnetic field,17 the gravity-driven Rayleigh−Taylor instability of suspended liquids,18  elastic solids,19  and electrostatically induced structure formations at the polymer−air interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='20  The patterns arise due to the competition of external fields (gravity, magnetic, or electrostatic), which amplify the interface deformation with surface tension and elastic energy, which have stabilizing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Linear stability analyses establish the dispersion relation between the amplitude growth rates of the normal modes of deformation and the wave numbers of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The modes with the highest growth rates dominate and determine the characteristic wavelength of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Hexagonal patterns arise due to the resonance of the dominant modes, whose wave vectors form an equilateral triangle in the reciprocal space, as shown by nonlinear stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='18,21 A natural candidate for the role of such a normally directed force in our system of soot particles on the surface of a resistively heated microtube is the thermophoretic force,    , which results from the difference of the average momentum transferred to the particle by molecules arriving  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  4  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The most characteristic morphology features of the carbon microtubes resistively heated  in the oxygen-deficient atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (a) Craters on the surface of a microtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (b,c) An array of pyramids at different magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (d) Almost hexagonal array of the pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (e) The interface between pyramid’s bottoms and the unchanged wall of a tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The white arrow : inner surface of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (f) Pyramids with thin filaments on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (g)  A magnified view of a crater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Note that the pyramids are pointing normally to the walls of the craters (h) A boundary between two craters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (i) Irregular multi-apex pyramids at the bondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' from the forward and the backward directions, with respect to the temperature gradient’s direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='22  More rapid molecules, which arrive from the region of higher temperature, transfer larger momentum on average than the molecules that arrive from the colder regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This results in a drag force on the particle, which drives it from hotter regions to colder ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Thermoelectric (a) (b) (c) (d) (e) (f) (g) (h) (i)  100umLm20um1 μm50μm1 μumumManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  5 phenomena, such as electrostatic charging of the particles due to thermionic loss of electrons at high temperatures may be also essential for the process of the micropyramids formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Results and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Figure 1 presents the most representative features of the micro-scale morphology  observed on the surface of a tube heated to                and at an oxygen concentration for 5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The surface of the tube, which was initially smooth, became covered by micro-craters (Figure 1a) whose concave bottoms are roofed by the arrays of the features which we call micro-pyramids, because their shape resembles this geometric form (Figure 1b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' No similar structure was found on the surface of the control tubes heated without air admixing (see Figure S1 of the Supporting Information, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids have sharp apices with submicrometric curvature radii, and the apices of some pyramids are split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The surface of the pyramids is not smooth, rather, it has porous aspect (see Figure 1c and Figure S2 of SI for a high-resolution image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids are often prolonged by a thread whose thickness does not exceed 100 nm   (Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' A fragment of  an extremity of an intentionally broken tube is shown in Figure 1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The image reveals the interface between the loose material of the pyramids and the monolithic structure of the tube wall, which was not yet etched by the oxygen-containing atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  The pyramids appear only on the outer surface of the tube but not on their interior walls, shown by the white arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Seen from the top, the pyramid arrays have nearly hexagonal coordination in the centers of the craters (Figure 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids are always oriented normally to the concave craters walls (Figure 1g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' At the boundaries between the craters, where the normal direction is not defined, the pyramids have irregular shapes characterized by split apices (Figures 1g,1i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  6  Figure 2 The inner structure of the pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (a) Pyramids after the application of Scotch tape to the surface of a tube (                        .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Some pyramids are partially destroyed, and some are completely removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Note that the soot pyramids cover the pyramidal elevations of the unchanged material of the tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (b) High-resolution SEM of the interior of a partially destroyed pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (c,d) HRTEM of the material of the miccropyramids, formed at          and         , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' A CNO particle is shown on the figure (c) by the white arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (e, f) the Raman spectra of the material of the pyramids, formed at          and , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' To collect the material of the pyramids for Raman spectroscopy characterization, a tube was touched by a narrow strip of Scotch tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Some of the pyramids were destroyed partially by this operation (Figure 2a), which allowed the examination of their inner structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' High-resolution SEM reveals that the pyramids consist of nanoparticles whose diameters do not exceed 30nm (Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The fact that the pyramids can be easily disrupted by touching signifies that the particles are not connected by covalent bonds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' rather, they are held together by van der Waals forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' A high-resolution TEM reveals that particles are the closed-shell ones, similar to the particles produced by the carbon arc method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='23 Some particles can be identified as multi-shell carbon nano-onions (CNO) (Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  The graphitization degree and the fraction of CNOs (a) (b) (c) (d) (e) (f)  Lim100nm10nm20nm600 D T=2000°C 1。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='/1,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content="69 400 G G 'n' a 200 2D 0 0 1000 2000 3000 4000 (cm-1)2000 D T=2380°C 1500 G L-/_=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='77 1000 e 2D 500 0 0 1000 2000 3000 4000 y(cmManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  7 increase at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' For a tube heated to T≈2380°C (Figure 2d) the quasi-totality of the particles, collected on the surface of the tubes, can be classified as CNO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The Raman spectra of the particles  (Figure 2e,f) have well-resolved G (graphitic) and D (disordered) bands,24 meaning that the particles contain a relatively low concentration of defects of the sp2- coordination of carbon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The ratio of the intensities of the G and the D peaks,  , grows with temperature, implying the improving the sp2-coordination of the CNO particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The mechanism of formation of the soot particles on the surface of the resistively heated graphitized microtubes is not yet clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The physico-chemical conditions at the surface of the microtubes are different from the conditions of the well-understood processes of soot production as  hydrocarbon flames,25,26   spark discharges27 and laser ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='28  In the flames, the process starts by the gas-phase chemical reactions leading to the formation of polycyclic aromatic hydrocarbons (PAH), followed by the particles nucleation via the PAH dimerization, particle growth by the surface reactions, and particles coagulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' It is unlikely that the carbonized microtubes can be the source of  PAH precursors, because at temperatures above \uf0bb1000°K carbon films are almost completely dehydrogenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='29 In the cases of spark discharge and laser ablation, soot nuclei are formed due to evaporation of a small amount of carbon electrode or carbon target material, and condensation of the supersaturated vapor in the neutral gas flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The soot nuclei coagulate, forming the soot particles, which aggregate eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  This mechanism is also unlikely be relevant to the formation of the soot particles in our system, because the temperatures of the resistive heating of the microtubes is well below of graphite sublimation point (around 4000°K at the ambient pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='30 To the best of our knowledge, soot formation on the surface of graphitized materials, exposed to the temperatures close to 2000°C and oxygen concentrations of a few hundreds of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' was  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  8 not reported to the date, despite a great deal of literature on oxidation of graphite materials, especially for the nuclear applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='31 Typically, temperature at which graphite oxidation was studied did not exceed 1000°K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' At these temperatures, the basal graphite plane is inert to molecular oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The graphite oxidation proceeds from the zig-zag and armchair active sites of the graphite crystallites, attacked by molecular oxygen, with the CO and CO2 desorbtion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Oxidation of multilayer graphene flakes suspended on amorphous carbon grid in air and heated by laser to temperatures close to 2000°K resulted in the layer-by-layer thinning of the flakes, assuming that flakes are eroded in a highly anisotropic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Yet, no formation of carbon nanoparticles, such as soot, was reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='32 In order to gain a deeper insight into the conditions of the soot particles formation in our system, we have made a control experiment, in which a 25μm-thick pyrolytic graphite film was resistively heated to 2000°C in the Ar atmosphere at oxygen concentration 600 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The sample was eroded in course of 20min of the treatment (Figure S3), but no soot and pyramids formation was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' In particular, the graphite stripe remained shiny, while the carbonized chitosan tubes became violet black upon etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This means that the mechanism of the soot formation in our system requires a relatively high number of defects of the sp2-coordination of the carbon material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  1800°C  1900°C  2000°C  2100°C  2200°C  (\uf06dm)  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Average distance between the pyramide centers as the function of resistive heating temperature, for  The size of the pyramids and their number per unit surface depend strongly on temperature of the resistive heating (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Within the temperature interval 1800°C – 2200°C, in which the  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  9  Figure 3 Voronoi analysis of the pyramids arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Color coding of the Voronoi cells coordination: brown – 4- edged cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' blue – 5-edged cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' green- 6-edged cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' pink – 7-edged cells, and orange – 8-edged cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (a) Superposition of the SEM image of a crater bottom for a tube heated at         , and                 during 4 hours,  and the Voronoi diagram, calculated by the pyramids vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (b) Coloring of the Voronoi diagram cells according to the number of the cells edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (c) Distribution of the cells by the number of their edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (d-f) The same as (a-c), respectively, for a tube heated at          ,               , during 2 hours (see figure S5, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' micropyramids arrays can be unambiguously identified, the average distance between the pyramids centers decreases by ~6 times as temperature grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids also became more « slim », and their apices are more often continued by a thin filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The crater-like features imply that the pyramids do not appear simultaneously on the surface of the tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Indeed, we have found that initially small, isolated craters, having an almost perfectly circular shape and containing a few pyramids, appear on the surface of the tubes a few dozens of minutes after the beginning of the resistive heating (Figure S4a,b, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' During the experiment, the craters became wider and deeper, and the number of pyramids in them increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' New craters (a) (f) (c) (b) (d) (e)  Co,~300p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='8 Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2 0 4 5 6 8 NumberofVoronoicelledges-02 ~600p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='8 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2 0 4 8 NumberofVonoicell edgesO O O OManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  10 appeared and grew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Eventually, all the surfaces became covered by the craters, which touched each other (Figure S4c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The characteristic time of the pyramids formation can be strongly reduced by increasing the oxygen concentration in the vessel atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Thus, the almost regular arrays of the pyramids were observed on the tubes heated at         ,                after 120 min of resistive heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' A few craters were observed on the surface of the tubes, but the majority of the pyramids were formed outside them (Figure S5, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Apparently, the rate of the pyramids formation affects their mutual coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This fact follows from the analysis of the Voronoi diagrams of the SEM images of the pyramids arrays produced at at           and at the relative oxygen concentrations                 and               (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' By definition, the Voronoi cell of a vortex is defined by the ensemble of the points which are closer to a given vortex than to any other vortex of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The edges of the polygons correspond roughly to the faces of the pyramids, and the vortices of the polygons correspond to the edges of the pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' In our analysis, the vortices were defined as the extremities of the micropyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' At the lower oxygen concentration, the absolute majority (~80%) of the Voronoi cells have 6 edges, while at the higher oxygen concentration, the fraction of the 6-coordinated cells drops to around 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Further increase of oxygen concentration to                 led to rapid consumption of the microtubes (typically, within 10-20 min) and irregular shapes of the pyramids, as well as poor degree of their hexagonal arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The hypothetical mechanism of the pyramids growing and the propagation of the pyramids arrays is shown on the Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' We suppose that a pyramid’s formation starts from a local defect  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  11 on the surface of a tube, such as a local microscale elevation (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Soot particles, generated at the bottom of the defect, roll up along its walls in the vertex direction, increasing the defect’s height and transforming it into a pyramide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Simultaneously, local cavities are formed at the bottom of the pyramide, because of the particles excavation from the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The particles start to roll along the walls of the cavities, forming new pyramids at their borders (Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This process is repeated, leading to formation of the pyramid’s arrays (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Similar process may start from a defect in form of a micro- cavity on the surface of a tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The growing of the pyramids seems to be limited by their sputtering from the pyramid’s vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The formation of the concave “craters” signifies that the pyramids accelerate somehow the transformation of the graphitized material in soot, because the older pyramids have propagated deeper in the walls of the tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This is probably because the migration of the soot particles onto the pyramid’s walls unmasks the tube’s surface and makes it more accessible to the molecules of oxygen present in the gas mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The height of the pyramids in the largest craters is much smaller than the craters’ depth (see Figure 1g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This means that the soot nanoparticles, which constitute the pyramids, are constantly removed from the pyramids and replaced by new particles, which are produced by the interaction of oxygen with the tube’s walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  To clarify the fate of the eliminated particles, we placed a tungsten wire close to a microtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' In 1 hour of resistive heating at         , the so-called graphene flowers8,9 formed on the surface of the wire (Figure S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Thus, the carbon material did not burn  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Hypothetical mechanism of pyramids growing and pyramids arrays propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  (a) (b) CManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  12  Figure 5 Normal view (a) and side view (b) of the micropyramids produced at and  stabilized by heating at          without air admixing during 30min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The sintering of soot particles rigidifies the pyramids, so that they are not smashed or disrupted when touched (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Figure 2a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' rather, they are wholly unrooted (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' completely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' rather, it sputtered into the space around the tubes, where it can be sedimented on a support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids can be stabilized by sintering soot particles in an oxygen-free atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This operation makes them mechanically stable (Figure 5), so that the pyramids are wholly unrooted rather than smashed, when they are touched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The sintering procedure increases also the height to base aspect ratio of the pyramids, and cleans up the space between them from the soot particles, so that the pyramids are transformed into needle-like structures, well separated from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' We suppose that the effect of the « cleaning up » of the space between the pyramids is due to the fact that, in the absence of oxygen, soot particles are not produced anymore in course of the erosion of the material of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' But they are not sintered immediately, and therefore migrate to the top of the pyramids, increasing their height to width aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  The particles which are inside the pyramids (and not on their surface) are less mobile, and for them the probability of sintering is higher, therefore these particles constitute the rigid « skeleton » of the pyramids, making them more robust mechanically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (a) (b) (c)  3um2 μmManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  13 The ensemble of the experimental facts presented above allows us to suppose the existence of a force normally directed to the interface between the soot layer and the material of the tube, which is not yet transformed into soot by its interaction with oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' A natural candidate for the role of such a force is thermophoretic force,    , which results from the difference of the average momentum transferred to the particle by molecules arriving from the forward and the backward directions, respecting the temperature gradient’s direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' More rapid molecules, which arrive from the region of higher temperature, transfer larger momentum on average than the molecules that arrive from the colder regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This results in a drag force on the particle, which drives it from hotter regions to colder ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The fact that the pyramids do not appear inside the tubes, where the temperature is presumably uniform, seems to support the hypothesis that the temperature gradient giving rise to thermophoretic force is the factor that causes the formation of pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The intensity of     can be estimated from the following considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' At , the mean free path of argon atoms is  , where is the Boltzmann constant,   is the pressure of the gas, and            is the diameter of the argon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' From the HRTEM images (Figure 2c,d), one can evaluate the characteristic diameter of the soot particles as       .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Since the Knudsen number  is large, the  thermophoretic  force  can  be  calculated  in  the  free  molecular  limit22  as  where                  is argon gas thermal conductivity at the given temperature,33      is the mass of argon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The temperature gradient at the filament’s surface is usually evaluated with the assumption that there exists a layer of almost immovable gas around the filament (the so-called stagnant layer of Langmuir).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='34  Coherent anti- Stokes Raman scattering measurements35  and computer simulations36 estimate this gradient to for tungsten filaments heated to         at the normal pressure of the  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  14 nitrogen surrounding the filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Assuming that, for argon, the temperature gradient is on the same order of magnitude, one can estimate the thermophoretic force acting on a particle of the characteristic size        as                 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' It is several orders of magnitude smaller than the typical adhesion force of soot particles measured by atomic force microscopy to be in the range of a few nN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='37  Therefore, it is hardly possible that the thermophoretic force alone can detach the soot particles from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Conversely, it is likely that, at elevated temperatures, the average kinetic energy of the particles, , takes over the van der Waals adhesion energy, allowing the rearrangement of the mutual position of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The thermophoretic force field might then bias these rearrangements so they have the effect of migrating the particles outward from the tube’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' At the tips of the pyramids, the particles have a relatively small number of neighbors and can be “evaporated” into the hot gas surrounding the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  This scenario is in line with current theories about the formation of nanoparticle aggregates, which assume that the coagulation efficiency of the particles depends on both their size and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='38,39  At sufficiently high temperatures, smaller soot particles are likely to bounce off rather than adhere to each other because of the relatively high kinetic energy compared with the interaction energy of the particles (the so-called thermal rebound effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='38 simulated the coagulation of soot particles in frames of the Hamaker model, in which the effective interaction of two spherical atomic clusters is the sum of all pairwise Lennard−Jones interactions of the atoms belonging to opposite clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' As follows from their model, the van der Waals interaction cannot hold together two spherical soot particles of diameters       when the temperature exceeds         .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' These values are uncertain, since the soot particles are mostly aspherical at this temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Moreover, in close packing, each particle has more than one neighbor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' therefore, it needs more kinetic energy to escape from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Also, the formation of  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  15 covalent bonds between some particles is not excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Nevertheless, it seems plausible that at high temperatures, which are reached in the resistively heated tubes, the soot particles may move with respect to each other, at least on the surface of the pyramids, where they have a smaller average number of contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' It may, then, be possible to make an analogy between the ensemble of soot particles at high temperatures and a layer of suspended viscous fluid whose interface is deformed by gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='18 This analogy is supported by the existence of thin filaments by which the extremities of some pyramids are prolonged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Such filaments are unlikely to be formed by the random aggregation of soot particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Rather, they look as the result of the viscous flow in a force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Another argument for the liquid-like behavior of the soot layer on the surface of resistively heated microtubes is the circular growth of pyramid clusters (Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The already formed pyramids seem to induce the formation of subsequent generations of pyramids in their neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Similar behavior was observed for the arrays of pending droplets in the layers of suspended viscous fluid film,18 in which the concentric rings of hexagonally coordinated droplet arrays grow around the initial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Thermoelectric phenomena may be also relevant to the pyramids formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  A grounded carbon spherical nanoparticle of the diameter        loses one electron every         at (       )  and every         at          (       ) , as calculated by the Richardson−Dushman equation for the thermoelectric current,  , where is the surface of the particle,                   is the Richardson’s constant, and               is the work function for carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='40,41 Filippov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' considered the possibility of an electrostatic breakdown (Coulomb explosion) of soot agglomerates heated by laser pulses to high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='40  If the analogy between the soot layer at high temperature and a layer of viscous liquid having a finite surface tension is correct, then charging particles on  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  16 the surface of the layer may lead to the development of electrohydrodynamic instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='42 However, unlike the particles in the isolated soot clusters, the particles on the surface of the tubes constantly restore their electroneutrality due to the electron flux from the grounded tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Unfortunately, in the absence of literature and data on the conductivity and effective permittivity of the soot layer at high temperatures, it is impossible to make reliable estimates of the surface charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Theoretically, it might be possible to discriminate, which of the two factor is more relevant to the pyramids formation, by heating the carbonized tubes in an oven, thus eliminating the temperature gradient at the tubes surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' But, the maximal temperature, at which the alumina ovens can operate in continuous regime does not exceed 1600° - 1700°C, which is essentially below of the lowest temperature, at which we succeeded to observe the pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The graphitic ovens can reach much higher temperatures (up to 3000°C), but they are incompatible with the oxygen- containing atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The arrays of the carbon micropyramids may be explored as the field emitter arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='9,10,43   As a first proof of concept we present here preliminary results to test the capacity of field emission properties of individual pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' We used a micro-nanoprobing station mounted in a scanning electron microscope, see Figure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  The nanoprobes are standard tungsten tips without any coating (such as for example gold to improve the workfunction of the tip probe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The measurements are taken at 10-6 mbar at the ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' An example of current-voltage (I-V) characteristic of a pyramid, which is separated from the tip by a 200nm wide gap, is shown in Figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The observed emission curves are very similar to those observed in a previous work devoted to the emission from carbon nanotube films and approximated by the classical Fowler– Nordheim (FN) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='43   The first few sweeps show a noisy curves (blue, red, green lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  17  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The measurement of the tunneling current from the micropyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (a) The set-up: four micro-nano- manipulator IMINA® mounted around a cryo stage (600K-77K), Kammrath® (b) SEM image of the nanoprobe in front of a pyramid  (d=200nm) and the emitted current  vs applied potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The current increases with a threshold bias of -75V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (c) SEM image and  anomalous (non-FN) emitted current versus bias curve for two pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (d) The I-V curve for a pyramide annealed at 2300°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  After several sweeps the I-V characteristics became regular (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This is usually attributed to a “conditioning” of the emitter, such as oxygen desorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='43   The threshold of electron emission is -75V and the measured emitted current reach 60nA for a bias of -100V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' It should be said that the emissive characteristics of the pyramids are highly variable and may demonstrate completely different I-V curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The Figure 6c shows two type of I-V curve (a) (b) (c) (d)  70 60 50 Current (nA) 40 30 20 10 d=200nm 20 0 -20 -40 -60 -80 -100 Voltage (V)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='06 d=200nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='04 Current (nA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='01 0 -50 -100 -150 -200 Voltage (V)60- 50- Current (nA) 40 30 20- 10- d=800nm 0 0 -50 -100 -150 -200 Voltage (V)Tip 1μmTipTip 5umManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  18 recorded for two other pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The shape of the curve is very different than the expected FN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The current increases slowly up to a sharp increase at different bias voltage with a saturation at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='5nA which is a value of one order of magnitude lower than for the  pyramid shown on the Figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Possibly, the morphology of the pyramids changes during the I-V measurement, however a plateau of emissive current is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The FN-like emission curves were detected for the micropyramids annealed at 2300°C (Figure 6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' From these measurements follows that the carbon micropyramids do can be considered as the field emission sources, yet their emissive characteristics are very variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' It is not surprising since, according to the FN theory, the tunneling currents are highly dependent on the shape of the extremities of the emitters (mathematically, it is expressed by the local field enhancement factor), and the extremities of the pyramids are all different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' We need much more studies to understand the behavior of individual pyramids, to probe the emissivity with lower tip radius curvature (these experiments were done with 500nm but tip with 100 and 50 nm) to see the effect of the pyramids shapes and morphology, as well as the coating on the tungsten tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Conclusion and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Hexagonal arrays of micropyramids composed of soot nanoparticles are observed on the surface of graphitized microtubes resistively heated to                in the Ar atmosphere containing a small (~300-600p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=') fraction of oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The few micrometer tall and wide micropyramids are arranged in quasi-regular hexagonal arrays on the surface of the microtubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids are pointing normally to the walls of the craters, except the borders between the craters, where the normal direction is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids are formed only on the external walls of the microtubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' HRTEM reveals that the increasing fraction soot particles is morphologically identical to the carbon nano-onions, when resistive heating temperature grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  19 The pyramids are soft, and can be easily destroyed by touching them, but they can be rigidified by the resistive heating without oxygen admixing in the Ar stream passing through the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The mechanism of the pyramids formation and their self-assembly in the quasi-hexagonal arrays is not yet clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The fact that the pyramids are always directing parallel to the local normals to the tube surface indicates on the existence of some factor which favors the drift of the nanoparticles in these directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The thermophoretic force, arising as the consequence of a strong temperature gradient in the vicinity of the tube, is a natural candidate on the role of such a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Under this assumption, one can draw an analogy with the gravity-driven Rayleigh−Taylor instability of a thin suspended liquid layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' It is also possible that the electrostatic charging of the soot particle layer due to thermoelectric emission contributes to the pyramids formation and their assembly in the hexagonal arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Regular mutual arrangement and the sharp extremities of the micropyramids allow to consider them as perspective micro-structures for advanced applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' In our study, we have demonstrated that the individual carbon micropyramids emit the tunneling current upon application of local electrical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' This indicates on the principal possibility to create field emitter arrays45 on the base of the micropyramids ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  To achieve this goal, a reliable approach should be found for the formation of the pyramids arrays on the flat substrates and on sufficiently large surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The pyramids ensembles on the flat substrates may be then investigated also for the creation of superhydrophobic15 and bactericidal16  coatings, as well as the ultra-black materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='14 Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The experimental setup for the generation of the soot micropyramid arrays is shown schematically in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' An electro-conductive hollow-core microtube derived from a  chitosan  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  20 self-rolled microtube, made according to the procedure given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' [44], was placed inside quartz vessel 1, where it was suspended on crocodile-like electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' An electrical current was generated in the tube by a DC power supply 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The vessel was filled with an air/argon mixture at normal pressure             with the use of valves 3 and 4, guided by a multichannel gas flow controller 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The experimental setup to produce arrays of self-assembled soot micropyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  After passing through the system, the gas mixture was rejected in atmosphere 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The relative concentration,    , of oxygen was varied between 300 and 3,000 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The temperature of the microtube was measured by Modline 5 Raytec optical pyrometer 7 conjugated with a computer 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Typically, the formation of well-developed pyramid patterns requires several dozens of minutes              , but at a relatively high O2 concentration, this time period was shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' After a joule heating experiment, the filament was gently dismounted and examined by scanning electron microscopy (SEM) and transmission electron microscopy (TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Soot particles transferred onto Scotch tape were examined by Raman spectroscopy (Horiba Labram 300,  =632nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The field emission was measured with the use of a nanoprobing station mounted  2 8 Air 5 1 Ar 4Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  21 in a scanning electron microscope FEG-MEB XL30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The station consisted of four micro- nanomanipulators IMINA®  mounted around a Kammrath® cryostage (600K-77K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Panasonic pyrolytic graphite sheets were purchased at Farnell Electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Acknowledgements We thank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Vidal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Knopf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Rety, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Beda (all-IS2M CNRS) for their assistance with electron microscopy imaging and the carbonization of chitosan microtubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The measurements of the tunneling currents were done with the financial support by the CNRS, the Région Grand Est and FEDER (EU) through the NanoteraHertz project, and the French National Research Agency (ANR) through the MIXES project (Grant ANR-19-CE09-0028).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The Supporting Information Available : Figure S1, a control experiment on resistive heating of a tube without air admixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Figure S2, high resolution SEM images of the micropyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Figure S3, a control experiment, etching of a pyrolytic graphite film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Figure S4, the grow of “craters” on the surface of the resistively heated microtubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Figure S5, the morphology of a microtube, resistively heated at          and oxygen relative concentration              .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Figure S6, graphene flowers grown on the surface of a tungsten wire suspended in the vicinity of a resistively heated tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  22  References  [1]  Harris, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' 2021, 172, 26-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' [13]  Huang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' RSC  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='2c04395  23 Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' 2016, 6, 16745-16750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' Ding, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='  He, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' Express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' 2015,  23, 20115- 20123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' 2014, 15, 89-104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='  Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Nobbs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' Biointerfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' 2002, 4, 2028-2037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' [27]  Hagen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='  Rinkenburger,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Gunther, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Bockhorn, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Niessner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Suntz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Loukou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Trimis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Haisch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Spark-Discharge Generated Soot: Varying Nanostructure and Reactivity Against Oxidation with Molecular Oxygen by Synthetic Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' 2020, 143, 43-  Manuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  24 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Utry, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Pintér, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Kiss-Albert, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Puskas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Tapai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Kecskeméti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Smausz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Hopp, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Bozoki, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Konya, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Szabo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Microphysical Properties of Carbonaceous Aerosol Particles Generated by Laser Ablation of a Graphite Target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Atmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Angot, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Aréou, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Dürbeck, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Jacob, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Martin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='  Pardanaud, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Roubin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' Schwarz-Selinger, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Characterization of Temperature-Induced Changes in Amorphous Hydrogenated Carbon Thin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Diam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='  Graphite Sublimation Temperatures, Carbon Arcs and Crystallite Erosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' 1974, 12, 111-141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='  Contescu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Smith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='  Strydom, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Windes, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='  Understanding the Reaction of Nuclear Graphite with Molecular Oxygen: Kinetics, Transport, and Structural Evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Alaferov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content='  Vaz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Perim, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Autrero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Paupitz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Galvao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' Moshkalov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' 1988,  159, 129-148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' [42]  Firouznia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Saintillan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Electrohydrodynamic Instabilities in Freely Suspended Viscous Films Under Normal Electric Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' 2021,  6, 103703 (17pp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' [43]  Di Bartolomeo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Scarfato, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Giubileo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Bobba, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Biasiucci, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Cucolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Santucci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Passacantando, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  A Local Field Emission Study of Partially Aligned Carbon-Nanotubes by Atomic Force Microscope Probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' 2007,  45, 2957-2971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' [44]  Beda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Yamada, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Egunov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Matei Ghimbeu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Malval, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Saito, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Luchnikov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Carbon Microtubes Derived from Self-Rolled Chitosan Acetate Films and Graphitized by Joule Heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' 2019,  54, 11345–11356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' [45]  Guibileo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Di Bartolomeo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Iemmo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Luongo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Urban, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Field Emission from Carbon Nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' 2018,  8, 526 (21pp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  TOC figure  200um 5Lm Self-assemblyofsootnanoparticlesintoregulararraysofmicropyramidswasobserved onthesurfaceofcarbonmicrotubesresistivelyheated inoxygen-deficientatmosphereManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  26 Self-assembly of soot nanoparticles on the surface of resistively heated carbon microtubes in near-hexagonal arrays of micropyramids - Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Valeriy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Luchnikov,1* Yukie Saito,2* Luc Delmotte,1 Joseph Dentzer,1 Emmanuel Denys,1 Vincent Malesys,1 Ludovic Josien,1 Laurent Simon,1 Simon Gree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1  1Université de Haute-Alsace, CNRS, IS2M, UMR 7361, Mulhouse F-68057, France 2Graduate School of Agricultural and Life Sciences, The University of Tokyo, Tokyo, 113- 8657, Japan  Corresponding authors: Valeriy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Luchnikov, email: valeriy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='luchnikov@uha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='fr and Yukie Saito, email: aysaito@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='ecc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='u  tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='jp  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Control experiment on resistive heating of a tube without air admixing, at         .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (a,b) Surface of the tube, at different magnifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (c) An impurity escaping the microtube surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (d) Cross-section view of a surface of a joule-heated microtube, showing the traces of impurities which have migrated on the surface and which are responsible for the surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (a) (b) (c) (d)  500um20 um20ur5umManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  27  Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Micropyramids, obtained at          (high resolution SEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  5um1 μmManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  28  Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The control experiment – etching of a pyrolytic graphite film at 2000°C and oxygen relative concentration 600p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' during 20 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' No soot formation is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' Grow of the “craters” on the surface of the microtubes resistively heated at          and oxygen relative concentration                 (a,b): a tube surface after 80 min of heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (c,d) the same (a) (b) (c) (d) (a) (b)  500um50μm500um50um1 mm100umManuscript accepted to ACS Nano https://pubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='2c04395  29 tube surface, after 320 minutes of heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' The morphology of a microtube, resistively heated at          and oxygen relative concentration                 (after 120 minutes of heating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (a) Graphene flowers grown on the surface of a tungsten wire suspended in a carbon tube resistively heated to 2300°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content=' (b) Soot particles between the “petals” of the GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
+page_content='  (a)  (b)  (b)  (a)  100nm200μm10μm1 μm' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E4T4oBgHgl3EQfVQyb/content/2301.05023v1.pdf'}
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+Finding and Counting Patterns in Sparse Graphs
+Balagopal Komarath �
+IIT Gandhinagar, India
+Anant Kumar �
+IIT Gandhinagar, India
+Suchismita Mishra �
+Universidad Andrés Bello, Chile
+Aditi Sethia �
+IIT Gandhinagar, India
+Abstract
+We consider algorithms for finding and counting small, fixed graphs in sparse host graphs. In the
+non-sparse setting, the parameters treedepth and treewidth play a crucial role in fast, constant-space
+and polynomial-space algorithms respectively. We discover two new parameters that we call matched
+treedepth and matched treewidth. We show that finding and counting patterns with low matched
+treedepth and low matched treewidth can be done asymptotically faster than the existing algorithms
+when the host graphs are sparse for many patterns. As an application to finding and counting fixed-
+size patterns, we discover �O(m3)-time 1, constant-space algorithms for cycles of length at most 11 and
+�O(m2)-time, polynomial-space algorithms for paths of length at most 10.
+2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms; The-
+ory of computation → Graph algorithms analysis
+Keywords and phrases Subgraph Detection and Counting, Homomorphism Polynomials, Treewidth
+and Treedepth, Matchings
+1
+Introduction
+Given simple graphs G, called the pattern, and H, called the host, a fundamental compu-
+tational problem is to find or count occurrences of G in H. What does it mean for G to
+occur in H? The three most common notions of occurrence are characterized by mappings
+φ : V(G) �→ V(H). We say:
+1. If {u, v} ∈ E(G) implies {φ(u), φ(v)} ∈ E(H) and φ is one-to-one, then we say that
+φ witnesses a subgraph isomorphic to G in H. The subgraph is obtained by taking the
+vertices and edges in the image of φ. The number of G-subgraphs of H is just the number
+of such subgraphs G′ of H.
+2. If {u, v} ∈ E(G) is equivalent to {φ(u), φ(v)} ∈ E(H) and φ is one-to-one, then φ wit-
+nesses an induced subgraph isomorphic to G in H. The induced subgraph is obtained by
+taking the vertices and all edges induced by those vertices in the image of φ.
+3. If {u, v} ∈ E(G) implies {φ(u), φ(v)} ∈ E(H), then we say that φ is a homomorphism from
+G to H. Note that unlike a subgraph isomorphism, φ is not required to be one-to-one.
+1
+�O hides factors that are logarithmic in the input size.
+arXiv:2301.02569v1  [cs.DS]  6 Jan 2023
+
+2
+Finding and Counting Patterns in Sparse Graphs
+For any of these notions, the detection problem is clearly in NP. All three of them are also
+straightforward generalizations of the NP-hard problem CLIQUE. Therefore, the existence
+of efficient algorithms for finding or counting patterns under any of these notions is unlikely
+in general.
+The class of pattern detection and counting problems remain interesting even if we restrict
+our attention to fixed pattern graphs. Williams [21] showed that the improved algorithms
+for finding triangles could be used to find faster algorithms for even NP-complete problems
+such as MAX2SAT. For fixed pattern graphs of size k, the brute-force search algorithm is
+as follows: Iterate over all k-tuples over V(H) and check whether G occurs in the induced
+subgraph of H on the vertices in that k-tuple. This algorithm takes θ(nk) time and constant
+space. Therefore, when we restrict our attention to fixed patterns, we seek improvements
+over this running time preferably keeping the space usage low. There are two broad tech-
+niques that reduce the running-time: the usage of fast matrix multiplication algorithms as a
+sub-routine and the exploitation of structural properties of pattern graphs.
+If A is the adjacency matrix of the graph, then Nešetˇril and Poljak [17] showed that one can
+obtain an O(nω)-time algorithm for counting triangles using the identity trace(A3) = 6∆,
+where ∆ is the number of triangles in the graph, where ω < 2.38 is the matrix multiplication
+exponent. Using a simple reduction, they extended this to an algorithm to count 3k-cliques
+in O(nkω)-time. They also showed that we can use improved algorithms for counting k-
+cliques to count any k-vertex pattern. Later, Kloks, Kratsch and Müller [13] showed how
+to use fast rectangular matrix multiplication to obtain similar improvements to the running
+time for counting cliques of all sizes, not just multiples of three. Note that the improvements
+obtained by these algorithms are applicable to all k-vertex patterns. i.e., they do not use the
+pattern’s structure to obtain better algorithms. Since finding a k-clique requires nΩ(k)-time
+unless ETH is false, we need to exploit the structure of the pattern to obtain significantly
+better algorithms.
+For patterns sparser than cliques, the run-time can be significantly improved over even fast
+matrix multiplication based (pattern finding) algorithms. The crucial idea is to exploit the
+structure of the pattern graph. A k-walk polynomial is a polynomial where the monomials
+correspond to walks that are k vertices long. For example, a walk (u, v, w, x) will correspond
+to the monomial xuvxvwxwx and a walk (u, v, u, v) to the monomial x3
+uv
+2. Williams [22]
+showed that we can detect k-paths in graphs by (1) computing the k-walk polynomial and
+(2) checking whether it has multilinear monomials. We can compute the k-walk polynomial
+in linear-time using a simple dynamic programming algorithm and then multilinear monomi-
+als can be detected with high probability by evaluating this polynomial over an appropriate
+ring where the randomly chosen elements satisfy a2 = 0. This yields is a O(2k(n+m))-time
+algorithm for finding k-vertex paths as subgraphs in n-vertex, m-edge host graphs.
+We now consider the problem of counting sparse patterns. For counting k-paths as sub-
+graphs, the best known algorithm by Curticapean, Dell and Marx [4] takes only O(f(k)n0.174k+o(k))-
+time for some function f. Coming to fixed pattern graphs, Alon, Yuster and Zwick [1] gave
+O(nω)-time algorithms for counting cycle subgraphs of length at most 8 using an algorithm
+that combines fast matrix multiplication and exploitation of the structure of the pattern. No-
+tice that this is the same as the time required for counting triangles (3-cliques).
+2 We write uv to denote the edge {u, v}.
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+3
+The notion of graph homomorphisms was shown to play a crucial role in all the above im-
+proved algorithms for finding and counting non-clique subgraphs. More specifically, Fomin,
+Lokshtanov, Raman, Rao, and Saurabh [11] showed how the efficient construction of ho-
+momorphism polynomials (see Definition 17), a generalization of k-walk polynomials, can
+be used to detect subgraphs with small treewidth efficiently. Their algorithm can be seen
+as a generalization of Williams’s algorithm [22] for k-paths to arbitrary graphs. Similarly,
+Curticapean, Dell and Marx [4] showed that efficient algorithms for counting subgraphs
+can be derived from efficient algorithms for counting homomorphisms of graphs of small
+treewidth. Their algorithm can be seen as a generalization of the cycle-counting algorithms
+of Alon, Yuster, and Zwick [1].
+Algorithms for finding and counting patterns in sparse host graphs are also studied. An
+additional parameter, m, the number of edges in the host graph, is taken into account for
+the design and analysis of these algorithms. In the worst-case, m could be as high as
+�n
+2
+�
+,
+and hence, an O(nt)-time algorithm and an O(mt/2)-time algorithm for some t have the
+same asymptotic time complexity. However, it is common in practice that m = o(n2). For
+example, if the host graph models a road network, then m = O(n), where the constant
+factor is determined by the maximum number of roads at any intersection. In such cases, an
+O(mt/2)-time algorithm is asymptotically better than an O(nt)-time algorithm.
+The broad themes of using fast matrix multiplication and/or structural parameters of the pat-
+tern to obtain improved algorithms are still applicable in the setting of sparse host graphs.
+Using fast matrix multiplication, Eisenbrand and Grandoni [8] showed that we can count
+k-cliques in O(mkω/6)-time. Kloks, Kratsch and Müller [13] showed that K4 subgraphs can
+be counted in O(m(ω+1)/2)-time. Again, since ω < 3, this is better than the O(m2)-time
+given by the brute-force algorithm. Using structural parameters of the pattern, Kowaluk,
+Lingas, and Lundell [16] obtained many improved algorithms in the sparse host graph set-
+ting. For example, their methods obtain an algorithm that runs in O(m4)-time for counting
+P10 as subgraphs. In this work, we obtain an �O(m2)-time algorithm for counting P10 (See
+Theorems 12,13,14,15 for similar improvements).
+The model of computation that we consider is the unit-cost RAM model. In particular, we can
+store labels of vertices and edges in the host graph in a constant number of words3. In this
+model, algorithms based on fast matrix multiplication and/or treewidth mentioned above
+use polynomial space. However, the brute-force search algorithm uses only constant space
+as it only needs to store k vertex labels at a time (Recall that we regard k as a constant.).
+How much speed-up can we obtain while preserving constant space usage? The graph para-
+meter treedepth plays a crucial role in answering this question. It is well known that we
+can count the homomorphisms from a pattern of treedepth d in O(nd)-time while using
+only constant space (See Komarath, Rahul, and Pandey [14] for a construction of arithmetic
+formulas counting them. These arithmetic formulas can be implicitly constructed and eval-
+uated in constant space.). Since all k-vertex patterns except k-clique has treedepth strictly
+less than k, this immediately yields an improvement over the running-time of brute-force
+while preserving constant space usage. In this work, we improve upon the treedepth-based
+algorithms for sparse host graphs where the pattern graph is a cycle of length at most 11
+(See Theorem 2).
+3 In the TM model or the log-cost RAM model, storing labels of vertices would take O(log n) space.
+
+4
+Finding and Counting Patterns in Sparse Graphs
+1.1
+Connection to arithmetic circuits for graph homomorphism
+polynomials
+A popular sub-routine in these algorithms is an algorithm by Diaz, Serna and Thilikos [6]
+that efficiently counts the number of homomorphisms from a pattern of small treewidth
+to an arbitrary host graph. Indeed, it can be shown that this algorithm can be easily gen-
+eralized to efficiently construct circuits for homomorphism polynomials instead of counting
+homomorphisms. Bläser, Komarath and Sreenivasaiah [2] showed that efficient construc-
+tions for homomorphism polynomials can even be used to detect induced subgraphs in some
+cases. They also show that many of the faster induced subgraph detection algorithms, such
+finding four-node subgraphs by Williams et al. [23] and five-node subgraphs by Kowaluk,
+Lingas, and Lundell [16] can be described as algorithms that efficiently construct these
+homomorphism polynomials. Therefore, arithmetic circuits for graph homomorphism poly-
+nomials provide a unifying framework for describing almost all the fast algorithms that
+we know for finding and counting subgraphs and finding induced subgraphs. Can we im-
+prove these algorithms by finding more efficient ways to construct arithmetic circuits for
+homomorphism polynomials? Unfortunately, it is known that for the type of circuit that is
+constructed, i.e., circuits that do not involve cancellations, the existing constructions are the
+best possible for all pattern graphs, as shown by Komarath, Pandey, and Rahul [14]. The situ-
+ation is similar for constant space algorithms. The best known algorithms can be expressed
+as divide-and-conquer algorithms that evaluate small formulas constructed by making use
+of the graph parameter treedepth. Komarath, Pandey, and Rahul[14] also showed that the
+running-time of these algorithms match the best possible formula size for all pattern graphs.
+These arithmetic circuit lower bounds serve as a technical motivation for considering sparse
+host graphs, in addition to the practical motivation mentioned earlier.
+1.2
+Our findings
+In this paper, we study algorithms for finding and counting patterns in host graphs that
+work well especially when the host is sparse. We discover algorithms that are (1) strictly
+better than the brute-force algorithm, (2) strictly better than the best-known algorithms
+when the host graph is sparse, (3) close to the best-known algorithms when the host graphs
+are dense. Our algorithms are based on two new structural graph parameters – the matched
+treedepth and matched treewidth. (See 19 and 21 for formal definitions). We show that they
+can be used to obtain improved running times for algorithms that use constant space and
+polynomial space respectively. Our algorithms are summarized in Table 1. In the table, the
+parameter m is the number of edges in the host graph. We denote using mtw the matched
+treewidth of the pattern and using mtd the matched treedepth of the pattern. The notation
+�O hides factors that are poly-logarithmic in the input (the host graph) size.
+We now explain the relevance of our new parameters; state our algorithms, the relationships
+between various graph parameters, and some structural characterizations that we prove in
+this paper in the rest of this section.
+1.2.0.1
+Treedepth and matched treedepth
+The matched treedepth of a graph is closely related to its treedepth. The constant space
+algorithm based on treedepth is essentially an divide-and-conquer algorithm over a elim-
+ination tree of the pattern graph that executes a brute-force search over each root-to-leaf
+path in the elimination tree. Therefore, it runs in time O(nd). We exploit the fact that the
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+5
+Pattern
+Type
+Problem
+Time
+Space
+Remarks
+Ck
+Subgraph
+Counting
+�O(m3)
+O(1)
+k ⩽ 11
+Pk
+Subgraph
+Counting
+�O(m2)
+�O(m2)
+k ⩽ 10
+Ck
+Subgraph
+Counting
+�O(m2)
+�O(m2)
+k ⩽ 9
+Any
+Homomorphism
+Counting
+�O(m⌈mtd/2⌉)
+O(1)
+Any
+Homomorphism
+Counting
+�O(m⌈(mtw+1)/2⌉)
+�O(m⌈(mtw+1)/2⌉)
+C6
+Induced subgraph
+Detection
+�O(m2)
+�O(m2)
+Pk
+Induced subgraph
+Detection
+�O(m⌈(k−2)/2⌉)
+�O(m⌈(k−2)/2⌉)
+Table 1 Pattern counting and detection algorithms for sparse host graphs.
+elimination tree is matched, which forces an additional constraint that the vertices in each
+root-to-leaf path has to be covered by a matching. This allows the brute-force part of the
+algorithm to discover all d vertices on the path using only d/2 edges. The central algorithm
+that we use to obtain constant space algorithms is given below:
+▶ Theorem 1. Let G be a graph with mtd(G) = d, then given an m-edge graph H as input, we
+can count the number of homomorphisms from G to H in �O(m⌈d/2⌉)-time and constant space.
+It is well-known that the number of G-subgraphs, for any G, can be expressed as a linear
+combination of the number of homomorphisms from a related set of graphs called the spasm
+of G. The spasm of G contains exactly all graphs that can be obtained by iteratively merging
+the independent sets in G. (See 26 for formal definition). Although the treedepth of the
+spasm of C11 is bounded by 6, however, the matched treedepth is not necessarily bounded
+by the treedepth. We analyze all graphs in the spasm of C10 and C11 (there are 501 such
+graphs) and show that the matched treedepth of each graph is at most 6. This yields the
+following algorithm:
+▶ Theorem 2. Given an m-edge graph H as input, we can count the number of Ck, where
+k ⩽ 11, as subgraphs in �O(m3)-time and constant space.
+For comparison, the brute-force algorithm takes O(m6)-time and constant space; and the
+treedepth based algorithm takes O(n6)-time and constant space.
+As seen from the proof of our algorithm for counting C11, the spasm of a pattern can contain
+a large number of graphs even for relatively small patterns. Therefore, it would be nice
+to have theorems that upper-bound the matched treedepth. Unfortunately, the property
+mtd(G) ⩽ k is not even subgraph-closed unlike treedepth. For example, it can be proved
+that mtd(K4 − e) = 3 but mtd(C4) = 4. However, interesting structural observations can still
+be made for matched treedepth. The following is a theorem that upper-bounds matched
+treedepth in terms of treedepth.
+▶ Theorem 3. For any graph G, mtd(G) ⩽ 2 · td(G) − 2.
+Theorem 3 implies that our constant-space algorithms from Theorem 1 for counting homo-
+morphisms are asymptotically faster for all patterns, where the inputs are sparse host graphs,
+when compared to the treedepth-based algorithm.
+The following theorem shows that the time complexity for counting homomorphisms of a
+pattern is lower-bounded by the time complexity for counting all of its induced subgraphs.
+▶ Theorem 4. Let G be a graph and G′ is a connected, induced subgraph of G, then:
+
+6
+Finding and Counting Patterns in Sparse Graphs
+1. mtd(G′) ⩽ mtd(G) if mtd(G) is even.
+2. mtd(G′) ⩽ mtd(G) + 1 if mtd(G) is odd.
+In light of the importance of matched treedepth, it becomes crucial that we understand this
+structural parameter as much as possible. The graphs of treedepth 2 are exactly the class
+of star graphs. This is also the class of graphs with matched treedepth 2. However, for the
+graph C4, we have td(C4) = 3 and mtd(C4) = 4. So it is interesting to know what are exactly
+the graphs where treedepth and matched treedepth coincide. The following theorem should
+be viewed as giving us a preliminary understanding of the relationship between these two
+parameters.
+▶ Theorem 5. Let G be a graph such that td(G) = 3. Then mtd(G) = 3 if and only if G is
+(C4, P6, T3,3)-free.
+The graph T3,3 is the (3, 3) tadpole graph (See Figure 2).
+1.2.0.2
+Treewidth and matched treewidth
+The treewidth-based dynamic programming algorithm of Díaz, Serna, and Thilikos [6] can
+be strengthened to output an arithmetic circuit that computes the homomorphism polynomial
+for the pattern. An arithmetic circuit is a directed acyclic graph where each internal node
+is labeled + or ×, each leaf is labeled by a variable or a field constant, and there is a
+designated output node. Such a graph computes a polynomial over the underlying field in
+a natural fashion. We find that by using a dynamic programming algorithm over matched
+tree decompositions, we can improve the size of the arithmetic circuit for sparse host graphs.
+Our central theorem is given below:
+▶ Theorem 6. Let G be a graph with mtw(G) = t, then given an m-edge host graph H as
+input, we can construct an arithmetic circuit computing the homomorphism polynomial from
+G to H in time �O(m⌈(t+1)/2⌉).
+For graphs where matched treewidth and treewidth coincide, the running time for counting
+homomorphisms is a quadratic improvement on the algorithm by Díaz, Serna, and Thilikos
+[6] for sparse graphs. Therefore, this is also the best possible improvement one can hope
+to get without improving upon the algorithm by Díaz, Serna and Thilikos [6]. What is the
+worst case? The following theorem implies that the resulting algorithm cannot be worse on
+sparse host graphs.
+▶ Theorem 7. For any graph G, we have mtw(G) ⩽ 2 · tw(G) + 1.
+Unfortunately, unlike for treewidth, the parameter mtw(G) is not monotone over the sub-
+graph partial order. We first observe an explicit graph family with lower tw and larger mtw.
+Consider the complete bipartite graph Kn,n on n vertices. Notice that tw(Kn,n) = n.
+▶ Proposition 8. mtw(Kn,n) = 2n − 2 for all n > 1.
+The following observation shows that there exists supergraphs of Kn,n with lower mtw than
+that of Kn,n.
+▶ Observation 9. Consider the supergraph G of Kn,n such that V(G) = V(H), and there are
+edges in one partition of Kn,n such that the independent set of size n becomes a path on n
+vertices. Note that although mtw(Kn,n) = 2n − 2, but mtw(G) = n.
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+7
+We show how we can use structural theorems about matched treewidth to prove algorithmic
+upper bounds. For example, to count the number of P10 subgraphs, we only have to show
+that all graphs in the spasm of P10 have low matched treewidth. The spasm of P10 is a
+large set that contains more than 300 graphs. Indeed, it is possible to analyze the matched
+treewidth for each of these graphs individually. However, it would be better if we have
+theorems that eliminate such tedious work.
+We derive some structural theorems for low values of matched treewidth.
+Graphs with
+matched treewidth 1 are exactly trees. We also show tw(C5) = 2 and mtw(C5) = 3.
+We characterize the matched treewidth of partial 2-trees using forbidden induced minors
+(See Definition 24) wherever possible.
+We show that C5 is exactly the obstruction that
+forces higher matched treewidth for partial 2-trees.
+▶ Theorem 10. For any partial 2-tree G, the graph G is C5-induced-minor-free if and only if
+mtw(G) = 2.
+Notice that tw(G) = 2 yields O(n3)-time algorithms for counting homomorphisms. Even
+if mtw(G) = 3, we obtain �O(m2)-time algorithms for counting homomorphisms which is
+an improvement for sparse graphs. Does all treewidth 2 graphs have matched treewidth
+at most 3? No. The graph X in Figure 1 has treewidth 2 and matched treewidth 4 (See
+Observation 35). In fact, we can prove that X is exactly the obstruction that forces treewidth
+2 graphs to have matched treewidth 4.
+u0
+u1
+u2
+u3
+u4
+u5
+u′
+1
+u′
+3
+u′
+5
+Figure 1 The graph X.
+▶ Theorem 11. For any partial 2-tree G, the graph G is X-induced-minor-free if and only if
+mtw(G) ⩽ 3.
+This theorem implies that all X-induced-minor-free, treewidth 2 patterns have �O(m2)-time
+homomorphism counting algorithms. This is an improvement for sparse host graph even
+over the fast matrix multiplication based algorithm given by Curticapean, Dell, and Marx
+[9] for counting homomorphisms from treewidth 2 graphs that runs in O(nω)-time. Since
+the spasm of P10 does not contain any treewidth 4 graph or graph with an X-induced minor,
+we can show that there is an �O(m2)-time algorithm for counting subgraph isormophisms of
+all paths on at most 10 vertices by showing that all treewidth 3 graphs in the spasm of P10
+has matched treewidth 3. There are only 18 such graphs. Analyzing their matched treewidth
+yields the following theorem:
+
+8
+Finding and Counting Patterns in Sparse Graphs
+▶ Theorem 12. Given an m-edge graph H as input, we can count the number of Pk subgraphs,
+where k ⩽ 10, in �O(m2)-time.
+To the best of our knowledge, the best known path counting algorithms take Ω(n4) time
+for paths on 10 vertices. Therefore, our algorithm is a significant improvement for sparse
+host graphs and no worse than the best known algorithm for dense host graphs. An easy
+corollary of the proof of this result is given below:
+▶ Theorem 13. Given an m-edge graph H as input, we can count the number of cycles of
+length at most 9 in �O(m2)-time.
+These cycle counting algorithms are an improvement on sparse graphs over the O(nω)-time
+algorithms for cycles of length at most 8 given by Alon, Yuster and Zwick [1].
+We also show how to use our improved homomorphism polynomial construction algorithm
+to speed up detection of induced subgraphs. In particular, we show the following:
+▶ Theorem 14. Given an m-edge host graph as input, we can find an induced C6 or report
+that none exists in �O(m2)-time.
+This algorithm is no worse than the O(n4) time algorithm that can be derived using the
+techniques by Bläser, Komarath, and Sreenivasaiah [2]. For sparse graphs, our algorithm
+provides a quadratic improvement. We also show the following:
+▶ Theorem 15. Given an m-edge host graph as input, we can find an induced Pk or report
+that none exists in �O(m(k−2)/2)-time.
+This is also a quadratic improvement over the O(nk−2) time algorithm given by Bläser, Ko-
+marath, and Sreenivasaiah [2] when the host graph is sparse. These algorithms are obtained
+by analyzing the matched treewidth and the automorphism structure of a set of graphs de-
+rived from the pattern.
+From a technical standpoint, we see our algorithms as a natural combination of pattern
+detection and counting algorithms that work well on sparse host graph such as the �O(m) al-
+gorithm for counting k-walks, the �O(mk/2) algorithm for counting k-cliques, and �O(m(k−1)/2)
+time algorithm for detecting induced Kk−e by Vassilevska [19] and insights that improve the
+running-time on dense graphs by exploiting structural parameters treedepth and treewidth.
+We do not make use of fast matrix multiplication in any of our algorithms. Such algorithms,
+called combinatorial algorithms, are also of general interest to the community.
+1.3
+Related work
+Algorithms for counting induced subgraphs are related to the problems that we consider but
+we do not consider any algorithms for it. This problem seems to be much harder. It is
+conjectured by Floderus, Kowaluk, Lingas, Lundell [9] that counting induced subgraphs for
+any k-vertex pattern graph is as hard as counting k-cliques for sufficiently large k. Several
+works have considered the parameterized complexity of counting subgraphs (See [5, 4, 18,
+10, 7]) where the primary goal is to obtain a dichotomy of easy vs hard based on structural
+graph parameters. Some works have also considered restrictions on host graphs such as d-
+degeneracy [3]. The papers on parameterized complexity primarily chooses to focus on the
+growth-rate of the exponent for a family of patterns such as k-paths, k-cycles, or k-cliques
+and not the exact exponent for small graphs as we do in this paper.
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+9
+2
+Preliminaries
+We consider simple graphs. We refer the reader to Douglas West’s textbook [20] for basic
+definitions in graph theory. We use the following common notations for some well-known
+graphs: Pk for k-vertex paths, Ck for k-cycles, Kk for k-cliques, Kk − e for k-clique with one
+edge missing, Km,n for complete bipartite graphs. A k-star is a (k + 1)-vertex graph with
+a vertex u adjacent to vertices v1, . . . , vk and no other edges. A star graph is a k-star for
+some k. For a graph G and S ⊆ V(G), we denote by G[S] the subgraph of G induced by the
+vertices in S.
+▶ Definition 16. Given two graphs G and H, a graph homomorphism from G to H is a map
+φ : V(G) → V(H) such that if uv ∈ E(G), then φ(u)φ(v) ∈ E(H).
+We denote by Hom(G, H), the set of all homomorphisms from G to H.
+▶ Definition 17. Given two graphs G and H, a homomorphism polynomial is an associated
+polynomial HomG[H] such that there is a one-to-one correspondence between its monomials
+and the homomorphisms from G to H. We define:
+HomG[H] =
+�
+φ∈Hom(G,H)
+�
+u∈V(G)
+yφ(u)
+�
+{u,v}∈E(G)
+x{φ(u),φ(v)}
+Note that H has a subgraph isomorphic to G if and only if PG has a multilinear monomial.
+We say that a graph G′ is (G1, . . . , Gm)-free if no induced subgraph of G′ is isomorphic to
+Gi for any i. We denote the complement of a graph G by G. We have V(G) = V(G) and the
+edges of G are exactly the non-edges of G and vice versa.
+We assume that all pattern graphs are connected. Since our primary algorithms are all based
+on counting homomorphisms, this does not lose generality as the number of homomorph-
+isms from a disconnected pattern is just the product of the number of homomorphisms from
+its components.
+▶ Definition 18. An elimination tree (T, r) of a connected graph G is a tree rooted at r ∈ V(G),
+where the sub-trees of r are recursively elimination trees of the connected components of the
+graph G \ r. The elimination tree of an empty (no vertices or edges) graph is the empty tree.
+The depth of an elimination tree (T, r) is defined as the maximum number of vertices over all
+root-to-leaf paths in T. The treedepth of a graph G, denoted td(G), is the minimum depth
+among all possible elimination trees of G.
+Intuitively, treedepth measures the closeness of a given graph to star graphs which are
+exactly the connected graphs having treedepth 2.
+We introduce a related notion called
+matched treedepth that seems to be helpful when designing algorithms for finding or count-
+ing patterns in sparse host graphs.
+▶ Definition 19. A matched elimination tree for a graph G is an elimination tree such that
+the following conditions are true for any root-to-leaf path (v1, . . . , vk):
+If k is even, then v1v2, v3v4, . . . , vk−1vk is a matching in G.
+If k is odd, then there is some i such that E′ = {v1v2, . . . , vi−1vi, vivi+1, . . . , vk−1vk} and
+E′ ⊆ E(G). We have that E′ \ {vi−1vi, vivi+1} is a matching on (k − 3)/2 vertices.
+The matched treedepth of a graph G, denoted mtd(G), is the minimum depth among all pos-
+sible matched elimination trees of G.
+
+10
+Finding and Counting Patterns in Sparse Graphs
+The matched treedepth is always finite (See Proposition 25).
+▶ Definition 20. A tree decomposition of a graph G is a pair (T, B(t)t∈T) where T is a tree
+and B(t), called a bag, is a collection of subset of vertices of G corresponding to every node
+t ∈ T.
+(Connectivity Property) For all v ∈ V(G), there is a node t ∈ V(T) such that v ∈ B(t) and
+all such nodes t that contain v form a connected component in T.
+(Edge Property) For all e ∈ E(G), there is a node t ∈ V(T) such that e ⊆ B(t).
+The width of a tree decompostion (T, B) is defined as the maximum bag size minus one, that is,
+maxt∈T |B(t) − 1|. The treewidth of a graph G, tw(G), is the minimum possible width among
+all possible tree decompositions of G.
+Intuitively, treewidth measures the closeness of the given graph to trees which are exactly
+the graphs with treewidth 1.
+We introduce a related notion, called matched treewidth,
+closely related to treewidth, that seems to be useful for designing dynamic programming
+algorithms over sparse host graphs.
+▶ Definition 21. A matched tree decomposition for a graph G is a tree decomposition where
+for every bag in the tree decomposition, the subgraph of G induced by the vertices in that bag
+has either a perfect matching or a matching where exactly one vertex v in the bag is unmatched
+and v is adjacent to some vertex in the matching. We call such bags matched. The matched
+treewidth of a graph G, mtw(G), is the minimum possible width among all possible matched
+tree decompositions of G.
+Matched treewidth is finite for all graphs (See Proposition 31). This is not trivial unlike
+treewidth because a single bag tree decomposition that contains all the vertices in the graph
+need not be matched.
+We call a tree decomposition reduced if no bag B is a subset of another bag. Given any tree
+decomposition T, we can obtain a reduced tree decomposition T ′ such that the width of T ′
+is at most the width of T. Moreover, all bags in T ′ are also bags in T. This implies that if T
+is matched, then T ′ is matched as well.
+There are several equivalent characterizations for treewidth. Below, we state the ones that
+we use in this paper.
+▶ Definition 22. A k-tree is a graph formed by starting with a (k + 1)-clique and repeatedly
+adding a vertex connected to exactly k vertices of the existing (k+1)-clique. A partial of a graph
+G is a graph obtained by deleting edges from G. The set of all graphs with treewidth at most k
+is exactly the class of partial k-trees.
+We can construct a standard tree decomposition T for any k-tree as follows: The root bag of
+T contains the vertices in the initial (k + 1)-clique. Let S be a k-sized subset of this clique
+such that a new vertex v is added to the k-tree by connecting it to all vertices in S. Then,
+we add a sub-tree to T that will be a standard tree decomposition of the k-tree constructed
+using S ∪ {v} as the starting (k + 1)-clique.
+A chordal completion of a graph G is a super-graph G′ of G such that G′ has no induced
+cycles of length more than 3. A chordal completion that minimizes the size of the largest
+clique is called minimum chordal completion. The treewidth of a graph G is the size of the
+largest clique in its minimum chordal completion.
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+11
+Two paths P1 and P2 from u to v are internally disjoint if and only if P1 and P2 do not have
+any common internal vertex.
+A graph G is 2-connected or biconnected, if for any x ∈ V(G), G−x is connected. Equivalently,
+for any two vertices in G, there are at least 2 internally disjoint paths in G.
+▶ Definition 23. A series-parallel graph is a triple (G, s, t) where s and t are vertices in G.
+This class is recursively defined as follows:
+An edge {s, t} is a series-parallel graph.
+(series composition) If (G1, s1, t1) and (G2, s2, t2) are series-parallel graphs, then the graph
+obtained by identifiying s2 with t1 is also series-parallel.
+(parallel composition) If (G1, s1, t1) and (G2, s2, t2) are series-parallel graphs, then the
+graph obtained by identifiying s1 with s2 and t1 with t2 is also series-parallel.
+A graph has treewidth at most 2 is if and only if all of its biconnected components are series-
+parallel graphs.
+▶ Definition 24. A graph G is said to be a minor of a graph G′ if G can be obtained from
+G′ either by deleting edges/vertices, or by contracting the edges. (The operation of contraction
+merges two adjacent vertices u and v in the graph and removes the edge (u, v).) If G is obtained
+from an induced subgraph of G′ by contracting the edges, then it is said to be an induced minor
+of G′.
+A graph G is called G′-induced-minor-free (G′-minor-free) if G′ is not an induced minor
+(resp. minor) of G.
+An edge subdivision is an operation which deletes the edge (u, v) and adds a new vertex
+w and the edges (u, w) and (w, v). A graph G′ obtained from G by a sequence of edge
+subdivisions is said to be a subdivision of G.
+2.1
+Representation of graphs
+We assume the following time complexities for basic graph operations. Any representation
+that satisfies these is sufficient.
+Given u and v, it can be checked in �O(1)-time whether uv is an edge.
+Iterating over all the edges xy ∈ E(H) ordered by x can be done in �O(m)-time, where m
+is the number of edges in H.
+These requirements are satisfied by the following adjacency-list representation. To represent
+a graph H, we store a red-black tree T that contains non-isolated vertices of H where vertices
+are ordered according to their labels. Consider a node in this tree that corresponds to a
+vertex u. This node stores another red-black tree Tu that stores all neighbors of u in H. Now,
+to check whether uv is an edge or not, we perform a lookup for u in T followed by a lookup
+for v in Tu if u was found. We can iterate over all edges xy ordered by x by performing an
+inorder traversal of T where for each node u, we perform an inorder traversal of Tu.
+If the pattern does not contain any isolated vertices, then we can ignore isolated vertices in
+the host graph as well. If the pattern is G = G′ + v, where v is an isolated vertex and G′ is
+any graph, then the number of homomorphisms from G to H is obtained by multiplying the
+
+12
+Finding and Counting Patterns in Sparse Graphs
+number of homomorphisms from G′ to H by n, where n is the number of vertices in H. This
+can be calculated by simply storing the number of vertices of H in the data structure.
+3
+Matched treedepth
+In this section, we introduce algorithms that count homomorphisms and subgraphs effi-
+ciently in constant space on sparse host graphs. The central theorem in this section is given
+below.
+▶ Theorem 1. Let G be a graph with mtd(G) = d, then given an m-edge graph H as input, we
+can count the number of homomorphisms from G to H in �O(m⌈d/2⌉)-time and constant space.
+Proof. The algorithm is given in Algorithm 1. We can compute the result needed by calling
+COUNT-HOM-MTD(G, E, r, H, φ), where E is an elimination tree for G of depth d, r is the
+root vertex in E, and φ is the empty homomorphism. For simplicity of presentation, we
+assume that each root-to-leaf path in E has an even number of vertices. Odd number of
+vertices in a root-to-leaf path is handled similarly.
+We assume that the host graph H is represented using a symmetric adjacency list represent-
+ation. This is mainly to ensure that we can iterate over all edges xy in H ordered by x in
+Line 3.
+First, we prove that the algorithm is correct. We claim that the call COUNT-HOM-MTD
+(G, E, v, H, σ) where the parameters are as specified in the algorithm returns the number of
+homomorphisms from Gv to H that extends σ. This is proved by an induction on the height
+of v in E. Since v is a top node, the base case is when the height is 2. In this case, Gv is a
+star graph and it is easy to see that the algorithm works. We now prove the inductive case.
+The variable t computes the final answer. Denote by su,x for vertex x in H the number of
+homomorphisms from Gu to H that extends τ = σ∪{v �→ x}. Notice that since uv ∈ E(G), to
+extend τ, the vertex u must be mapped to some y such that xy ∈ E(H). Therefore, iterating
+over all such y is sufficient. Notice that we can compute t as �
+τ
+�
+u su,x. However, this
+would need storing |V(G)||V(H)| variables. By iterating over the edges of H ordered by x, we
+can afford to reuse a single su for different x instead of keeping a separate su,x for each x.
+Now, we show that su correctly computes su,x once the main loop finishes with an x. By the
+inductive hypothesis, the variable cw is the number of homomorphisms from Gw to H that
+extends σ′ = σ ∪ {v �→ x, u �→ y}. Therefore, we have su,x = �
+y
+�
+w cw. Line 12 correctly
+computes this into su. Line 15 correctly updates t once an x is finished. Finally, we reset su
+to 0 before processing the next x.
+Now, we prove the running-time and space usage of the algorithm. Notice that the depth of
+the recursion is bounded by the depth of the elimination tree E and each level of recursion
+stores only constantly many variables. Therefore, the space usage is constant. The main
+loop in Line 3 runs for 2m iterations. The inner-loops only have a constant number of
+iterations. Therefore, the recursive calls are made only O(m) times. We process two levels
+of the elimination tree in a level. Therefore, the total running-time is given by t(d) ⩽
+O(m)t(d − 2) + �O(1) = �O(md/2).
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+13
+Algorithm 1 COUNT-HOM-MTD(G, E, v, H, σ)
+Require: G - The pattern graph
+Require: E - Matched elimination tree for G
+Require: v - A top vertex in E
+Require: H - The host graph
+Require: σ - A partial homomorphism from the ancestors of v to H
+t ← 0
+su ← 0 for all children u of v
+for all edges xy ∈ E(H) ordered by x do
+for all children u of v in E do
+σ′ ← σ ∪ {v �→ x, u �→ y}
+if σ′ is an invalid homomorphism then
+continue
+end if
+for all children w of u do
+cw ← COUNT-HOM-MTD(G, E, w, H, σ′)
+end for
+su ← su + �
+w cw
+end for
+if xy is the last edge on x then
+t ← t + �
+u su
+su ← 0 for all u
+end if
+end for
+return t
+◀
+A fundamental question regarding matched treedepth is whether it is finite for all graphs. It
+is, as the following proposition shows.
+▶ Proposition 25. For any graph G, we have mtd(G) is at most 1 + the number of vertices in
+the smallest maximal matching in G.
+Proof. We partition V(G) into a maximal matching M = (v1w1, . . . , vmwm) and an inde-
+pendent set {u1, . . . , uk}. A matched elimination tree T for G can be constructed as follows:
+Put the vertices v1, w1, . . . , vm, wm on a root-to-leaf path in that order. Let us call this path
+the spine. For each vertex ui, make the lowest vertex in the spine adjacent to ui in G its
+parent in T. It is easy to see that T is a matched elimination tree and its depth is at most 1 +
+the number of vertices in M.
+◀
+Although, Proposition 25 proves an upper-bound for matched treedepth. It is not very useful
+from an algorithmic perspective as it is easy to see that there is an O(mk+1)-time, constant
+space algorithm for counting patterns with maximal matchings of k edges. Any smallest
+maximal matching in C11, for example, has 4 edges. However, the proof of Theorem 2 given
+below shows that we can do better. The algorithm is based on the well-known technique of
+expressing subgraph count as a linear combination of homomorphism counts.
+
+14
+Finding and Counting Patterns in Sparse Graphs
+▶ Definition 26. Let I be the set of all the independent sets in the graph G. For some I ∈
+I, Merge(G, I) is the graph obtained by merging the vertices of I. Then
+Spasm(G) = {G} ∪
+�
+I∈I
+Spasm(Merge(G, I))
+For any pattern graph G, it turns out that the number of subgraph isomorphisms from G to
+a host graph H are just the linear combination of all possible graph homomorphisms from
+SpasmG to H. That is, there exists constants αG′ ∈ Q such that:
+SubG[H] =
+�
+G′
+αG′HomG′[H]
+(1)
+where G′ ranges over all graphs in Spasm(G). This equation is used to count subgraphs by
+many authors (See for example [1, 4]).
+▶ Theorem 2. Given an m-edge graph H as input, we can count the number of Ck, where
+k ⩽ 11, as subgraphs in �O(m3)-time and constant space.
+Proof. We analyzed all graphs in Spasm(C11) and Spasm(C10) and concluded that each of
+them has matched treedepth at most 6. A pdf that contains all these graphs and their corres-
+ponding matched elimination trees can be found at (https://github.com/anonymous1203/Spasm).
+For seeing that the stated algorithms exist for smaller cycles, observe that Spasm(Ck) ⊂
+Spasm(Ck+2) for k ⩾ 3.
+◀
+The class of treedepth 2 graphs is exactly the star graphs. These graphs have an O(n2)-
+time, constant space, treedepth-based algorithm. This class of graphs is also the graphs of
+matched treedepth 2. It is easy to see that the number of homomorphisms from star graphs
+can also be counted in O(m)-time and constant space using the following observation. Let
+d1, . . . , dn be the degrees of vertices in the graph. Then, the count is given by the expression
+�
+i dk
+i . Note that this is an asymptotic improvement for sparse graphs. We now show that
+this asymptotic improvements exists for all patterns. First, we need a restricted form of
+elimination trees.
+▶ Definition 27. An elimination tree T is connected if for every node u in T and a child v of
+u in T, u is adjacent to some node in Tv
+Now, we show that connected elimination trees are optimal.
+▶ Lemma 28. Every connected graph G has a connected elimination tree of depth td(G).
+Proof. We show how to construct a connected elimination tree T ′ from an elimination tree
+T without increasing its depth. Let T ′ = T initially. Suppose there exists some node u in T ′
+that violates the property (If not, we are done). Then, there exists a child v of u in T ′ such
+that there is no edge in G between u and any node in T ′
+v. Let w be a node in T ′
+v such that
+w is adjacent to some proper ancestor x of u. Such a w must exist because G is connected.
+Now, in T ′, remove the subtree T ′
+v and make it a subtree of the node x. Repeat this process
+until all nodes in T ′ satisfy the required property. This process must terminate since at each
+step, we reduce the number of nodes violating the property by at least one. This process
+cannot increase the depth of T ′ because the only modification is to move a subtree upwards
+to be a subtree of a proper ancestor of its parent.
+◀
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+15
+▶ Theorem 3. For any graph G, mtd(G) ⩽ 2 · td(G) − 2.
+Proof. We start with a connected elimination tree T with depth d of G and show how to
+construct a matched elimination tree of G from T. We use induction on d. For d = 2, the
+tree is already matched and has depth 2 = 2 · 2 − 2.
+Our construction is iterative and top-down. Each iteration ensures that the current top-most
+node in the elimination tree is adjacent, in the graph G, to all its children and that the
+elimination tree is connected.
+Each iteration consists of two phases iteratively executed until the desired property is satis-
+fied. The first phase ensures that the root node r is adjacent in G to all its children in T. If r
+has a child v that is not adjacent in G to r, then since T is connected, there is some node w
+in Tv such that rw ∈ E(G). Then, we make w a child of r in T, delete w from Tv, and make
+all children of w children of parent of w. The resulting tree is an elimination tree of depth
+at most d + 1. After this phase, the root node is adjacent in G to all its children, the tree’s
+depth has increased by at most one. However, it may not be connected.
+In the second phase, we use the construction of Lemma 28 to make the tree connected
+without increasing the depth. Observe that the construction will keep the existing children
+of root as is and may add new chlidren to r that are not adjacent to r in G. Suppose a new
+node u was added as a new child to r in this second phase. The height of subtree rooted at u
+is at most d−1. Therefore, the tree (r, Tu) obtained by attaching u to r has height at most d.
+We can now execute phase 1 on all the trees (r, Tu) for all such u without increasing depth
+beyond d + 1. This process must eventually terminate as we add at least one new child to
+the root every time.
+At the end of the iteration, consider a grandchild x of r. If it is a leaf, since the tree is
+connected, x must be adjacent in G to its parent in T and we are done. Otherwise, the
+subtree Tx is a tree of depth at least 2 and at most d − 1 that is connected. So by the
+induction hypothesis, we obtain that Tx is a matched elimination tree of depth at most
+2(d − 1) − 2. This means that the original tree is converted to a matched elimination tree of
+depth at most 2 + 2(d − 1) − 2 = 2d − 2 as required.
+◀
+▶ Corollary 29. Suppose G has treedepth d. Given an m-edge host graph as input, we can
+count the number of homomorphisms from G to the host graph in �O(md−1)-time and constant
+space.
+Notice that that above algorithm is asymptotically better than the treedepth based algorithm
+for all patterns on sparse host graphs.
+We now prove that the matched treedepth of an induced subgraph cannot be much larger
+compared to that of the graph containing it. In other words, we can obtain lower bounds
+on the matched treedepth of a graph by obtaining lower bounds on the matched treedepth
+of any of its induced subgraphs. We prove the theorem for connected subgraphs. But note
+that the matched treedepth of a disconnected graph is the maximum of its connected com-
+ponents.
+▶ Theorem 4. Let G be a graph and G′ is a connected, induced subgraph of G, then:
+1. mtd(G′) ⩽ mtd(G) if mtd(G) is even.
+2. mtd(G′) ⩽ mtd(G) + 1 if mtd(G) is odd.
+
+16
+Finding and Counting Patterns in Sparse Graphs
+Proof. We start with a matched elimination tree T of even (The odd case is similar) depth
+d for G and construct a matched elimination tree for G′. First, we delete all nodes in the
+elimination tree that are in G but not in G′. If a node u in T has parent v that was deleted,
+then we make u a child of the closest ancestor of v that is still in T. The final forest thus
+obtained is a tree T ′ because G′ is connected. We assume without loss of generality that T ′
+is a connected elimination tree.
+Suppose r′ is the root of T ′. We will now modify T ′ into a matched elimination tree. We will
+first analyze paths in T ′ that correspond to even length root-to-leaf paths in T. If T ′ is not
+matched on this path, then there exists a node u closest to r′ such that u is not connected in
+G′ to a child v in T ′. This is only possible if u was either matched to a child w of u in T or
+its parent w in T and w is not in G′. Therefore, we have distT ′(r′, u) + depth(T ′
+v) < depth(T)
+and this means we can afford to increase the length of any path that passes through edge uv
+by 1. Since T ′ is connected, we can now apply the transformation in the proof of Theorem 3
+to match u with one of its descendants in T ′
+v. The depth is still at most d because this
+transformation increases the depth by at most 1 In effect, the increase in depth in this branch
+of the tree by pulling up a descendant is compensated by the fact that the unmatched vertex
+was introduced by deleting a vertex in this branch.
+We can iteratively apply the above construction to make T ′ a matched elimination tree while
+keeping T ′ connected. However, applying the above transformation may introduce a node u
+in T ′ that is not matched to a child v because v’s parent w was pulled up in the tree to match
+with some other vertex. Such u also satisfy the inequality distT ′(r′, u)+depth(T ′
+v) < depth(T).
+Why? Any path in the tree T ′ that passed through both u and v earlier had to pass through
+w. But, the fact that w was pulled up implies that these paths had length strictly less than
+depth(T) by the argument in the previous paragraph. And shifting w to a position earlier in
+the path cannot increase its length.
+For root-to-leaf paths in T of odd length, there might be a matched P3 on vertices uvw such
+that v is not in G′. In this case too, by the same argument, the transformations increase the
+depth to at most d+1 (In this case, we may have to pull up two distinct vertices for matching
+u and w.). When d is even, increasing the length of such paths by 1 does not increase the
+depth of the tree. When d is odd, increasing the length of such paths by 1 increases the
+depth by atmost 1.
+◀
+C4
+P6
+T3,3
+Figure 2 Forbidden graphs for mtd(.) ⩽ 3
+Theorem 3 implies that all graphs G with td(G) = 3 also have mtd(G) ⩽ 4. It is easy to see
+that C4, P6, and T3,3 given in Figure 2 have td(.) = 3 and mtd(.) = 4. By Theorem 4, it is
+also possible that some super graph of these graphs have mtd(.) = 3. However, in the below
+theorem we show that this cannot happen and that the graphs G with td(G) = mtd(G) = 3
+are exactly the graphs where we forbid these graphs as induced subgraphs.
+▶ Theorem 5. Let G be a graph such that td(G) = 3. Then mtd(G) = 3 if and only if G is
+(C4, P6, T3,3)-free.
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+17
+Proof. (Proof for the “if” direction) Let G be a connected graph with td(G) = 3 such that
+G is (C4, P6, T3,3)-free. We will construct a matched elimination tree for G of depth 3 from
+its elimination tree.
+Consider an elimination tree T of G with depth 3, rooted at some vertex r. Let {v1, v2, . . . vk}
+be the children of r in T. Notice that for any child c of vi (for any i), if c is not adjacent to
+vi in G then it must be adjacent to r in G (else, G is not connected). We make all such c,
+which are not adjacent to vi in G, a child of r instead of vi, in the elimination tree T. Note
+that this neither violates any property of elimination tree nor does it increase the depth of
+elimination tree. Also, after doing this modification, we can further assume that all the
+children of vi in T are adjacent to vi in G.
+Now, if rvi ∈ E(G) for all i ∈ [k], then T itself is a matched elimination tree and we are
+done. Suppose there exists an i such that rvi /∈ E(G). Since G is connected, a path must
+exists from r to vi. Since T has depth 3, this path must be a P3 and therefore r and vi has
+at least one common neighbor. Any common neighbour u of r and vi must be a child of vi
+in T. Moreover, if r and vi have two common neighbors, say u1 and u2, then ru1vu2r is an
+induced C4. So r and vi must have exactly one common neighbor. We now consider the
+following cases:
+r has exactly one child v1.
+Let c1, c2, . . . , ck be the children of v1. Recall that rv1 /∈ E(G), and they have exactly
+one common neighbor in G, say c1. Then, we construct a matched elimination tree T ′ as
+follows: The root of T ′ is c1, vertices r and v1 become the children of c1, the children of
+v1 in T ′ are {c2, . . . , ck} The tree T ′ is a valid matched elimination tree of depth 3 for G.
+r has multiple children, say {v1, . . . vℓ} for some ℓ > 1.
+We split this into two cases.
+(r is not adjacent to exactly one of its children, say v1)
+The vertex v1 is not a leaf in T since G is connected. If v1 has exactly one child,
+then, since it must be a common neighbor c of r and v1, we swap c and v1, and this
+does not increase the depth of T. Else, suppose v1 has at least two children, namely
+{c1, c2, . . . ck} (where the common neighbor with r is c1). We now argue that all the
+other children {v2, . . . vl} of r must be leaves in T. If not, then say v2 has a child c3,
+then we encounter an induced P6, namely (c2, v1, c1, r, v2, c3) (if rc3 ̸∈ E(G)) or an
+induced T3,3 (if rc3 ∈ E(G)), which is a contradiction. So, {v2, . . . vl} are leaf vertices.
+Now, we can convert T to T ′ by rooting it at c1 instead of r. The children of c1 in T ′
+are precisely r and v1, the children of r are precisely all the leaves {v2, . . . vl}, and the
+children of v1 are all the same except c1, that is, {c2, . . . ck}. It is easy to see that T ′ is
+a matched elimination tree of depth 3.
+(r is not adjacent to at least two of its children, say v1 and v2 and maybe more)
+Due to the connectivity of G, the vertices v1, v2 cannot be leaves in T. If v1 has exactly
+one child c, then c must be a common neighbor of r and v1. We swap v1 and c in
+T. This will either fall into the previous case or we can assume v1 has more than one
+child. Let c1, c2 be two children of v1, where c1 is the common neighbor of r and v1.
+Then we get an induced P6, namely, (c2, v1, c1, r, c, v2), where c is a common neighbor
+of r and v2. (Such a c must exist as (r, v2 /∈ E(G)).
+
+18
+Finding and Counting Patterns in Sparse Graphs
+(Proof for the “only if” direction) Suppose G is a connected graph with td(G) = 3 such that
+mtd(G) = 3. We will show that G cannot contain C4, P6, or T3,3 as induced subgraphs. We
+will prove that any connected induced subgraph G′ of G has matched treedepth at most 3.
+Let T be the matched elimination tree of G of depth 3. Let r be the root of T. If r is not in
+G′, then since G′ is connected, all vertices in G′ must come from a single sub-tree of r in T.
+In that case, that sub-tree witnesses a matched elimination tree of depth at most 2 for G′. If
+r is in G′, then G′ may be obtained by deleting some level 1 and level 2 vertices in T. We
+will construct a matched elimination tree for G′ from T. If a level 2 vertex is not in G′, we
+simply delete it from T as well. If a level 1 vertex v is not in G′, then for each u ∈ V(G′) that
+is also a child of v in T, there must be an edge ru in G′ since G′ is connected. Therefore, we
+make u a child of r in the matched elimination tree for G′.
+◀
+By Theorem 1 and Theorem 3, we can conclude that every pattern G with td(G) = 3 has
+an �O(m2) algorithm for counting homomorphisms from G to m-edge sparse host graphs.
+Therefore, the presence of C4, P6, or T3,3 as induced subgraphs in G does not affect the
+running-time of Algorithm 1. If we consider patterns G with td(G) = 4, then we have
+examples such as P8 where td(P8) = 4 and mtd(P8) = 5. Therefore, we can obtain only an
+�O(m3) algorithm for counting homomorphisms from P8. It would be interesting to prove
+theorems similar to Theorem 5 for higher treedepth, say 4. But, we do not even know the
+exact set of forbidden induced subgraphs for treedepth 4 [12] so this seems difficult.
+4
+Matched treewidth
+In this section, we introduce algorithms that count homomorphisms and subgraphs effi-
+ciently on sparse host graphs by using polynomial space. The central theorem in this section
+is given below.
+▶ Theorem 6. Let G be a graph with mtw(G) = t, then given an m-edge host graph H as
+input, we can construct an arithmetic circuit computing the homomorphism polynomial from
+G to H in time �O(m⌈(t+1)/2⌉).
+Proof. Let T be a matched tree decomposition of G. We fix an arbitrary assignment of edges
+and vertices of G to bags of T such that each vertex and edge is assigned to exactly one bag
+that contains it. Let B be a bag in T. We consider the matched matching M in B as a sequence
+of edges (a1b1, . . . , akbk) by arbitrarily ordering them. Given edges u1v1, . . . , ukvk in the
+host graph H such that σ(ai) = ui and σ(bi) = vi is a valid partial homomorphism on the
+vertices of B, we define the following monomials:
+EdgeMon(B, u1v1, . . . , ukvk) =
+�
+e
+xσ(e)
+VertexMon(B, u1v1, . . . , ukvk) =
+�
+v
+yσ(v)
+where e and v range over all edges and vertices assigned to B. We also define Mon(B, u1v1, . . . ,
+ukvk) as the product of EdgeMon(B, u1v1, . . . , ukvk) and VertexMon(B, u1v1, . . . , ukvk).
+Let B and B′ be bags in T such that B′ is the parent of B. We arbitrarily order the vertices
+to obtain a sequence X(B ∩ B′) of the vertices in B ∩ B′ (We may assume that the vertices
+are labeled from [|V(G)|] and choose increasing order). We define MapGate(B, B′, u1, . . . ,
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+19
+uk) where X(B ∩ B′) = (a1, . . . , ak) as the named gate which corresponds to the partial
+homomorphism σ(ai) = ui, 1 ⩽ i ⩽ k.
+Algorithm 2 constructs the required circuit. Notice that all operations within the loops in
+Line 9, 20, 36 runs in O(log n) time. The loops itself executes for O(m⌈(t+1)/2⌉) iterations
+since any matched matching on t + 1 vertices contains at most ⌈(t + 1)/2⌉ edges. The loops
+in Line 2, 7, and 18, executes for O(1) iterations since the pattern graph G is fixed. Notice
+that we iterate over pairs uv that correspond to edges instead of edges {u, v} because the
+order determines the homomorphism.
+When a hash table lookup for a MapGate(.) gate fails. We add it to that table with an initial
+value if the gate occurred on the left-hand side and replace it with 0 if it occurs on the right-
+hand side. We now prove the correctness of the circuit using parse trees. Notice that the only
++ gates in the circuit are MapGate(. . . ). For this proof, we can think of r as MapGate(R, φ)
+where R is the root bag in T with an empty bag as parent. Therefore, the monomials of
+the final polynomial correspond to the choices at these gates while building the parse tree.
+For some bag B with parent B′ in T and vertices x1, . . . , xk in G, the inputs to the gate
+MapGate(B, B′, x1, . . . , xk) correspond to valid partial homomorphisms on the vertices of B
+that map ai to xi for all ai ∈ X(B ∩ B′). We identify these gates with the bag B. Now,
+given a homomorphism σ from G to H, we build its parse tree by choosing for each such
+gate, the restriction of σ to the vertices in that bag. We have to prove that this choice will
+be consistent with the images x1, . . . , xk at each gate. Indeed, this is vacuously true at the
+root gate (the list is empty). For an arbitrary a ∈ V(G), consider the topmost bag B where
+a first appears. In the partial homomorphism chosen at B, we can freely fix the image of the
+vertex a to that of in σ. By construction, this choice is then propagated when moving to its
+child gates corresponding to bags where a is present (See Lines 15, 30, and Lines 42 ). Also,
+if a child bag of B does not contain the vertex a, then the vertex a will not appear in that
+subtree too. It is easy to see that this parse tree computes the correct monomial. Since the
+circuit is monotone, this proves that all monomials that correspond to homomorphisms are
+present in the polynomial. For the other direction, we have to argue that only monomials
+that correspond to homomorphisms are present in the polynomial. Indeed, any parse tree
+corresponds to a sequence of choices of partial homomorphisms at each gate. We argue
+that these partial homomorphisms must be consistent with each other and therefore can
+be combined into a valid homomorphism. This is because T is a tree decomposition and
+therefore any a ∈ V(G) must occur in a (connected) subtree of T. The construction of the
+circuit ensures that once the image of vertex a is determined in a parse tree, it is correctly
+propagated to all partial homomorphisms where a is a member of the domain. Furthermore,
+since each vertex and edge in G are assigned to a unique bag, their images appear exactly
+once in the monomial. This completes the proof.
+◀
+▶ Remark 30. An anonymous reviewer on an earlier draft of this paper commented that
+the graph parameter generalized hypertree width, denoted ghw, may yield similar running-
+times. Indeed, we have verified that �O(mghw)-time algorithms exist and that ghw ⩽ ⌈(mtw +
+1)/2⌉ and the running-times coincide in the worst-case. To the best of our knowledge, the
+parameter ghw has not been analyzed in the context of fast algorithms for small patterns. It
+is primarily used for designing efficient algorithms on hypergraphs with high treewidth and
+low ghw. We believe analyzing mtw is also useful since it is sometimes more fine-grained
+than ghw. i.e., the class ghw = k may contain graphs from both classes mtw = 2k − 2 and
+
+20
+Finding and Counting Patterns in Sparse Graphs
+Algorithm 2 Computing HomG[H]
+1: Let T be a near-perfect tree decomposition of G.
+2: for each bag B in T do
+3:
+for each child B′ of B in T do
+4:
+Initialize empty hash table T(B,B′).
+5:
+end for
+6: end for
+7: for each non-root leaf bag B in T do
+8:
+Let B′ be the parent of B in T.
+9:
+for (u1v1, . . . , ukvk) ∈ E(H)k do
+10:
+Let σ be σ(ai) = ui, σ(bi) = vi.
+11:
+if σ is not a valid partial homomorphism from G to H then
+12:
+Skip this iteration
+13:
+end if
+14:
+Let x1, . . . , xk′ be the images of vertices in X(B ∩ B′) in σ.
+15:
+MapGate(B, B′, x1, . . . , xk′) += Mon(B, u1v1, . . . , ukvk)
+16:
+end for
+17: end for
+18: for each non-root bag B in T in a bottom-up order do
+19:
+Let M = (a1b1, . . . , akbk).
+20:
+for (u1v1, . . . , ukvk) ∈ E(H)k do
+21:
+Let σ be σ(ai) = ui, σ(bi) = vi.
+22:
+if σ is not a valid partial homomorphism from G to H then
+23:
+Skip this iteration
+24:
+end if
+25:
+Let B′ be the parent of B in T.
+26:
+Let B1, . . . , Bs be the children of B in T.
+27:
+Let x1, . . . , xk′ be the images of vertices in X(B ∩ B′) in σ.
+28:
+Let wi1, . . . , wiki be the images of vertices in X(B ∩ Bi) in σ.
+29:
+MapGate(B, B′, x1, . . . , xk′) +=
+30:
+Mon(B, u1v1, . . . , ukvk) �s
+i=1 MapGate(B, Bi, wi1, . . . , wiki)
+31:
+end for
+32: end for
+33: Let B be the root in T
+34: Let B1, . . . , Bs be the children of B in T.
+35: r ← 0
+36: for (u1v1, . . . , ukvk) ∈ E(H)k do
+37:
+Let σ be σ(ai) = ui, σ(bi) = vi.
+38:
+if σ is not a valid partial homomorphism from G to H then
+39:
+Skip this iteration
+40:
+end if
+41:
+Let wi1, . . . , wiki be the images of vertices in X(B ∩ Bi) in σ.
+42:
+r += Mon(B, u1v1, . . . , ukvk) �s
+i=1 MapGate(Bi, B, wi1, . . . , wiki)
+43: end for
+44: return r
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+21
+mtw = 2k − 1.
+First, we make some fundamental observations about mtw.
+We can relate it to matched treedepth as we relate treewidth to treedepth.
+▶ Proposition 31. For any graph G, we have mtw(G) ⩽ mtd(G) + 1.
+Proof. Let E be the optimal matched elimination tree for G. We construct a matched tree
+decomposition T for G from E as follows: For each path from root to leaf in E from the
+leftmost path to the rightmost path, construct bags that contain all vertices in those paths.
+Then, join those bags into a path by adding an edge between bags B (corresponds to path
+to leaf u) and B′ (corresponds to the path to leaf v) if and only if v is the next leaf in E from
+u when leaves are ordered from left to right.
+◀
+Now, we prove that matched treewidth cannot be much higher than treewidth.
+▶ Theorem 7. For any graph G, we have mtw(G) ⩽ 2 · tw(G) + 1.
+Proof. Let T be a tree decomposition of G of width k. We will describe a procedure to
+convert T to a matched tree decomposition. The construction is top-down. The final tree
+will have the property the vertices in each non-leaf bag will induce a perfect matching in G.
+Let Br be the root bag in T. The following procedure will be applied to Br.
+1. Find maximal matching M in Br.
+2. For each v ∈ Br \ M such that NG(v) ⊆ Br. Observe that since M is a maximal matching
+NG(v) ⊆ V(M) as well. We delete v from Br and add a leaf bag Bv as a child of Br. The
+bag Bv will contain v and the vertices in M. Since v is adjacent to at least one vertex in
+M, this bag is matched.
+3. For each v ∈ Br \ M such that NG(v) ⊈ Br. We can find a u such that u /∈ Br and
+u ∈ NG(v). We choose such a u that is in a bag B that is at a minimal distance from Br
+in T. We add u to Br and all the bags in the path from B to Br in T. We then modify
+M = M ∪ {uv}.
+Notice that each iteration in step 2 and step 3 reduces the unmatched vertices in Br by 1 and
+only adds leaf bags to T that are matched. In addition, if Br originally had x vertices, after
+this procedure it will have at most 2x vertices as we add at most one vertex corresponding
+to each of the original x vertices. Therefore, the size of modified Br is at most 2k + 2 and
+the graph induced by vertices in Br has a perfect matching.
+Now, consider an arbitrary bag B such that all its ancestors are bags with a perfect matching.
+Let Bp be the parent of B in T and let M be the perfect matching on the vertices in Bp. We
+now apply the following procedure on B.
+1. For each v ∈ B ∩ Bp, let u be the partner of v in M. If u is in B, then we match v to u
+in B as well. If not, then we add u to B and match v to u in B. This does not violate
+any properties of tree decompositions. Notice that if v was added to B in step 3 of the
+procedure for the root bag, then its partner u must be in B as well.
+2. For v ∈ B\Bp, we apply steps 2 and 3 in the procedure described for the root bag. These
+steps only modify the bags in the subtree of T rooted at B and will only add to T leaf
+
+22
+Finding and Counting Patterns in Sparse Graphs
+bags that are matched. Moreover, at the end of this step, the vertices in bag B induce a
+graph that has a perfect matching.
+Observe that the size of a bag B can at most double from its original size since we add at
+most one vertex for each vertex originally in the bag. This is true for all the newly added
+leaf bags as well. Therefore, we have constructed a matched tree decomposition of width at
+most 2k + 1.
+◀
+We now show an explicit family that has close to the worst-case relation between treewidth
+and matched treewidth.
+▶ Proposition 8. mtw(Kn,n) = 2n − 2 for all n > 1.
+Proof. Let T be just an edge with vertex set {Kn,n \ {a}, Kn,n \ {a′}} for some a and a′ that
+are in the same part. One can easily verify that T is matched tree decomposition of Kn,n.
+Thus mtw(Kn,n) ⩽ 2n − 2.
+Let T be an arbitrary matched tree decomposition of Kn,n. Root T at a leaf bag, say X and
+let X1 be the only child of X. If X ⊆ X1, then we can delete X from T. Therefore, there exists
+a vertex u ∈ X of Kn,n such that u ̸∈ X1. So N(u) ⊆ X. Assume wlog that u ∈ A. Then, we
+get B ⊆ X. Now to match the n vertices in B, we need at least n − 1 vertices from A in the
+bag X. So mtw(Kn,n) ⩾ 2n − 2.
+◀
+It is easy to see that tw(Kn,n) = n. Therefore, its matched treewidth is only 3 less than the
+worst case 2n + 1.
+We now use the Algorithm 2 and Definition 26 to count paths in cycles in sparse host graphs.
+Instead of analyzing all graphs in Spasm(P10), we completely characterize the matched
+treewidth of graphs with treewidth 2. This will simplify the case analysis required for prov-
+ing algorithmic upper bounds for many small pattern graphs.
+4.1
+Matched treewidth of partial 2-trees
+In this section, we study the matched treewidth of partial 2-trees. A summary is given in
+Table 2. From Theorem 7, we have mtw(G) ⩽ 5 when tw(G) = 2 which yields the last row.
+The graph Y (See Figure 4) satisfies tw(Y) = 2 and mtw(Y) = 5. But Y is also an induced
+subgraph of Z (See Figure 5) which satisfies tw(Z) = 2 and mtw(Z) = 4. Therefore, a
+forbidden induced minor subgraph characterization is not applicable for this case. We now
+prove the remaining two characterizations.
+mtw ⩽ .
+Forbidden induced minor
+2
+C5
+3
+X
+4
+Not applicable.
+5
+None.
+Table 2 Matched treewidth of partial 2-trees.
+▶ Theorem 10. For any partial 2-tree G, the graph G is C5-induced-minor-free if and only if
+mtw(G) = 2.
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+23
+Proof. (Proof for the “if” direction) Suppose for contradiction that there is a partial 2-tree G
+such that mtw(G) = 2 and G has a C5-induced-minor. Let T be a matched tree decomposition
+of G with width 2. Since C5 is an induced minor, we can obtain C5 from G by deleting
+vertices and contracting edges. For all v that is deleted, delete v from all bags of T. Similarly,
+for all edges uv that are contracted, replace u and v consistently in all the bags of T by one
+of u or v. We obtain a (not necessarily matched) tree decomposition T ′ of C5. Assume wlog
+that T ′ is a reduced tree decomposition. Since T has width at most 2, any bag in T ′ contains
+at most 3 vertices. If a bag in T ′ has 3 vertices, then they already form a P3 since it must
+have been so in T. There cannot be a bag of one vertex in T ′ because it is reduced and
+C5 is connected. We claim that a bag of size 2 in T ′ must contain some u and v such that
+uv ∈ E(C5).
+▷ Claim 32.
+Let u and v be two vertices in C5 that are not adjacent. Then, the tree
+decomposition T ′ cannot contain the bag {u, v}.
+Proof. Suppose for contradiction that T ′ contains the bag B = {1, 3}. The other cases are
+symmetric. Root the tree T ′ at that bag. We now analyze various cases.
+1. (B has one child) Since 1 and 3 are adjacent to other vertices, the child of B must contain
+both 1 and 3, contradicting the fact that T ′ is reduced.
+2. (B has more than one child) We split this case sub-cases.
+a. (The edges 15 and 34 are covered in distinct subtrees (Say T1 and T2) of B) Since
+45 ∈ E(C5), this edge must be covered. The bag that covers 45 must have a path
+containing 5 to the bag covering 15 in T1 due to the connectivity of 5. Similarly, this
+bag must have a path containing 4 to the bag covering 34 in T2 due to the connectivity
+of 4. But, this is impossible since B contains only 1 and 3.
+b. (The edges 15 and 34 are covered in the same subtree of B) This case is split into
+further sub-cases.
+i. (The bag B, and the bags containing 15 and 34 occur on the same path in T ′)
+Suppose the path is from B to the bag containing 3 and 4 via the bag B′ containing
+1 and 5. The other case is similar. Then, the bag B′ must contain 3 to maintain the
+connectivity of 3. Now, we have B′ ⊃ B, a contradiction.
+ii. (The bags containing 15 and 34 has a common ancestor that is a proper descendant
+of B) Let B′ be this common ancestor. This bag B′ must contain 3 as it lies on the
+path from B to the bag containing 3 and 4. Also, this bag B′ must contain 1 as
+this lies on the path from B to the bag containing 1 and 5. But, then B′ ⊇ B, a
+contradiction.
+◀
+This completes the proof of the "if" direction.
+(Proof of the “only if” direction) We use proof by contradiction. Suppose G is a counter-
+example on minimum number of vertices.
+▷ Claim 33.
+The graph G is 2-connected.
+
+24
+Finding and Counting Patterns in Sparse Graphs
+Proof. Suppose G has a cut vertex v. By deleting v, we obtain smaller graphs G1, . . . , Gm
+for some m > 1. Any cycle in G is also a cycle in Gi for some i. Since G is minimal, each
+Gi has a matched tree decomposition of width at most 2, say Ti. For all i, let Bi be a bag in
+Ti that contains v. Add edges between B1 and Bi for all 1 < i ⩽ m. This is a matched tree
+decomposition for G of width 2, a contradiction.
+◀
+We take a 2-tree G′ that is a super-graph of G and has the same vertex set as G.
+▷ Claim 34.
+Let uv be an edge in G′ but not in G. Then u and v have a common neighbor
+in G.
+Proof. Since G is 2-connected, there are two internally disjoint paths P = uu1 · · · ukv and
+P′ = uv1 · · · vℓv in G between u and v. We may assume that P and P′ are induced paths
+in G. If k or ℓ is 1, then u and v has a common neighbor. So we assume k and ℓ are at
+least 2. Now PP′ is a cycle of length at least 6 and it must have a chord (otherwise, this is a
+C5-induced-minor in G.). Therefore, there is some i and j such that ui is adjacent to vj. the
+edges on P, P′, together with this chord is a K4-minor in G′, a contradiction.
+◀
+Let T be a standard tree decomposition of G′. This is a matched tree decomposition for G.
+We show that this is also a matched tree decomposition for G. It is enough to show that a
+set {u, v, w} which forms a triangle in G′ induces either a P3 or a triangle in G. Assuming
+the contradiction, we get that v (say) is not adjacent to both u and w. By Claim 34, we
+obtain vertices x that is a common neighbor of u and v in G, and y that is a common
+neighbor of v and w in G. Since G′ is K4-minor-free, we have x ̸= y, wx ̸∈ E(G), xy /∈ E(G),
+and uy ̸∈ E(G). If uw ∈ E(G), then uwyvxu is an induced C5 in G, a contradiction. So
+uw ̸∈ E(G), and by Claim 34, we obtain a vertex z that is a common neighbor of u and
+w in G. Again, since G′ is K4-minor-free, we have z ̸= x, z ̸= y, zx ̸∈ E(G), zv /∈ E(G),
+and zy ̸∈ E(G). Then uzwyvxu is an induced C6 in G, a contradiction. This completes the
+proof.
+◀
+We now show that the graph X (See Figure 1) is a partial 2-tree that has matched treewidth
+more than 3.
+▶ Lemma 35. mtw(X) ⩾ 4
+Proof. Consider a reduced, matched tree decomposition T of X. Suppose for contradiction
+that X has width strictly less than 4. Consider a leaf bag Bl in T. The bag Bl must contain
+u0 or u2 or u4 as all edges in X are incident on one of these vertices. All cases are symmetric
+so wlog, we can assume Bl is a super-set of {u0, u1}. Consider the tree T to be rooted at Bl.
+As neither u0 nor u1 is pendant and T is reduced, it follows that Bl must contain at least one
+other vertex. Let Bp be the bag adjacent to Bl. We claim that {u0, u2} or {u0, u4} is contained
+in Bp. If u0 /∈ Bp, then {u1, u′
+1, u5, u′
+5} ⊆ Bl and therefore |Bl| ⩾ 5. Suppose u1 ∈ Bp. Since
+T is reduced, this means that some vertex other than u0 and u1 was in Bl but not in Bp. If
+this vertex is u2 or u4, then |Bl| ⩾ 5. If it is one of the other vertices, either u2 or u4 is in
+Bl and Bp. If u1 /∈ Bp, then Bl must contain u2 and therefore so must Bp. Now, we assume
+wlog that both u0 and u2 are in the bag Bp. If Bp also contains u4, then we are done as the
+size of matched bag would be at least 5.
+Let Bc be a descendant bag of Bp in T such that Bc is the root of the subtree of T that
+contains u4. Again, if {u0, u2, u4} ⊆ Bc, then |Bc| ⩾ 5 and we are done. So assume wlog
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+25
+that u2 /∈ Bc. If u0 /∈ Bc, then Bc ⊇ {u5, u′
+5, u3, u′
+3} and we are done. This is because these
+vertices are common neighbors of u4 with u0 or common neighbors of u4 with u2 and Bc is
+the root bag of the sub-tree where u4 appears in T. So u0 is also in Bc. But Bc also contains
+u3 and u′
+3 and Bc must contain at least one more vertex to match u0.
+◀
+We now prove that forbidding X-induced-minor (for partial 2-trees) exactly gives us the class
+of graphs with mtw(.) ⩽ 3.
+▶ Definition 36. A minimum chordal completion ˜G of some graph G is said to be special if no
+independent set of size 3 in G induces a clique in ˜G. We write smcc instead of special minimum
+chordal completion.
+Note that treewidth of a graph is same as the treewidth of its smcc (if it exists).
+▶ Theorem 11. For any partial 2-tree G, the graph G is X-induced-minor-free if and only if
+mtw(G) ⩽ 3.
+Proof. (Proof for the “if” direction) Let G be a graph with a matched tree decomposition U
+of width 3. Suppose for contradication that G has an X-induced-minor. Then, we can also
+obtain X by contracting some edges of an induced subgraph of X′ of G. Let T ′ be the tree
+decomposition of X′ obtained from U by deleting vertices in the set S from all bags. Since X
+can be obtained from X′ by contracting some edges, it will have a set of three vertices, say
+u0, u2, u4, that form an independent set and internally vertex-disjoint paths Qi+1, Q′
+i+1 from
+ui to ui+2 4. We define the map f : V(X′) �→ V(X) is such that f(ui) = ui for i ∈ {0, 2, 4}, f
+maps all internal vertices in the path Qi+1 to ui+1 and Q′
+i+1 to u′
+i+1. The map f corresponds
+to the edge contraction that we use to obtain X from X′. Let T be the tree-decomposition of
+X obtained from T ′ by applying f to all vertices in all bags. We can remove bags of size 0
+and size 1 from T using standard techniques.
+Now we modify T to obtain a matched tree decomposition of width 3 for X thereby deriving
+a contradiction. Every bag B in T has a corresponding bag P in T ′. Note that if uv is an edge
+in X′, then f(u)f(v) is an edge in X, whenever f(u) ̸= f(v). If B has size 4, then since U is
+a matched tree decomposition, the bag P must also be matched. Therefore, if all bags in T ′
+has size 4, we are done. Otherwise, there is a bag B in T that has size less than three. We
+root the tree T at B and modify T in a top-down fashion. We need the following claim to
+prove the correctness of this procedure.
+▷ Claim 37.
+Let B′ be some bag in T of size 3, then the vertices in B′ cannot form an
+independent set of size 3 in X.
+Proof. If the corresponding bag P to B′ in U has size 4, then since U is a matched tree
+decomposition, the bag P is matched. So B′ was obtained by the deletion of a vertex or the
+contraction of an edge. Both of which will retain at least one edge in the bag.
+If the corresponding bag P has size 3, then since U is a matched tree composition, the
+subgraph induced by B′ has P3 as a subgraph.
+◀
+So we can assume the following:
+4 All indices i for ui and u′
+i in this proof are modulo 6.
+
+26
+Finding and Counting Patterns in Sparse Graphs
+1. Every bag in T of size 4 is matched.
+2. Every bag in T of size 3 has at least one edge.
+3. There are no bags in T of size 0 or size 1.
+We now process a bag B, starting from the root, assuming that all ancestors of B are already
+matched. We split the procedure into two cases.
+(|B| = 3) Let B = {u, v, w}, uv is an edge, and w is not adjacent to u or v. If w is present in
+the parent of B, then it is matched to some w′. We add w′ to B as well. If w is not present
+in the parent, then we choose a nearest descendant B′ of B in T that contains a neighbor w′
+of w. Every bag in the path from B to B′ contains w. We add w′ to every bag in this path.
+Now, the bag B is matched. Also, every other bag that we modified (in the path from B to
+B′) now has size 3 and has at least one edge or has size 4 and is matched. So all properties
+are preserved.
+(|B| = 2) Let B = {u, v} and uv is not an edge. If u is present in B’s parent in T, then it is
+matched to some u′ there. We add u′ to B. Otherwise, we choose a nearest descendant B′
+of B in T that contains a neighbor u′ of u. We add u′ to all bags in the path from B to B′.
+B is now a bag of size 3 with at least one edge. Also, every other bag that we modified now
+has size 3 and has at least one edge or has size 4 and is matched. Now, if B is matched we
+are done. Otherwise, we apply the previous case to B.
+(Proof for the “only if” direction) We now prove a series of lemmas that will prove the
+theorem.
+▶ Lemma 38. Let G be a graph with tw(G) ⩽ 2 and ˜G is a smcc of G. If there exists three
+internally disjoint paths from a vertex u to v in G, then uv is an edge in ˜G
+Proof. Suppose for contradiction u is not adjacent to v. Let Q1, Q2, Q3 be three internally
+disjoint paths from u to v. Let Q′
+1, Q′
+2 be the shortest path from u to v in ˜G[V(Q1)] and
+˜G[V(Q2)], respectively. Note that both of them are of length at least 3. Since ˜G is chordal,
+the cycle obtained from Q′
+1 and Q′
+2 has a chord say xy. Note that {x, y} ∩ {u, v} = ∅. Without
+loss of generality, we may assume that x is in Q′
+1. So y is in Q′
+2. Therefore, Q′
+1, Q′
+2 and Q3
+gives a minor of K4 in ˜G. This is a contradiction.
+◀
+▶ Lemma 39. Let G be a X-induced-minor-free connected graph with tree-width 2. Then, there
+exists a smcc ˜G of G.
+Proof. Suppose for contradiction that there exists a minimal counter-example G with tw(G) ⩽
+2 that is X-induced-minor-free but does not have a smcc. Since G is minimal, it has to be
+biconnected and by the characterization of treewidth 2 graphs, we can assume that G is
+series-parallel. We will use the series-parallel graph characterization to prove that G does
+not exist.
+Let s and t be the source and terminal of G. Since G is a counter example, it is not an edge.
+Suppose (G, s, t) is the series composition of (G1, s1, t1) with (G2, s2, t2) (t1 is identified with
+s2). Since G is a minimal counter example, there exists smccs ˜
+G1 of G1 and ˜
+G2 of G2. Let ˜G
+be the graph obtained from ˜
+G1 and ˜
+G2 by identifying t1 with s2. The graph ˜G is a smcc of
+G, a contradiction. Hence, the graph (G, s, t) is the parallel composition of strictly smaller
+graphs (G1, s1, t1) and (G2, s2, t2).
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+27
+▷ Claim 40.
+Let G be the parallel composition of two smaller graphs G1 and G2. Further-
+more, assume that G1 is the series composition of two or more graphs G′
+i for 1 ⩽ i ⩽ m.
+Then, for all 1 ⩽ i ⩽ m, if G′
+i is not an edge, then s′
+it′
+i is also not an edge.
+Proof. Suppose there exists 1 ⩽ i ⩽ m such that G′
+i is not an edge but s′
+it′
+i is an edge in G′
+i.
+So G′
+i is of order at least 3. There exists a smcc Gi of the graph obtained from G by deleting
+all the internal vertices of G′
+i (Since G is a minimum counter example). Again there exists
+a smcc ˜
+G′
+i of G′
+i. Define ˜G to be the graph obtained from Gi and ˜
+G′
+i by identifying the edge
+s′
+it′
+i. The graph ˜G is a smcc of G.
+◀
+Note that at least one of G1 or G2 is not an edge. We now split the proof into two exhaustive
+cases:
+(Case 1) Either G1 or G2 is an edge.
+Without loss of generality we may assume that G2 is an edge. So s1t1 is not an edge in
+G1 (as otherwise G1 = G). Suppose (G1, s1, t1) is a parallel composition of (G3, s3, t3)
+and (G4, s4, t4). Since s1t1 ̸∈ E(G1), we have s3t3 ̸∈ E(G3) and s4t4 ̸∈ E(G4). Therefore,
+both G3 and G4 has at least one internal vertex. Let G′
+3 be the graph obtained from
+G3 by adding the edge s3t3 and G′
+4 be the graph obtained from G4 by adding the edge
+s4t4. Since G is a minimum counter example, there exists smccs ˜
+G′
+3 of G′
+3 and ˜
+G′
+4 of
+G′
+4. Note that G is obtained from G′
+3 and G′
+4 by identifying edges s3t3 and s4t4. So the
+graph ˜G obtained from ˜
+G′
+3 and ˜
+G′
+4 by identifying those edges is a smcc of G. This is a
+contradiction.
+We may assume that (G1, s1, t1) is a series composition of smaller graphs (G′
+1, s′
+1, t′
+1),
+(G′
+2, s′
+2, t′
+2), . . . , (G′
+m, s′
+m, t′
+m) for some m ⩾ 2 such that for all 2 ⩽ i ⩽ m, the vertex
+t′
+i−1 is identified with s′
+i and for all 1 ⩽ i ⩽ m, (G′
+j, s′
+j, t′
+j) is either an edge or a parallel
+composition of two smaller graphs.
+Note that G is not a cycle since G is a counter example. So, there exists some i such that
+the graph G′
+i is not an edge. Therefore, by Claim 40, we have that s′
+it′
+i is not an edge.
+▷ Claim 41.
+There does not exist 1 ⩽ j(̸= i) < k(̸= i) ⩽ m such that s′
+jt′
+j, s′
+kt′
+k are
+non-edges.
+Proof.
+Suppose s′
+jt′
+j, s′
+kt′
+k are non-edges, for some 1 < j(̸= i) < k(̸= i) ⩽ m. Without
+loss of generality, we may assume that i < j. So G′
+i is a parallel composition of two
+smaller graphs. Hence there exist two internally disjoints paths Q1, Q′
+1 (each of length
+at least 3), from s′
+i to t′
+i in G′
+i. Similarly there exist two internally disjoints paths Q2, Q′
+2
+(each of length at least 3), from s′
+j to t′
+j in G′
+j. Also there exist two internally disjoints
+paths Q3, Q′
+3 (each of length at least 3), from s′
+k to t′
+k in G′
+k. Again, there exists a
+path from t′
+i to s′
+j, through G′
+i+1, . . . G′
+j−1, a path from t′
+j to s′
+k through G′
+j+1, . . . G′
+k−1.
+Again we have a path from t′
+k to s′
+i through G′
+k+1, . . . G′
+m, s′
+mt′
+1, G′
+1, . . . G′
+i−1. This gives
+a subdivision of X. This a contradiction. Hence, the above claim is true.
+◀
+▷ Claim 42.
+There exists a smcc of G′
+i such that s′
+it′
+i is an edge in the completion.
+Proof.
+Since s′
+i is not adjacent to t′
+i, (G′
+i, s′
+i, t′
+i) is parallel composition of two smaller
+graphs (By Claim 40), say (G3, s3, t3) and (G4, s4, t4). Let G′ be the graph obtained from
+
+28
+Finding and Counting Patterns in Sparse Graphs
+G by contracting the edge st. This is an X-induced-minor-free graph. The minimality of
+G says that G′ has an smcc, say ˜
+G′. By Lemma 38, s′
+i is adjacent to t′
+i in ˜
+G′. Note that
+the graph induced by the vertices of G′
+i in ˜
+G′ is a smcc of G′
+i in which s′
+it′
+i is an edge.
+◀
+By Claim 41, the following cases are now exhaustive.
+(Case 1a) s′
+jt′
+j is an edge, for all 1 ⩽ j(̸= i) ⩽ m.
+By the above claim, there exists a smcc ˜
+G′
+i of G′
+i in which s′
+it′
+i is an edge. We construct
+an smcc of G as follows: It contains the smcc ˜
+G′
+k for all G′
+k and the edge st. We then
+add the edges ss′
+i and s′
+it′
+j for all 1 ⩽ j(̸= i) ⩽ m. These edges ensure that there are
+no chordless cycles through G1 and G2. This smcc gives us a contradiction.
+(Case 1b) There exists 1 ⩽ j(̸= i) ⩽ m such that s′
+i is not adjacent to t′
+j.
+Without loss of generality we may assume that i < j. By the claim 42, there exists a
+smcc ˜
+G′
+i, ˜
+G′
+j of G′
+i and s′
+j, respectively, such that s′
+it′
+i ∈ E( ˜
+G′
+i) and s′
+jt′
+j ∈ E( ˜
+G′
+j). We
+construct an smcc of G as follows: It contains the smcc ˜
+G′
+k for all G′
+k and the edge st.
+We then add edges t′
+is′
+k for all 1 ⩽ k < i, st′
+k for all i < k < j, and s′
+jt and s′
+jt′
+k for all
+j < k ⩽ m. This smcc gives us a contradiction.
+The above constructions are illustrated in Figure 3. The edges colored black are the
+edges in G. The red colored edges are those added to construct the smcc.
+case (1a)
+case (1b)
+Figure 3 Constructing smcc in case 1.
+(Case 2) Both G1 and G2 are not edges. Suppose G1 and G2 both are parallel composition
+of smaller graphs. So there exists two internally disjoints paths P, P′ from s1 to t1 in G1
+and two internally disjoints paths Q, Q′ from s2 to t2 in G2. Let G′ be the graph obtained
+from G1 by adding Q. Since, G is a minimum counter example, G′ has a smcc ˜
+G′. Again,
+P, P′, Q are three mutually internally disjoint paths from s1 to t1. By, lemma 38, s1t1 is
+an edge in ˜
+G′. The graph ˜
+G1 induced by V(G1) in ˜
+G′ is a smcc such that s1t1 is an edge
+in ˜
+G1. Similarly, we can show that there exists a smcc ˜
+G2 of G2 such that s1t2 is an edge
+in ˜
+G2. The graph obtained from ˜
+G1 and ˜
+G2 by identifying the edges s1 with s2 and t1
+with t2 is a smcc of G.
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+29
+We may assume that at least one of G1 or G2 is a series composition of graphs. We decom-
+pose G1 and G2 into series composed graphs repeatedly, until we cannot. i.e., all the indi-
+vidual components are parallel compositions or an edge. Let (G1, s1, t1) be series compos-
+itions of (G′
+1, s′
+1, t′
+1), (G′
+2, s′
+2, t′
+2), . . . , (G′
+m, s′
+m, t′
+m) and let (G2, s2, t2) be series composi-
+tions of (G′
+m+1, s′
+m+1, t′
+m+1), (G′
+m+2, s′
+m+2, t′
+m+2), . . . , (G′
+m+ℓ, s′
+m+ℓ, t′
+m+ℓ) in that order.
+▷ Claim 43.
+There exists a 1 ⩽ i ⩽ m + ℓ such that s′
+it′
+i is an edge.
+Proof.
+Suppose for contradiction that s′
+i is not adjacent to t′
+i, for all 1 ⩽ i ⩽ m + ℓ. So,
+the graph (G′
+1, s′
+1, t′
+1) is a parallel composition of two smaller graphs. Hence, there exists
+two internally disjoint paths, each of length at least 3, from s′
+1 to t′
+1 in G′
+1. Similarly,
+there are two internally disjoint paths, each of length at least 3, from s′
+i to t′
+i in G′
+i, for
+all 1 ⩽ i ⩽ m+ℓ. Since m+ℓ > 2, we get an X-induced-minor in G, a contradiction.
+◀
+Thus, there exists a 1 ⩽ i ⩽ m+ℓ such that s′
+it′
+i is an edge. By Claim 40, we can conclude
+that G′
+i is an edge. Without loss of generality, we may assume that 1 ⩽ i ⩽ m. Note that
+(G \ {s′
+it′
+i}, s′
+i, t′
+i), is the graph obtained by the series composition of G′
+i−1, G′
+i−2, . . . , G′
+1,
+G2, G′
+m, G′
+m−1, . . . , G′
+i+1 by identifying s′
+i−k, s′
+1, t2, t′
+j with t′
+i−k+1, s2, t′
+m and s′
+j, respect-
+ively, for 1 ⩽ k < i and i < j ⩽ m. Again, the graph (G, s′
+i, t′
+i) is the parallel composition
+of (G\{s′
+it′
+i}, s′
+i, t′
+i) with an edge. Now, we can use case (1) to get a contradiction. Hence,
+a minimum counter example does not exist.
+◀
+▶ Lemma 44. Let G be a graph with treewidth k and T be a tree decomposition of G such that
+no bag of size k + 1 of T is independent in G. Then, mtw(G) ⩽ 2tw(G) − 1.
+Proof. Observe that in the proof of Theorem 7, the construction yields a matched tree
+decomposition of width 2tw(G) − 1 if each bag of size tw(G) + 1 has at least one edge.
+◀
+By Lemma 39, every G such that tw(G) = 2 and G is X-induced-minor-free has an smcc. Now,
+a standard tree decomposition T of ˜G is also a tree decomposition of G with the additional
+property that every bag of size 3 has at least one edge. We apply Lemma 44 to construct a
+matched tree decomposition of width 3 for G.
+◀
+To complete the characterization of matched treewidth of partial 2-trees, we prove the fol-
+lowing lemma.
+▶ Lemma 45. mtw(Y) ⩾ 5
+Proof. The proof is similar to the proof of Lemma 35. We additionally use the fact that
+u1, u2, and u3 do not have any common neighbors. Suppose for contradiction that Y has
+a reduced, matched tree decomposition T of width strictly less than 5. Let Bl be a leaf in
+T. We first argue that Bl must contain one of u1, u2, or u3. If not, it contains only a subset
+of the other edges, say, like u4u5 (other cases are symmetric). Since those vertices are not
+pendant, the neighbor of Bl in T, which we call Bp, will be a superset of Bl. Suppose Bl
+contains u1 and u4 (rest of the cases are symmetric). Since T is reduced and has width less
+than 5, the bag Bp must contain both u1 and u2 (or u1 and u3, a symmetric case). Root T
+at Bl.
+
+30
+Finding and Counting Patterns in Sparse Graphs
+u1
+u7
+u5
+u2
+u3
+u12
+u14
+u9
+u11
+u4
+u6
+u8
+u10
+u15
+u13
+Figure 4 The graph Y.
+If u3 ∈ Bp, we are done since |Bp| ⩾ 6. Let Bc be the closest descendant of Bp that contains
+u3. Assume wlog that u2 /∈ Bc. Both u1 and u2 cannot be missing from Bc. In that case, all
+neighbors of u3 must be in Bc and that will imply |Bc| ⩾ 6. So u1 and u3 are both in Bc (the
+other case is symmetric). It must also contain a vertex v that is a neighbor of u1 since it is
+matched. Consider the path P from Bp to Bc in T. If u8 (u10) does not appear in this path,
+then u9 (u11 resp.) must appear on all bags in this path. Therefore, Bp will have to contain
+u3 to match u9 (or u11). So u8 and u10 must appear in this path.
+Let B1 be the first bag in P where both u8 and u10 have appeared. We split the proof into
+two cases.
+(Both u8 and u10 appear in B1) If u2 /∈ B1, then |B1| ⩾ 6 and we are done. Otherwise,
+B1 = {u1, v, u2, u8, u10} for some neighbor v of u1. Let B2 be the first bag from B1 to Bc
+where the edge u8u9 or u10u11 appears. Then, either B2 = {u1, v, u2, u10, u8, u9} (u10u11
+has not appeared) or B2 = {u1, v, u10, u11, u8, u9} (both edges appear simultaneously) and
+we are done.
+(One of u8 or u10 is missing from B1) This means either u9 or u11 is in B1. We have either
+B1 = {u1, v, u2, u10, u9, u3} or B1 = {u1, v, u2, u8, u11, u3} and we are done.
+◀
+Figure 5 is a super-graph of Y that has lower matched treewidth.
+4.2
+Finding and counting Subgraphs and induced subgraphs using
+matched treewidth
+▶ Theorem 12. Given an m-edge graph H as input, we can count the number of Pk subgraphs,
+where k ⩽ 10, in �O(m2)-time.
+Proof. There are more than 300 graphs in Spasm(P10). We have verified that all of them
+have mtw at most 3. To minimize the work, we can filter out all graphs in the spasm that
+has tw(G′) = 1 or tw(G′) = 2 and G′ is X-induced-minor-free. Observe that since X has
+9 vertices 12 edges, it cannot be an induced minor in any of the graphs in Spasm(P10).
+Also, none of the forbidden minors for treewidth 4 can appear in Spasm(P10).
+There-
+fore, we only need to analyze graphs of treewidth 3 in Spasm(P10).
+There are only 18
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+31
+u1
+u7
+u5
+u2
+u3
+u12
+u14
+u9
+u11
+u4
+u6
+u8
+u10
+u15
+u13
+v
+u1u4u5u6u7
+u1vu5u7u2
+u1vu2u8u9
+u1vu2u9u3
+u1vu2u3u11
+u1vu3u14u12
+u1vu2u10u11
+u1u13u12u14u15
+Figure 5 The graph Z and its matched tree decomposition of width 4.
+such graphs.
+They are listed in a pdf file in the repository associated with this paper
+(https://github.com/anonymous1203/Spasm).
+Since Spasm(Pk) ⊆ Spasm(Pk+1) for all k ⩾ 2, we can make the same claim for all paths on
+fewer than 10 vertices. We now use Equation 1 to compute the result.
+◀
+▶ Remark 46. Since K5 ∈ Spasm(P11), and treewidth of K5 is 4, the above method cannot
+yield an �O(m2) algorithm for Pk where k ⩾ 11.
+We note that the above proof also yields �O(m2) time algorithms for counting any pattern
+in Spasm(P10). This is because if G ∈ Spasm(P10), then Spasm(G) ⊆ Spasm(P10). In the
+following theorem, we point out an important class of graphs in Spasm(P10).
+▶ Theorem 13. Given an m-edge graph H as input, we can count the number of cycles of
+length at most 9 in �O(m2)-time.
+Proof. We observe that Spasm(Ck) ⊂ Spasm(P10) for k ⩽ 9.
+◀
+▶ Remark 47. Since K5 ∈ Spasm(C10), and treewidth of K5 is 4, the above method cannot
+yield an �O(m2) algorithm for Ck where k ⩾ 10.
+We can also use our efficient construction of arithemetic circuits for homomorphism poly-
+nomials for detecting small induced subgraphs. Brute-force search finds, or even counts
+C6 as induced subgraphs in an m-edge host graph in O(m3) time. Bläser, Komarath, and
+Sreenivasaiah [2] showed that efficient computation of homomorphism polynomials can be
+used to speed-up the detection of induced subgraphs. For example, their techniques can be
+used to show that detecting an induced C6 in an n-vertex host graph can be done in O(n4)
+time. In this section, we derive an �O(m2) algorithm for detecting induced C6 in an m-edge
+host graph. This algorithm is a natural analogue of the algorithm by Bla¨ser, Komarath and
+Sreenivasaiah [2].
+We now give some definitions that are necessary to understand how homomorphism poly-
+nomials are used in induced subgraph detection.
+
+32
+Finding and Counting Patterns in Sparse Graphs
+▶ Definition 48. We define the induced subgraph isomorphism polynomial for pattern graph
+G on n-vertex host graphs, denoted IndG, as follows. The variables of the polynomial are yv
+and x{u,v} for all u, v ∈ [n].
+IndG =
+�
+G′
+�
+v
+yv
+�
+e
+xe
+�
+f
+(1 − xf)
+where G′ ranges over all (not-necessarily induced) subgraphs of Kn isomorphic to G, v ranges
+over the vertices of G′, e ranges over the edges of G′, and f ranges over the edges in Kn between
+vertices in G′ but not in G′.
+We denote by IndG(E(H)), the polynomial obtained by substituting the adjacency in H for
+the edge variables. Note that the monomials of IndG(E(H)) are products of |V(G)| vertex
+variables and correspond to the induced subgraphs of H isomorphic to G. In addition, these
+monomials all have coefficient 1 because there can only be at most 1 induced subgraph
+isomorphic to G on any given k vertices. The induced subgraph polynomials and subgraph
+polynomials are related via the equation:
+IndG(E(H)) =
+�
+G′
+(−1)|E(G′)−E(G)|#SubG[G′]SubG′[H](xe = 1)
+(2)
+where G′ ranges over all k-vertex supergraphs of G and #SubG[G′] denotes the number of
+times G occurs as a subgraph in G′. We use the substitution (xe = 1) to denote that all edge
+variables in the polynomial are substituted with 1. Variants of this equation have been used
+by many authors for induced subgraph detection (See [23, 2, 16]).
+We now briefly describe how to use homomorphism polynomials to detect induced subgraph
+isomorphisms (See [2] for a more detailed description). Note that the IndG(E(H)) has the
+monomial xv1 · · · xvk if and only if v1, . . . , vk induces a G. Therefore, to check whether G
+occurs as an induced subgraph, we only have to test whether IndG(E(H)) is non-zero. Fur-
+thermore, the coefficient of every monomial is 1 because a k-vertex subgraph can contain at
+most one induced subgraph isomorphic to G. Therefore, whether H contains an induced sub-
+graph isomorphic to G can be reduced to whether IndG(E(H)) is non-zero modulo 2. The ad-
+vantage of computing over a ring of characteristic 2 is that it eliminates all SubG′[H](xe = 1)
+from the right-hand side of Equation 2 for which #SubG[G′] is even. However, we do not
+have efficient computations for subgraph polynomials. Here, we make use of the observation
+that SubG′[H](xe = 1) is equal to the multilinear part of
+1
+#Aut(G′)HomG′[H](xe = 1). There-
+fore, to test whether SubG′[H](xe = 1) is non-zero modulo 2, we need only test whether
+1
+#Aut(G′)HomG′[H](xe = 1) contains a multilinear term with an odd coefficient.
+To check the presence of multilinear terms with odd coefficients, we can randomly substitute
+elements that satisfy the equation x2 = 0 from group algebras over Z2 [15]. We stress
+that we only substitute these elements for the vertex variables. The edge variables are all
+always replaced by 0 or 1. The use of a characteristic-2 ring introduces another issue. We
+now cannot compute
+1
+#Aut(G′)HomG′[H](xe = 1) for graphs that have an even number of
+automorphisms, by finding the homomorphism polynomial and dividing by the number of
+automorphisms. The solution is to compute a polynomial that avoids these automorphisms
+in the multilinear part of HomG′[H](xe = 1) for each such G′ so that this division becomes
+unnecessary, while being careful not to introduce additional multilinear terms. This is the
+crux of the following proof.
+▶ Theorem 14. Given an m-edge host graph as input, we can find an induced C6 or report
+that none exists in �O(m2)-time.
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+33
+Proof. We describe how to compute polynomials for which the multilinear part is the same
+as HomG′[H](xe = 1) and the coefficient of all monomials are odd for all 6-vertex super-
+graphs of C6 that contain C6 an odd number times. The complete list is given in Figure 6.
+These computations involve modifying Algorithm 2 slightly for each such G′. We consider
+the case of C6. Each multilinear monomial in HomC6[H](xe = 1) has coefficient 12. By
+ensuring that only homomorphisms σ where σ(2) = min(σ(2), σ(3), σ(5), σ(6)) are present
+in the polynomial, we can ensure that all C6 subgraphs of H are counted exactly thrice, once
+for each choice of {σ(1), σ(4)}. This check can be done when the algorithm processes bag
+2365 (Figure 6) in Line 22. Notice that we need to iterate only over the edges present in H
+in the algorithm. This is because the other edge variables will be substituted with 0 anyway
+and those monomials will definitely vanish. This is crucial in ensuring that our algorithm
+remains �O(m2) and not �O(n4).
+We consider one more case from our list. The graph in the first row and third column in
+Figure 6 has four automorphisms (horizontal flip and vertical flip). We can ensure that
+these subgraphs are counted exactly once in the polynomial by ensuring that Line 22 in
+Algorithm 2 also checks that σ(3) < σ(6) (preventing horizontal flips) and σ(2) < σ(4)
+(preventing vertical flips). These checks can be done when the algorithm processes the bag
+1634 and 1234 respectively.
+Figure 6 shows the matched tree decompositions and the constraints on σ that can be used
+to apply Algorithm 2 to compute all these polynomials.
+◀
+▶ Theorem 15. Given an m-edge host graph as input, we can find an induced Pk or report
+that none exists in �O(m(k−2)/2)-time.
+Proof. We first prove that Pk has matched treewidth k−3. We only consider the case of odd
+k (the even case is similar). Let vertices in the path be 1, 2, . . . , 2j + 1. Then, our matched
+tree decomposition will have three bags. A root bag that excludes only {j, j+2}, a left child of
+the root bag that excludes {j, j+1}, and a right child of the root bag that excludes {j+1, j+2}.
+This is clearly a tree decomposition. To see that it is matched, at the root bag, we have the
+matched matching (1, 2j + 1), (2, 2j), . . . , (j − 1, j + 3), (j + 1, j + 3). On the left child, we
+can keep the rest of edges the same and match (j + 2, j − 1). Similarly, on the right child, we
+match (j, j + 3).
+Since Pk has two automorphisms, we also need to show that we can avoid one automorph-
+ism while computing the homomorphism polynomial. We note that the non-identity auto-
+morphism τ must have τ(1) = k and τ(k) = 1. Therefore, we can avoid this by always
+ensuring that σ(1) < σ(k) when building the homomorphism polynomial in Algorithm 2.
+We know that IndPk(E(H)) = SubPk[H](xe = 1) (mod 2). The theorem follows.
+◀
+5
+Acknowledgement:
+The research work of S. Mishra is partially funded by Fondecyt Postdoctoral grant 3220618
+of Agencia National de Investigatión y Desarrollo (ANID), Chile.
+
+34
+Finding and Counting Patterns in Sparse Graphs
+1
+2
+3
+4
+5
+6
+126
+2365
+345
+σ(2) = min(σ(2), σ(3), σ(5), σ(6))
+1
+2
+3
+4
+5
+6
+σ(2) < σ(6)
+1
+2
+3
+4
+5
+6
+126
+2365
+345
+1234
+1634
+1654
+σ(3) < σ(6), σ(2) < σ(4)
+1
+2
+3
+4
+5
+6
+1236
+3645
+1
+2
+3
+4
+5
+6
+6123
+6234
+654
+σ(4) < σ(6)
+2
+4
+6
+1
+2345
+1245
+1456
+σ(3) < σ(5)
+3
+5
+1
+2
+3
+4
+5
+6
+1256
+2356
+3456
+1
+2
+3
+4
+5
+6
+1264
+3264
+5264
+σ(2) < σ(4) < σ(6)
+1
+2
+3
+4
+5
+6
+6123
+6234
+6254
+σ(1) < σ(3)
+1
+2
+3
+4
+5
+6
+126
+2365
+345
+σ(2) = min(σ(2), σ(3), σ(5), σ(6))
+1
+2
+3
+4
+5
+6
+126
+2365
+345
+σ(2) < σ(5)
+1
+2
+3
+4
+5
+6
+1256
+2356
+3456
+σ(4) < σ(6)
+1
+2
+3
+4
+5
+6
+1263
+6134
+6534
+σ(5) < σ(6)
+1
+2
+3
+4
+6
+5
+σ(2) < σ(6)
+1265
+2564
+2534
+1
+2
+3
+4
+5
+6
+1426
+2645
+2345
+σ(1) < σ(4), σ(1) < σ(2) < σ(6)
+1
+2
+3
+4
+5
+6
+1426
+2645
+2345
+σ(2) < σ(4)
+1
+2
+3
+5
+6
+4
+σ(2) < σ(6)
+1265
+2564
+2534
+1
+2
+3
+5
+6
+4
+σ(2) < σ(6)
+1246
+2564
+2534
+Figure 6 Detecting Induced C6
+
+B. Komarath, A. Kumar, S. Mishra, and A. Sethia
+35
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+
diff --git a/s9E0T4oBgHgl3EQfrwEw/content/tmp_files/load_file.txt b/s9E0T4oBgHgl3EQfrwEw/content/tmp_files/load_file.txt
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@@ -0,0 +1,1580 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf,len=1579
+page_content='Finding and Counting Patterns in Sparse Graphs  Balagopal Komarath �  IIT Gandhinagar, India  Anant Kumar �  IIT Gandhinagar, India  Suchismita Mishra �  Universidad Andrés Bello, Chile  Aditi Sethia �  IIT Gandhinagar, India  Abstract  We consider algorithms for finding and counting small, fixed graphs in sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In the  non-sparse setting, the parameters treedepth and treewidth play a crucial role in fast, constant-space  and polynomial-space algorithms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We discover two new parameters that we call matched  treedepth and matched treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We show that finding and counting patterns with low matched  treedepth and low matched treewidth can be done asymptotically faster than the existing algorithms  when the host graphs are sparse for many patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' As an application to finding and counting fixed-  size patterns, we discover �O(m3)-time 1, constant-space algorithms for cycles of length at most 11 and  �O(m2)-time, polynomial-space algorithms for paths of length at most 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The-  ory of computation → Graph algorithms analysis  Keywords and phrases Subgraph Detection and Counting, Homomorphism Polynomials, Treewidth  and Treedepth, Matchings  1  Introduction  Given simple graphs G, called the pattern, and H, called the host, a fundamental compu-  tational problem is to find or count occurrences of G in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' What does it mean for G to  occur in H?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The three most common notions of occurrence are characterized by mappings  φ : V(G) �→ V(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We say:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If {u, v} ∈ E(G) implies {φ(u), φ(v)} ∈ E(H) and φ is one-to-one, then we say that  φ witnesses a subgraph isomorphic to G in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The subgraph is obtained by taking the  vertices and edges in the image of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The number of G-subgraphs of H is just the number  of such subgraphs G′ of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If {u, v} ∈ E(G) is equivalent to {φ(u), φ(v)} ∈ E(H) and φ is one-to-one, then φ wit-  nesses an induced subgraph isomorphic to G in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The induced subgraph is obtained by  taking the vertices and all edges induced by those vertices in the image of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If {u, v} ∈ E(G) implies {φ(u), φ(v)} ∈ E(H), then we say that φ is a homomorphism from  G to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that unlike a subgraph isomorphism, φ is not required to be one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  1  �O hides factors that are logarithmic in the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='02569v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='DS]  6 Jan 2023  2  Finding and Counting Patterns in Sparse Graphs  For any of these notions, the detection problem is clearly in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' All three of them are also  straightforward generalizations of the NP-hard problem CLIQUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, the existence  of efficient algorithms for finding or counting patterns under any of these notions is unlikely  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The class of pattern detection and counting problems remain interesting even if we restrict  our attention to fixed pattern graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Williams [21] showed that the improved algorithms  for finding triangles could be used to find faster algorithms for even NP-complete problems  such as MAX2SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For fixed pattern graphs of size k, the brute-force search algorithm is  as follows: Iterate over all k-tuples over V(H) and check whether G occurs in the induced  subgraph of H on the vertices in that k-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This algorithm takes θ(nk) time and constant  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, when we restrict our attention to fixed patterns, we seek improvements  over this running time preferably keeping the space usage low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' There are two broad tech-  niques that reduce the running-time: the usage of fast matrix multiplication algorithms as a  sub-routine and the exploitation of structural properties of pattern graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  If A is the adjacency matrix of the graph, then Nešetˇril and Poljak [17] showed that one can  obtain an O(nω)-time algorithm for counting triangles using the identity trace(A3) = 6∆,  where ∆ is the number of triangles in the graph, where ω < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='38 is the matrix multiplication  exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Using a simple reduction, they extended this to an algorithm to count 3k-cliques  in O(nkω)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' They also showed that we can use improved algorithms for counting k-  cliques to count any k-vertex pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Later, Kloks, Kratsch and Müller [13] showed how  to use fast rectangular matrix multiplication to obtain similar improvements to the running  time for counting cliques of all sizes, not just multiples of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that the improvements  obtained by these algorithms are applicable to all k-vertex patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=', they do not use the  pattern’s structure to obtain better algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since finding a k-clique requires nΩ(k)-time  unless ETH is false, we need to exploit the structure of the pattern to obtain significantly  better algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  For patterns sparser than cliques, the run-time can be significantly improved over even fast  matrix multiplication based (pattern finding) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The crucial idea is to exploit the  structure of the pattern graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A k-walk polynomial is a polynomial where the monomials  correspond to walks that are k vertices long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For example, a walk (u, v, w, x) will correspond  to the monomial xuvxvwxwx and a walk (u, v, u, v) to the monomial x3  uv  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Williams [22]  showed that we can detect k-paths in graphs by (1) computing the k-walk polynomial and  (2) checking whether it has multilinear monomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We can compute the k-walk polynomial  in linear-time using a simple dynamic programming algorithm and then multilinear monomi-  als can be detected with high probability by evaluating this polynomial over an appropriate  ring where the randomly chosen elements satisfy a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This yields is a O(2k(n+m))-time  algorithm for finding k-vertex paths as subgraphs in n-vertex, m-edge host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We now consider the problem of counting sparse patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For counting k-paths as sub-  graphs, the best known algorithm by Curticapean, Dell and Marx [4] takes only O(f(k)n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='174k+o(k))-  time for some function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Coming to fixed pattern graphs, Alon, Yuster and Zwick [1] gave  O(nω)-time algorithms for counting cycle subgraphs of length at most 8 using an algorithm  that combines fast matrix multiplication and exploitation of the structure of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' No-  tice that this is the same as the time required for counting triangles (3-cliques).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2 We write uv to denote the edge {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  3  The notion of graph homomorphisms was shown to play a crucial role in all the above im-  proved algorithms for finding and counting non-clique subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' More specifically, Fomin,  Lokshtanov, Raman, Rao, and Saurabh [11] showed how the efficient construction of ho-  momorphism polynomials (see Definition 17), a generalization of k-walk polynomials, can  be used to detect subgraphs with small treewidth efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Their algorithm can be seen  as a generalization of Williams’s algorithm [22] for k-paths to arbitrary graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Similarly,  Curticapean, Dell and Marx [4] showed that efficient algorithms for counting subgraphs  can be derived from efficient algorithms for counting homomorphisms of graphs of small  treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Their algorithm can be seen as a generalization of the cycle-counting algorithms  of Alon, Yuster, and Zwick [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Algorithms for finding and counting patterns in sparse host graphs are also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' An  additional parameter, m, the number of edges in the host graph, is taken into account for  the design and analysis of these algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In the worst-case, m could be as high as  �n  2  �  ,  and hence, an O(nt)-time algorithm and an O(mt/2)-time algorithm for some t have the  same asymptotic time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, it is common in practice that m = o(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For  example, if the host graph models a road network, then m = O(n), where the constant  factor is determined by the maximum number of roads at any intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In such cases, an  O(mt/2)-time algorithm is asymptotically better than an O(nt)-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The broad themes of using fast matrix multiplication and/or structural parameters of the pat-  tern to obtain improved algorithms are still applicable in the setting of sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Using fast matrix multiplication, Eisenbrand and Grandoni [8] showed that we can count  k-cliques in O(mkω/6)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kloks, Kratsch and Müller [13] showed that K4 subgraphs can  be counted in O(m(ω+1)/2)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Again, since ω < 3, this is better than the O(m2)-time  given by the brute-force algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Using structural parameters of the pattern, Kowaluk,  Lingas, and Lundell [16] obtained many improved algorithms in the sparse host graph set-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For example, their methods obtain an algorithm that runs in O(m4)-time for counting  P10 as subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In this work, we obtain an �O(m2)-time algorithm for counting P10 (See  Theorems 12,13,14,15 for similar improvements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The model of computation that we consider is the unit-cost RAM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In particular, we can  store labels of vertices and edges in the host graph in a constant number of words3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In this  model, algorithms based on fast matrix multiplication and/or treewidth mentioned above  use polynomial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, the brute-force search algorithm uses only constant space  as it only needs to store k vertex labels at a time (Recall that we regard k as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  How much speed-up can we obtain while preserving constant space usage?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The graph para-  meter treedepth plays a crucial role in answering this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It is well known that we  can count the homomorphisms from a pattern of treedepth d in O(nd)-time while using  only constant space (See Komarath, Rahul, and Pandey [14] for a construction of arithmetic  formulas counting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' These arithmetic formulas can be implicitly constructed and eval-  uated in constant space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since all k-vertex patterns except k-clique has treedepth strictly  less than k, this immediately yields an improvement over the running-time of brute-force  while preserving constant space usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In this work, we improve upon the treedepth-based  algorithms for sparse host graphs where the pattern graph is a cycle of length at most 11  (See Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  3 In the TM model or the log-cost RAM model, storing labels of vertices would take O(log n) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  4  Finding and Counting Patterns in Sparse Graphs  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='1  Connection to arithmetic circuits for graph homomorphism  polynomials  A popular sub-routine in these algorithms is an algorithm by Diaz, Serna and Thilikos [6]  that efficiently counts the number of homomorphisms from a pattern of small treewidth  to an arbitrary host graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Indeed, it can be shown that this algorithm can be easily gen-  eralized to efficiently construct circuits for homomorphism polynomials instead of counting  homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Bläser, Komarath and Sreenivasaiah [2] showed that efficient construc-  tions for homomorphism polynomials can even be used to detect induced subgraphs in some  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' They also show that many of the faster induced subgraph detection algorithms, such  finding four-node subgraphs by Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' [23] and five-node subgraphs by Kowaluk,  Lingas, and Lundell [16] can be described as algorithms that efficiently construct these  homomorphism polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, arithmetic circuits for graph homomorphism poly-  nomials provide a unifying framework for describing almost all the fast algorithms that  we know for finding and counting subgraphs and finding induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Can we im-  prove these algorithms by finding more efficient ways to construct arithmetic circuits for  homomorphism polynomials?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Unfortunately, it is known that for the type of circuit that is  constructed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=', circuits that do not involve cancellations, the existing constructions are the  best possible for all pattern graphs, as shown by Komarath, Pandey, and Rahul [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The situ-  ation is similar for constant space algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The best known algorithms can be expressed  as divide-and-conquer algorithms that evaluate small formulas constructed by making use  of the graph parameter treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, Pandey, and Rahul[14] also showed that the  running-time of these algorithms match the best possible formula size for all pattern graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  These arithmetic circuit lower bounds serve as a technical motivation for considering sparse  host graphs, in addition to the practical motivation mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='2  Our findings  In this paper, we study algorithms for finding and counting patterns in host graphs that  work well especially when the host is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We discover algorithms that are (1) strictly  better than the brute-force algorithm, (2) strictly better than the best-known algorithms  when the host graph is sparse, (3) close to the best-known algorithms when the host graphs  are dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Our algorithms are based on two new structural graph parameters – the matched  treedepth and matched treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (See 19 and 21 for formal definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We show that they  can be used to obtain improved running times for algorithms that use constant space and  polynomial space respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Our algorithms are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In the table, the  parameter m is the number of edges in the host graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We denote using mtw the matched  treewidth of the pattern and using mtd the matched treedepth of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The notation  �O hides factors that are poly-logarithmic in the input (the host graph) size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We now explain the relevance of our new parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' state our algorithms, the relationships  between various graph parameters, and some structural characterizations that we prove in  this paper in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='1  Treedepth and matched treedepth  The matched treedepth of a graph is closely related to its treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The constant space  algorithm based on treedepth is essentially an divide-and-conquer algorithm over a elim-  ination tree of the pattern graph that executes a brute-force search over each root-to-leaf  path in the elimination tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, it runs in time O(nd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We exploit the fact that the  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  5  Pattern  Type  Problem  Time  Space  Remarks  Ck  Subgraph  Counting  �O(m3)  O(1)  k ⩽ 11  Pk  Subgraph  Counting  �O(m2)  �O(m2)  k ⩽ 10  Ck  Subgraph  Counting  �O(m2)  �O(m2)  k ⩽ 9  Any  Homomorphism  Counting  �O(m⌈mtd/2⌉)  O(1)  Any  Homomorphism  Counting  �O(m⌈(mtw+1)/2⌉)  �O(m⌈(mtw+1)/2⌉)  C6  Induced subgraph  Detection  �O(m2)  �O(m2)  Pk  Induced subgraph  Detection  �O(m⌈(k−2)/2⌉)  �O(m⌈(k−2)/2⌉)  Table 1 Pattern counting and detection algorithms for sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  elimination tree is matched, which forces an additional constraint that the vertices in each  root-to-leaf path has to be covered by a matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This allows the brute-force part of the  algorithm to discover all d vertices on the path using only d/2 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The central algorithm  that we use to obtain constant space algorithms is given below:  ▶ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph with mtd(G) = d, then given an m-edge graph H as input, we  can count the number of homomorphisms from G to H in �O(m⌈d/2⌉)-time and constant space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  It is well-known that the number of G-subgraphs, for any G, can be expressed as a linear  combination of the number of homomorphisms from a related set of graphs called the spasm  of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The spasm of G contains exactly all graphs that can be obtained by iteratively merging  the independent sets in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (See 26 for formal definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Although the treedepth of the  spasm of C11 is bounded by 6, however, the matched treedepth is not necessarily bounded  by the treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We analyze all graphs in the spasm of C10 and C11 (there are 501 such  graphs) and show that the matched treedepth of each graph is at most 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This yields the  following algorithm:  ▶ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge graph H as input, we can count the number of Ck, where  k ⩽ 11, as subgraphs in �O(m3)-time and constant space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  For comparison, the brute-force algorithm takes O(m6)-time and constant space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' and the  treedepth based algorithm takes O(n6)-time and constant space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  As seen from the proof of our algorithm for counting C11, the spasm of a pattern can contain  a large number of graphs even for relatively small patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, it would be nice  to have theorems that upper-bound the matched treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Unfortunately, the property  mtd(G) ⩽ k is not even subgraph-closed unlike treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For example, it can be proved  that mtd(K4 − e) = 3 but mtd(C4) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, interesting structural observations can still  be made for matched treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The following is a theorem that upper-bounds matched  treedepth in terms of treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any graph G, mtd(G) ⩽ 2 · td(G) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Theorem 3 implies that our constant-space algorithms from Theorem 1 for counting homo-  morphisms are asymptotically faster for all patterns, where the inputs are sparse host graphs,  when compared to the treedepth-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The following theorem shows that the time complexity for counting homomorphisms of a  pattern is lower-bounded by the time complexity for counting all of its induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph and G′ is a connected, induced subgraph of G, then:  6  Finding and Counting Patterns in Sparse Graphs  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' mtd(G′) ⩽ mtd(G) if mtd(G) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' mtd(G′) ⩽ mtd(G) + 1 if mtd(G) is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  In light of the importance of matched treedepth, it becomes crucial that we understand this  structural parameter as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The graphs of treedepth 2 are exactly the class  of star graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is also the class of graphs with matched treedepth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, for the  graph C4, we have td(C4) = 3 and mtd(C4) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So it is interesting to know what are exactly  the graphs where treedepth and matched treedepth coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The following theorem should  be viewed as giving us a preliminary understanding of the relationship between these two  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph such that td(G) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then mtd(G) = 3 if and only if G is  (C4, P6, T3,3)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The graph T3,3 is the (3, 3) tadpole graph (See Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='2  Treewidth and matched treewidth  The treewidth-based dynamic programming algorithm of Díaz, Serna, and Thilikos [6] can  be strengthened to output an arithmetic circuit that computes the homomorphism polynomial  for the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' An arithmetic circuit is a directed acyclic graph where each internal node  is labeled + or ×, each leaf is labeled by a variable or a field constant, and there is a  designated output node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Such a graph computes a polynomial over the underlying field in  a natural fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We find that by using a dynamic programming algorithm over matched  tree decompositions, we can improve the size of the arithmetic circuit for sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Our central theorem is given below:  ▶ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph with mtw(G) = t, then given an m-edge host graph H as  input, we can construct an arithmetic circuit computing the homomorphism polynomial from  G to H in time �O(m⌈(t+1)/2⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  For graphs where matched treewidth and treewidth coincide, the running time for counting  homomorphisms is a quadratic improvement on the algorithm by Díaz, Serna, and Thilikos  [6] for sparse graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, this is also the best possible improvement one can hope  to get without improving upon the algorithm by Díaz, Serna and Thilikos [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' What is the  worst case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The following theorem implies that the resulting algorithm cannot be worse on  sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any graph G, we have mtw(G) ⩽ 2 · tw(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Unfortunately, unlike for treewidth, the parameter mtw(G) is not monotone over the sub-  graph partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We first observe an explicit graph family with lower tw and larger mtw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Consider the complete bipartite graph Kn,n on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice that tw(Kn,n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' mtw(Kn,n) = 2n − 2 for all n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The following observation shows that there exists supergraphs of Kn,n with lower mtw than  that of Kn,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Observation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Consider the supergraph G of Kn,n such that V(G) = V(H), and there are  edges in one partition of Kn,n such that the independent set of size n becomes a path on n  vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that although mtw(Kn,n) = 2n − 2, but mtw(G) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  7  We show how we can use structural theorems about matched treewidth to prove algorithmic  upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For example, to count the number of P10 subgraphs, we only have to show  that all graphs in the spasm of P10 have low matched treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The spasm of P10 is a  large set that contains more than 300 graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Indeed, it is possible to analyze the matched  treewidth for each of these graphs individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, it would be better if we have  theorems that eliminate such tedious work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We derive some structural theorems for low values of matched treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Graphs with  matched treewidth 1 are exactly trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We also show tw(C5) = 2 and mtw(C5) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We characterize the matched treewidth of partial 2-trees using forbidden induced minors  (See Definition 24) wherever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We show that C5 is exactly the obstruction that  forces higher matched treewidth for partial 2-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any partial 2-tree G, the graph G is C5-induced-minor-free if and only if  mtw(G) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Notice that tw(G) = 2 yields O(n3)-time algorithms for counting homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Even  if mtw(G) = 3, we obtain �O(m2)-time algorithms for counting homomorphisms which is  an improvement for sparse graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Does all treewidth 2 graphs have matched treewidth  at most 3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The graph X in Figure 1 has treewidth 2 and matched treewidth 4 (See  Observation 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In fact, we can prove that X is exactly the obstruction that forces treewidth  2 graphs to have matched treewidth 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  u0  u1  u2  u3  u4  u5  u′  1  u′  3  u′  5  Figure 1 The graph X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any partial 2-tree G, the graph G is X-induced-minor-free if and only if  mtw(G) ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  This theorem implies that all X-induced-minor-free, treewidth 2 patterns have �O(m2)-time  homomorphism counting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is an improvement for sparse host graph even  over the fast matrix multiplication based algorithm given by Curticapean, Dell, and Marx  [9] for counting homomorphisms from treewidth 2 graphs that runs in O(nω)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since  the spasm of P10 does not contain any treewidth 4 graph or graph with an X-induced minor,  we can show that there is an �O(m2)-time algorithm for counting subgraph isormophisms of  all paths on at most 10 vertices by showing that all treewidth 3 graphs in the spasm of P10  has matched treewidth 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' There are only 18 such graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Analyzing their matched treewidth  yields the following theorem:  8  Finding and Counting Patterns in Sparse Graphs  ▶ Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge graph H as input, we can count the number of Pk subgraphs,  where k ⩽ 10, in �O(m2)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  To the best of our knowledge, the best known path counting algorithms take Ω(n4) time  for paths on 10 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, our algorithm is a significant improvement for sparse  host graphs and no worse than the best known algorithm for dense host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' An easy  corollary of the proof of this result is given below:  ▶ Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge graph H as input, we can count the number of cycles of  length at most 9 in �O(m2)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  These cycle counting algorithms are an improvement on sparse graphs over the O(nω)-time  algorithms for cycles of length at most 8 given by Alon, Yuster and Zwick [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We also show how to use our improved homomorphism polynomial construction algorithm  to speed up detection of induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In particular, we show the following:  ▶ Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge host graph as input, we can find an induced C6 or report  that none exists in �O(m2)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  This algorithm is no worse than the O(n4) time algorithm that can be derived using the  techniques by Bläser, Komarath, and Sreenivasaiah [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For sparse graphs, our algorithm  provides a quadratic improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We also show the following:  ▶ Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge host graph as input, we can find an induced Pk or report  that none exists in �O(m(k−2)/2)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  This is also a quadratic improvement over the O(nk−2) time algorithm given by Bläser, Ko-  marath, and Sreenivasaiah [2] when the host graph is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' These algorithms are obtained  by analyzing the matched treewidth and the automorphism structure of a set of graphs de-  rived from the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  From a technical standpoint, we see our algorithms as a natural combination of pattern  detection and counting algorithms that work well on sparse host graph such as the �O(m) al-  gorithm for counting k-walks, the �O(mk/2) algorithm for counting k-cliques, and �O(m(k−1)/2)  time algorithm for detecting induced Kk−e by Vassilevska [19] and insights that improve the  running-time on dense graphs by exploiting structural parameters treedepth and treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We do not make use of fast matrix multiplication in any of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Such algorithms,  called combinatorial algorithms, are also of general interest to the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='3  Related work  Algorithms for counting induced subgraphs are related to the problems that we consider but  we do not consider any algorithms for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This problem seems to be much harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It is  conjectured by Floderus, Kowaluk, Lingas, Lundell [9] that counting induced subgraphs for  any k-vertex pattern graph is as hard as counting k-cliques for sufficiently large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Several  works have considered the parameterized complexity of counting subgraphs (See [5, 4, 18,  10, 7]) where the primary goal is to obtain a dichotomy of easy vs hard based on structural  graph parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Some works have also considered restrictions on host graphs such as d-  degeneracy [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The papers on parameterized complexity primarily chooses to focus on the  growth-rate of the exponent for a family of patterns such as k-paths, k-cycles, or k-cliques  and not the exact exponent for small graphs as we do in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  9  2  Preliminaries  We consider simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We refer the reader to Douglas West’s textbook [20] for basic  definitions in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We use the following common notations for some well-known  graphs: Pk for k-vertex paths, Ck for k-cycles, Kk for k-cliques, Kk − e for k-clique with one  edge missing, Km,n for complete bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A k-star is a (k + 1)-vertex graph with  a vertex u adjacent to vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , vk and no other edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A star graph is a k-star for  some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For a graph G and S ⊆ V(G), we denote by G[S] the subgraph of G induced by the  vertices in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given two graphs G and H, a graph homomorphism from G to H is a map  φ : V(G) → V(H) such that if uv ∈ E(G), then φ(u)φ(v) ∈ E(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We denote by Hom(G, H), the set of all homomorphisms from G to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given two graphs G and H, a homomorphism polynomial is an associated  polynomial HomG[H] such that there is a one-to-one correspondence between its monomials  and the homomorphisms from G to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We define:  HomG[H] =  �  φ∈Hom(G,H)  �  u∈V(G)  yφ(u)  �  {u,v}∈E(G)  x{φ(u),φ(v)}  Note that H has a subgraph isomorphic to G if and only if PG has a multilinear monomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We say that a graph G′ is (G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , Gm)-free if no induced subgraph of G′ is isomorphic to  Gi for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We denote the complement of a graph G by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We have V(G) = V(G) and the  edges of G are exactly the non-edges of G and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We assume that all pattern graphs are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since our primary algorithms are all based  on counting homomorphisms, this does not lose generality as the number of homomorph-  isms from a disconnected pattern is just the product of the number of homomorphisms from  its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' An elimination tree (T, r) of a connected graph G is a tree rooted at r ∈ V(G),  where the sub-trees of r are recursively elimination trees of the connected components of the  graph G \\ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The elimination tree of an empty (no vertices or edges) graph is the empty tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The depth of an elimination tree (T, r) is defined as the maximum number of vertices over all  root-to-leaf paths in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The treedepth of a graph G, denoted td(G), is the minimum depth  among all possible elimination trees of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Intuitively, treedepth measures the closeness of a given graph to star graphs which are  exactly the connected graphs having treedepth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We introduce a related notion called  matched treedepth that seems to be helpful when designing algorithms for finding or count-  ing patterns in sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A matched elimination tree for a graph G is an elimination tree such that  the following conditions are true for any root-to-leaf path (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , vk):  If k is even, then v1v2, v3v4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , vk−1vk is a matching in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  If k is odd, then there is some i such that E′ = {v1v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , vi−1vi, vivi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , vk−1vk} and  E′ ⊆ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We have that E′ \\ {vi−1vi, vivi+1} is a matching on (k − 3)/2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The matched treedepth of a graph G, denoted mtd(G), is the minimum depth among all pos-  sible matched elimination trees of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  10  Finding and Counting Patterns in Sparse Graphs  The matched treedepth is always finite (See Proposition 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A tree decomposition of a graph G is a pair (T, B(t)t∈T) where T is a tree  and B(t), called a bag, is a collection of subset of vertices of G corresponding to every node  t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (Connectivity Property) For all v ∈ V(G), there is a node t ∈ V(T) such that v ∈ B(t) and  all such nodes t that contain v form a connected component in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (Edge Property) For all e ∈ E(G), there is a node t ∈ V(T) such that e ⊆ B(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The width of a tree decompostion (T, B) is defined as the maximum bag size minus one, that is,  maxt∈T |B(t) − 1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The treewidth of a graph G, tw(G), is the minimum possible width among  all possible tree decompositions of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Intuitively, treewidth measures the closeness of the given graph to trees which are exactly  the graphs with treewidth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We introduce a related notion, called matched treewidth,  closely related to treewidth, that seems to be useful for designing dynamic programming  algorithms over sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A matched tree decomposition for a graph G is a tree decomposition where  for every bag in the tree decomposition, the subgraph of G induced by the vertices in that bag  has either a perfect matching or a matching where exactly one vertex v in the bag is unmatched  and v is adjacent to some vertex in the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We call such bags matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The matched  treewidth of a graph G, mtw(G), is the minimum possible width among all possible matched  tree decompositions of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Matched treewidth is finite for all graphs (See Proposition 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is not trivial unlike  treewidth because a single bag tree decomposition that contains all the vertices in the graph  need not be matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We call a tree decomposition reduced if no bag B is a subset of another bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given any tree  decomposition T, we can obtain a reduced tree decomposition T ′ such that the width of T ′  is at most the width of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Moreover, all bags in T ′ are also bags in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This implies that if T  is matched, then T ′ is matched as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  There are several equivalent characterizations for treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Below, we state the ones that  we use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A k-tree is a graph formed by starting with a (k + 1)-clique and repeatedly  adding a vertex connected to exactly k vertices of the existing (k+1)-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A partial of a graph  G is a graph obtained by deleting edges from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The set of all graphs with treewidth at most k  is exactly the class of partial k-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We can construct a standard tree decomposition T for any k-tree as follows: The root bag of  T contains the vertices in the initial (k + 1)-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let S be a k-sized subset of this clique  such that a new vertex v is added to the k-tree by connecting it to all vertices in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then,  we add a sub-tree to T that will be a standard tree decomposition of the k-tree constructed  using S ∪ {v} as the starting (k + 1)-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  A chordal completion of a graph G is a super-graph G′ of G such that G′ has no induced  cycles of length more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A chordal completion that minimizes the size of the largest  clique is called minimum chordal completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The treewidth of a graph G is the size of the  largest clique in its minimum chordal completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  11  Two paths P1 and P2 from u to v are internally disjoint if and only if P1 and P2 do not have  any common internal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  A graph G is 2-connected or biconnected, if for any x ∈ V(G), G−x is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Equivalently,  for any two vertices in G, there are at least 2 internally disjoint paths in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A series-parallel graph is a triple (G, s, t) where s and t are vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  This class is recursively defined as follows:  An edge {s, t} is a series-parallel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (series composition) If (G1, s1, t1) and (G2, s2, t2) are series-parallel graphs, then the graph  obtained by identifiying s2 with t1 is also series-parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (parallel composition) If (G1, s1, t1) and (G2, s2, t2) are series-parallel graphs, then the  graph obtained by identifiying s1 with s2 and t1 with t2 is also series-parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  A graph has treewidth at most 2 is if and only if all of its biconnected components are series-  parallel graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A graph G is said to be a minor of a graph G′ if G can be obtained from  G′ either by deleting edges/vertices, or by contracting the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (The operation of contraction  merges two adjacent vertices u and v in the graph and removes the edge (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=') If G is obtained  from an induced subgraph of G′ by contracting the edges, then it is said to be an induced minor  of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  A graph G is called G′-induced-minor-free (G′-minor-free) if G′ is not an induced minor  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' minor) of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  An edge subdivision is an operation which deletes the edge (u, v) and adds a new vertex  w and the edges (u, w) and (w, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A graph G′ obtained from G by a sequence of edge  subdivisions is said to be a subdivision of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='1  Representation of graphs  We assume the following time complexities for basic graph operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Any representation  that satisfies these is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Given u and v, it can be checked in �O(1)-time whether uv is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Iterating over all the edges xy ∈ E(H) ordered by x can be done in �O(m)-time, where m  is the number of edges in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  These requirements are satisfied by the following adjacency-list representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' To represent  a graph H, we store a red-black tree T that contains non-isolated vertices of H where vertices  are ordered according to their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Consider a node in this tree that corresponds to a  vertex u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This node stores another red-black tree Tu that stores all neighbors of u in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Now,  to check whether uv is an edge or not, we perform a lookup for u in T followed by a lookup  for v in Tu if u was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We can iterate over all edges xy ordered by x by performing an  inorder traversal of T where for each node u, we perform an inorder traversal of Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  If the pattern does not contain any isolated vertices, then we can ignore isolated vertices in  the host graph as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If the pattern is G = G′ + v, where v is an isolated vertex and G′ is  any graph, then the number of homomorphisms from G to H is obtained by multiplying the  12  Finding and Counting Patterns in Sparse Graphs  number of homomorphisms from G′ to H by n, where n is the number of vertices in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This  can be calculated by simply storing the number of vertices of H in the data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  3  Matched treedepth  In this section, we introduce algorithms that count homomorphisms and subgraphs effi-  ciently in constant space on sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The central theorem in this section is given  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph with mtd(G) = d, then given an m-edge graph H as input, we  can count the number of homomorphisms from G to H in �O(m⌈d/2⌉)-time and constant space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The algorithm is given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We can compute the result needed by calling  COUNT-HOM-MTD(G, E, r, H, φ), where E is an elimination tree for G of depth d, r is the  root vertex in E, and φ is the empty homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For simplicity of presentation, we  assume that each root-to-leaf path in E has an even number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Odd number of  vertices in a root-to-leaf path is handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We assume that the host graph H is represented using a symmetric adjacency list represent-  ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is mainly to ensure that we can iterate over all edges xy in H ordered by x in  Line 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  First, we prove that the algorithm is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We claim that the call COUNT-HOM-MTD  (G, E, v, H, σ) where the parameters are as specified in the algorithm returns the number of  homomorphisms from Gv to H that extends σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is proved by an induction on the height  of v in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since v is a top node, the base case is when the height is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In this case, Gv is a  star graph and it is easy to see that the algorithm works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We now prove the inductive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The variable t computes the final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Denote by su,x for vertex x in H the number of  homomorphisms from Gu to H that extends τ = σ∪{v �→ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice that since uv ∈ E(G), to  extend τ, the vertex u must be mapped to some y such that xy ∈ E(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, iterating  over all such y is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice that we can compute t as �  τ  �  u su,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, this  would need storing |V(G)||V(H)| variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By iterating over the edges of H ordered by x, we  can afford to reuse a single su for different x instead of keeping a separate su,x for each x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Now, we show that su correctly computes su,x once the main loop finishes with an x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By the  inductive hypothesis, the variable cw is the number of homomorphisms from Gw to H that  extends σ′ = σ ∪ {v �→ x, u �→ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, we have su,x = �  y  �  w cw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Line 12 correctly  computes this into su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Line 15 correctly updates t once an x is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Finally, we reset su  to 0 before processing the next x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Now, we prove the running-time and space usage of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice that the depth of  the recursion is bounded by the depth of the elimination tree E and each level of recursion  stores only constantly many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, the space usage is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The main  loop in Line 3 runs for 2m iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The inner-loops only have a constant number of  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, the recursive calls are made only O(m) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We process two levels  of the elimination tree in a level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, the total running-time is given by t(d) ⩽  O(m)t(d − 2) + �O(1) = �O(md/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  13  Algorithm 1 COUNT-HOM-MTD(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ)  Require: G - The pattern graph  Require: E - Matched elimination tree for G  Require: v - A top vertex in E  Require: H - The host graph  Require: σ - A partial homomorphism from the ancestors of v to H  t ← 0  su ← 0 for all children u of v  for all edges xy ∈ E(H) ordered by x do  for all children u of v in E do  σ′ ← σ ∪ {v �→ x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' u �→ y}  if σ′ is an invalid homomorphism then  continue  end if  for all children w of u do  cw ← COUNT-HOM-MTD(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ′)  end for  su ← su + �  w cw  end for  if xy is the last edge on x then  t ← t + �  u su  su ← 0 for all u  end if  end for  return t  ◀  A fundamental question regarding matched treedepth is whether it is finite for all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It  is, as the following proposition shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any graph G, we have mtd(G) is at most 1 + the number of vertices in  the smallest maximal matching in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We partition V(G) into a maximal matching M = (v1w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , vmwm) and an inde-  pendent set {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , uk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A matched elimination tree T for G can be constructed as follows:  Put the vertices v1, w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , vm, wm on a root-to-leaf path in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let us call this path  the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For each vertex ui, make the lowest vertex in the spine adjacent to ui in G its  parent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It is easy to see that T is a matched elimination tree and its depth is at most 1 +  the number of vertices in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  Although, Proposition 25 proves an upper-bound for matched treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It is not very useful  from an algorithmic perspective as it is easy to see that there is an O(mk+1)-time, constant  space algorithm for counting patterns with maximal matchings of k edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Any smallest  maximal matching in C11, for example, has 4 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, the proof of Theorem 2 given  below shows that we can do better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The algorithm is based on the well-known technique of  expressing subgraph count as a linear combination of homomorphism counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  14  Finding and Counting Patterns in Sparse Graphs  ▶ Definition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let I be the set of all the independent sets in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For some I ∈  I, Merge(G, I) is the graph obtained by merging the vertices of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then  Spasm(G) = {G} ∪  �  I∈I  Spasm(Merge(G, I))  For any pattern graph G, it turns out that the number of subgraph isomorphisms from G to  a host graph H are just the linear combination of all possible graph homomorphisms from  SpasmG to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' That is, there exists constants αG′ ∈ Q such that:  SubG[H] =  �  G′  αG′HomG′[H]  (1)  where G′ ranges over all graphs in Spasm(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This equation is used to count subgraphs by  many authors (See for example [1, 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge graph H as input, we can count the number of Ck, where  k ⩽ 11, as subgraphs in �O(m3)-time and constant space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We analyzed all graphs in Spasm(C11) and Spasm(C10) and concluded that each of  them has matched treedepth at most 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A pdf that contains all these graphs and their corres-  ponding matched elimination trees can be found at (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='com/anonymous1203/Spasm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  For seeing that the stated algorithms exist for smaller cycles, observe that Spasm(Ck) ⊂  Spasm(Ck+2) for k ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  The class of treedepth 2 graphs is exactly the star graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' These graphs have an O(n2)-  time, constant space, treedepth-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This class of graphs is also the graphs of  matched treedepth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It is easy to see that the number of homomorphisms from star graphs  can also be counted in O(m)-time and constant space using the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let  d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , dn be the degrees of vertices in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, the count is given by the expression  �  i dk  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that this is an asymptotic improvement for sparse graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We now show that  this asymptotic improvements exists for all patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' First, we need a restricted form of  elimination trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' An elimination tree T is connected if for every node u in T and a child v of  u in T, u is adjacent to some node in Tv  Now, we show that connected elimination trees are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Every connected graph G has a connected elimination tree of depth td(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We show how to construct a connected elimination tree T ′ from an elimination tree  T without increasing its depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let T ′ = T initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose there exists some node u in T ′  that violates the property (If not, we are done).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, there exists a child v of u in T ′ such  that there is no edge in G between u and any node in T ′  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let w be a node in T ′  v such that  w is adjacent to some proper ancestor x of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Such a w must exist because G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Now, in T ′, remove the subtree T ′  v and make it a subtree of the node x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Repeat this process  until all nodes in T ′ satisfy the required property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This process must terminate since at each  step, we reduce the number of nodes violating the property by at least one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This process  cannot increase the depth of T ′ because the only modification is to move a subtree upwards  to be a subtree of a proper ancestor of its parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  15  ▶ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any graph G, mtd(G) ⩽ 2 · td(G) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We start with a connected elimination tree T with depth d of G and show how to  construct a matched elimination tree of G from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We use induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For d = 2, the  tree is already matched and has depth 2 = 2 · 2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Our construction is iterative and top-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Each iteration ensures that the current top-most  node in the elimination tree is adjacent, in the graph G, to all its children and that the  elimination tree is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Each iteration consists of two phases iteratively executed until the desired property is satis-  fied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The first phase ensures that the root node r is adjacent in G to all its children in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If r  has a child v that is not adjacent in G to r, then since T is connected, there is some node w  in Tv such that rw ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, we make w a child of r in T, delete w from Tv, and make  all children of w children of parent of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The resulting tree is an elimination tree of depth  at most d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' After this phase, the root node is adjacent in G to all its children, the tree’s  depth has increased by at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, it may not be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  In the second phase, we use the construction of Lemma 28 to make the tree connected  without increasing the depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Observe that the construction will keep the existing children  of root as is and may add new chlidren to r that are not adjacent to r in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose a new  node u was added as a new child to r in this second phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The height of subtree rooted at u  is at most d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, the tree (r, Tu) obtained by attaching u to r has height at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We can now execute phase 1 on all the trees (r, Tu) for all such u without increasing depth  beyond d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This process must eventually terminate as we add at least one new child to  the root every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  At the end of the iteration, consider a grandchild x of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If it is a leaf, since the tree is  connected, x must be adjacent in G to its parent in T and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Otherwise, the  subtree Tx is a tree of depth at least 2 and at most d − 1 that is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So by the  induction hypothesis, we obtain that Tx is a matched elimination tree of depth at most  2(d − 1) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This means that the original tree is converted to a matched elimination tree of  depth at most 2 + 2(d − 1) − 2 = 2d − 2 as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  ▶ Corollary 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose G has treedepth d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge host graph as input, we can  count the number of homomorphisms from G to the host graph in �O(md−1)-time and constant  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Notice that that above algorithm is asymptotically better than the treedepth based algorithm  for all patterns on sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We now prove that the matched treedepth of an induced subgraph cannot be much larger  compared to that of the graph containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In other words, we can obtain lower bounds  on the matched treedepth of a graph by obtaining lower bounds on the matched treedepth  of any of its induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We prove the theorem for connected subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' But note  that the matched treedepth of a disconnected graph is the maximum of its connected com-  ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph and G′ is a connected, induced subgraph of G, then:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' mtd(G′) ⩽ mtd(G) if mtd(G) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' mtd(G′) ⩽ mtd(G) + 1 if mtd(G) is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  16  Finding and Counting Patterns in Sparse Graphs  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We start with a matched elimination tree T of even (The odd case is similar) depth  d for G and construct a matched elimination tree for G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' First, we delete all nodes in the  elimination tree that are in G but not in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If a node u in T has parent v that was deleted,  then we make u a child of the closest ancestor of v that is still in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The final forest thus  obtained is a tree T ′ because G′ is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We assume without loss of generality that T ′  is a connected elimination tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Suppose r′ is the root of T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We will now modify T ′ into a matched elimination tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We will  first analyze paths in T ′ that correspond to even length root-to-leaf paths in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If T ′ is not  matched on this path, then there exists a node u closest to r′ such that u is not connected in  G′ to a child v in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is only possible if u was either matched to a child w of u in T or  its parent w in T and w is not in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, we have distT ′(r′, u) + depth(T ′  v) < depth(T)  and this means we can afford to increase the length of any path that passes through edge uv  by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since T ′ is connected, we can now apply the transformation in the proof of Theorem 3  to match u with one of its descendants in T ′  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The depth is still at most d because this  transformation increases the depth by at most 1 In effect, the increase in depth in this branch  of the tree by pulling up a descendant is compensated by the fact that the unmatched vertex  was introduced by deleting a vertex in this branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We can iteratively apply the above construction to make T ′ a matched elimination tree while  keeping T ′ connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, applying the above transformation may introduce a node u  in T ′ that is not matched to a child v because v’s parent w was pulled up in the tree to match  with some other vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Such u also satisfy the inequality distT ′(r′, u)+depth(T ′  v) < depth(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Any path in the tree T ′ that passed through both u and v earlier had to pass through  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' But, the fact that w was pulled up implies that these paths had length strictly less than  depth(T) by the argument in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' And shifting w to a position earlier in  the path cannot increase its length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  For root-to-leaf paths in T of odd length, there might be a matched P3 on vertices uvw such  that v is not in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In this case too, by the same argument, the transformations increase the  depth to at most d+1 (In this case, we may have to pull up two distinct vertices for matching  u and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' When d is even, increasing the length of such paths by 1 does not increase the  depth of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' When d is odd, increasing the length of such paths by 1 increases the  depth by atmost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  C4  P6  T3,3  Figure 2 Forbidden graphs for mtd(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=') ⩽ 3  Theorem 3 implies that all graphs G with td(G) = 3 also have mtd(G) ⩽ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It is easy to see  that C4, P6, and T3,3 given in Figure 2 have td(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=') = 3 and mtd(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=') = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By Theorem 4, it is  also possible that some super graph of these graphs have mtd(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=') = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, in the below  theorem we show that this cannot happen and that the graphs G with td(G) = mtd(G) = 3  are exactly the graphs where we forbid these graphs as induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph such that td(G) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then mtd(G) = 3 if and only if G is  (C4, P6, T3,3)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (Proof for the “if” direction) Let G be a connected graph with td(G) = 3 such that  G is (C4, P6, T3,3)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We will construct a matched elimination tree for G of depth 3 from  its elimination tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Consider an elimination tree T of G with depth 3, rooted at some vertex r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' vk}  be the children of r in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice that for any child c of vi (for any i), if c is not adjacent to  vi in G then it must be adjacent to r in G (else, G is not connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We make all such c,  which are not adjacent to vi in G, a child of r instead of vi, in the elimination tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note  that this neither violates any property of elimination tree nor does it increase the depth of  elimination tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Also, after doing this modification, we can further assume that all the  children of vi in T are adjacent to vi in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Now, if rvi ∈ E(G) for all i ∈ [k], then T itself is a matched elimination tree and we are  done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose there exists an i such that rvi /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since G is connected, a path must  exists from r to vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since T has depth 3, this path must be a P3 and therefore r and vi has  at least one common neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Any common neighbour u of r and vi must be a child of vi  in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Moreover, if r and vi have two common neighbors, say u1 and u2, then ru1vu2r is an  induced C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So r and vi must have exactly one common neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We now consider the  following cases:  r has exactly one child v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ck be the children of v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Recall that rv1 /∈ E(G), and they have exactly  one common neighbor in G, say c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, we construct a matched elimination tree T ′ as  follows: The root of T ′ is c1, vertices r and v1 become the children of c1, the children of  v1 in T ′ are {c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ck} The tree T ′ is a valid matched elimination tree of depth 3 for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  r has multiple children, say {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' vℓ} for some ℓ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We split this into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (r is not adjacent to exactly one of its children, say v1)  The vertex v1 is not a leaf in T since G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If v1 has exactly one child,  then, since it must be a common neighbor c of r and v1, we swap c and v1, and this  does not increase the depth of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Else, suppose v1 has at least two children, namely  {c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' ck} (where the common neighbor with r is c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We now argue that all the  other children {v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' vl} of r must be leaves in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If not, then say v2 has a child c3,  then we encounter an induced P6, namely (c2, v1, c1, r, v2, c3) (if rc3 ̸∈ E(G)) or an  induced T3,3 (if rc3 ∈ E(G)), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So, {v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' vl} are leaf vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Now, we can convert T to T ′ by rooting it at c1 instead of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The children of c1 in T ′  are precisely r and v1, the children of r are precisely all the leaves {v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' vl}, and the  children of v1 are all the same except c1, that is, {c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' ck}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It is easy to see that T ′ is  a matched elimination tree of depth 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (r is not adjacent to at least two of its children, say v1 and v2 and maybe more)  Due to the connectivity of G, the vertices v1, v2 cannot be leaves in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If v1 has exactly  one child c, then c must be a common neighbor of r and v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We swap v1 and c in  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This will either fall into the previous case or we can assume v1 has more than one  child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let c1, c2 be two children of v1, where c1 is the common neighbor of r and v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Then we get an induced P6, namely, (c2, v1, c1, r, c, v2), where c is a common neighbor  of r and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (Such a c must exist as (r, v2 /∈ E(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  18  Finding and Counting Patterns in Sparse Graphs  (Proof for the “only if” direction) Suppose G is a connected graph with td(G) = 3 such that  mtd(G) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We will show that G cannot contain C4, P6, or T3,3 as induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We  will prove that any connected induced subgraph G′ of G has matched treedepth at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let T be the matched elimination tree of G of depth 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let r be the root of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If r is not in  G′, then since G′ is connected, all vertices in G′ must come from a single sub-tree of r in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  In that case, that sub-tree witnesses a matched elimination tree of depth at most 2 for G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If  r is in G′, then G′ may be obtained by deleting some level 1 and level 2 vertices in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We  will construct a matched elimination tree for G′ from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If a level 2 vertex is not in G′, we  simply delete it from T as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If a level 1 vertex v is not in G′, then for each u ∈ V(G′) that  is also a child of v in T, there must be an edge ru in G′ since G′ is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, we  make u a child of r in the matched elimination tree for G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  By Theorem 1 and Theorem 3, we can conclude that every pattern G with td(G) = 3 has  an �O(m2) algorithm for counting homomorphisms from G to m-edge sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Therefore, the presence of C4, P6, or T3,3 as induced subgraphs in G does not affect the  running-time of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If we consider patterns G with td(G) = 4, then we have  examples such as P8 where td(P8) = 4 and mtd(P8) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, we can obtain only an  �O(m3) algorithm for counting homomorphisms from P8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It would be interesting to prove  theorems similar to Theorem 5 for higher treedepth, say 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' But, we do not even know the  exact set of forbidden induced subgraphs for treedepth 4 [12] so this seems difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  4  Matched treewidth  In this section, we introduce algorithms that count homomorphisms and subgraphs effi-  ciently on sparse host graphs by using polynomial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The central theorem in this section  is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph with mtw(G) = t, then given an m-edge host graph H as  input, we can construct an arithmetic circuit computing the homomorphism polynomial from  G to H in time �O(m⌈(t+1)/2⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let T be a matched tree decomposition of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We fix an arbitrary assignment of edges  and vertices of G to bags of T such that each vertex and edge is assigned to exactly one bag  that contains it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let B be a bag in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We consider the matched matching M in B as a sequence  of edges (a1b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , akbk) by arbitrarily ordering them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given edges u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk in the  host graph H such that σ(ai) = ui and σ(bi) = vi is a valid partial homomorphism on the  vertices of B, we define the following monomials:  EdgeMon(B, u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk) =  �  e  xσ(e)  VertexMon(B, u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk) =  �  v  yσ(v)  where e and v range over all edges and vertices assigned to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We also define Mon(B, u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' ,  ukvk) as the product of EdgeMon(B, u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk) and VertexMon(B, u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let B and B′ be bags in T such that B′ is the parent of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We arbitrarily order the vertices  to obtain a sequence X(B ∩ B′) of the vertices in B ∩ B′ (We may assume that the vertices  are labeled from [|V(G)|] and choose increasing order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We define MapGate(B, B′, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' ,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  19  uk) where X(B ∩ B′) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ak) as the named gate which corresponds to the partial  homomorphism σ(ai) = ui, 1 ⩽ i ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Algorithm 2 constructs the required circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice that all operations within the loops in  Line 9, 20, 36 runs in O(log n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The loops itself executes for O(m⌈(t+1)/2⌉) iterations  since any matched matching on t + 1 vertices contains at most ⌈(t + 1)/2⌉ edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The loops  in Line 2, 7, and 18, executes for O(1) iterations since the pattern graph G is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice  that we iterate over pairs uv that correspond to edges instead of edges {u, v} because the  order determines the homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  When a hash table lookup for a MapGate(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=') gate fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We add it to that table with an initial  value if the gate occurred on the left-hand side and replace it with 0 if it occurs on the right-  hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We now prove the correctness of the circuit using parse trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice that the only  + gates in the circuit are MapGate(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For this proof, we can think of r as MapGate(R, φ)  where R is the root bag in T with an empty bag as parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, the monomials of  the final polynomial correspond to the choices at these gates while building the parse tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  For some bag B with parent B′ in T and vertices x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , xk in G, the inputs to the gate  MapGate(B, B′, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , xk) correspond to valid partial homomorphisms on the vertices of B  that map ai to xi for all ai ∈ X(B ∩ B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We identify these gates with the bag B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Now,  given a homomorphism σ from G to H, we build its parse tree by choosing for each such  gate, the restriction of σ to the vertices in that bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We have to prove that this choice will  be consistent with the images x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , xk at each gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Indeed, this is vacuously true at the  root gate (the list is empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For an arbitrary a ∈ V(G), consider the topmost bag B where  a first appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In the partial homomorphism chosen at B, we can freely fix the image of the  vertex a to that of in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By construction, this choice is then propagated when moving to its  child gates corresponding to bags where a is present (See Lines 15, 30, and Lines 42 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Also,  if a child bag of B does not contain the vertex a, then the vertex a will not appear in that  subtree too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It is easy to see that this parse tree computes the correct monomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since the  circuit is monotone, this proves that all monomials that correspond to homomorphisms are  present in the polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For the other direction, we have to argue that only monomials  that correspond to homomorphisms are present in the polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Indeed, any parse tree  corresponds to a sequence of choices of partial homomorphisms at each gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We argue  that these partial homomorphisms must be consistent with each other and therefore can  be combined into a valid homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is because T is a tree decomposition and  therefore any a ∈ V(G) must occur in a (connected) subtree of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The construction of the  circuit ensures that once the image of vertex a is determined in a parse tree, it is correctly  propagated to all partial homomorphisms where a is a member of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Furthermore,  since each vertex and edge in G are assigned to a unique bag, their images appear exactly  once in the monomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  ▶ Remark 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' An anonymous reviewer on an earlier draft of this paper commented that  the graph parameter generalized hypertree width, denoted ghw, may yield similar running-  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Indeed, we have verified that �O(mghw)-time algorithms exist and that ghw ⩽ ⌈(mtw +  1)/2⌉ and the running-times coincide in the worst-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' To the best of our knowledge, the  parameter ghw has not been analyzed in the context of fast algorithms for small patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It  is primarily used for designing efficient algorithms on hypergraphs with high treewidth and  low ghw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We believe analyzing mtw is also useful since it is sometimes more fine-grained  than ghw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=', the class ghw = k may contain graphs from both classes mtw = 2k − 2 and  20  Finding and Counting Patterns in Sparse Graphs  Algorithm 2 Computing HomG[H]  1: Let T be a near-perfect tree decomposition of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2: for each bag B in T do  3:  for each child B′ of B in T do  4:  Initialize empty hash table T(B,B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  5:  end for  6: end for  7: for each non-root leaf bag B in T do  8:  Let B′ be the parent of B in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  9:  for (u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk) ∈ E(H)k do  10:  Let σ be σ(ai) = ui, σ(bi) = vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  11:  if σ is not a valid partial homomorphism from G to H then  12:  Skip this iteration  13:  end if  14:  Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , xk′ be the images of vertices in X(B ∩ B′) in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  15:  MapGate(B, B′, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , xk′) += Mon(B, u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk)  16:  end for  17: end for  18: for each non-root bag B in T in a bottom-up order do  19:  Let M = (a1b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , akbk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  20:  for (u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk) ∈ E(H)k do  21:  Let σ be σ(ai) = ui, σ(bi) = vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  22:  if σ is not a valid partial homomorphism from G to H then  23:  Skip this iteration  24:  end if  25:  Let B′ be the parent of B in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  26:  Let B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , Bs be the children of B in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  27:  Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , xk′ be the images of vertices in X(B ∩ B′) in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  28:  Let wi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , wiki be the images of vertices in X(B ∩ Bi) in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  29:  MapGate(B, B′, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , xk′) +=  30:  Mon(B, u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk) �s  i=1 MapGate(B, Bi, wi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , wiki)  31:  end for  32: end for  33: Let B be the root in T  34: Let B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , Bs be the children of B in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  35: r ← 0  36: for (u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk) ∈ E(H)k do  37:  Let σ be σ(ai) = ui, σ(bi) = vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  38:  if σ is not a valid partial homomorphism from G to H then  39:  Skip this iteration  40:  end if  41:  Let wi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , wiki be the images of vertices in X(B ∩ Bi) in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  42:  r += Mon(B, u1v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , ukvk) �s  i=1 MapGate(Bi, B, wi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , wiki)  43: end for  44: return r  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  21  mtw = 2k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  First, we make some fundamental observations about mtw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We can relate it to matched treedepth as we relate treewidth to treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Proposition 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any graph G, we have mtw(G) ⩽ mtd(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let E be the optimal matched elimination tree for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We construct a matched tree  decomposition T for G from E as follows: For each path from root to leaf in E from the  leftmost path to the rightmost path, construct bags that contain all vertices in those paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Then, join those bags into a path by adding an edge between bags B (corresponds to path  to leaf u) and B′ (corresponds to the path to leaf v) if and only if v is the next leaf in E from  u when leaves are ordered from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  Now, we prove that matched treewidth cannot be much higher than treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any graph G, we have mtw(G) ⩽ 2 · tw(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let T be a tree decomposition of G of width k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We will describe a procedure to  convert T to a matched tree decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The construction is top-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The final tree  will have the property the vertices in each non-leaf bag will induce a perfect matching in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let Br be the root bag in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The following procedure will be applied to Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Find maximal matching M in Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For each v ∈ Br \\ M such that NG(v) ⊆ Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Observe that since M is a maximal matching  NG(v) ⊆ V(M) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We delete v from Br and add a leaf bag Bv as a child of Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The  bag Bv will contain v and the vertices in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since v is adjacent to at least one vertex in  M, this bag is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For each v ∈ Br \\ M such that NG(v) ⊈ Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We can find a u such that u /∈ Br and  u ∈ NG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We choose such a u that is in a bag B that is at a minimal distance from Br  in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We add u to Br and all the bags in the path from B to Br in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We then modify  M = M ∪ {uv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Notice that each iteration in step 2 and step 3 reduces the unmatched vertices in Br by 1 and  only adds leaf bags to T that are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In addition, if Br originally had x vertices, after  this procedure it will have at most 2x vertices as we add at most one vertex corresponding  to each of the original x vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, the size of modified Br is at most 2k + 2 and  the graph induced by vertices in Br has a perfect matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Now, consider an arbitrary bag B such that all its ancestors are bags with a perfect matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let Bp be the parent of B in T and let M be the perfect matching on the vertices in Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We  now apply the following procedure on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For each v ∈ B ∩ Bp, let u be the partner of v in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If u is in B, then we match v to u  in B as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If not, then we add u to B and match v to u in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This does not violate  any properties of tree decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice that if v was added to B in step 3 of the  procedure for the root bag, then its partner u must be in B as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For v ∈ B\\Bp, we apply steps 2 and 3 in the procedure described for the root bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' These  steps only modify the bags in the subtree of T rooted at B and will only add to T leaf  22  Finding and Counting Patterns in Sparse Graphs  bags that are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Moreover, at the end of this step, the vertices in bag B induce a  graph that has a perfect matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Observe that the size of a bag B can at most double from its original size since we add at  most one vertex for each vertex originally in the bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is true for all the newly added  leaf bags as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, we have constructed a matched tree decomposition of width at  most 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  We now show an explicit family that has close to the worst-case relation between treewidth  and matched treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' mtw(Kn,n) = 2n − 2 for all n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let T be just an edge with vertex set {Kn,n \\ {a}, Kn,n \\ {a′}} for some a and a′ that  are in the same part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' One can easily verify that T is matched tree decomposition of Kn,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Thus mtw(Kn,n) ⩽ 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let T be an arbitrary matched tree decomposition of Kn,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Root T at a leaf bag, say X and  let X1 be the only child of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If X ⊆ X1, then we can delete X from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, there exists  a vertex u ∈ X of Kn,n such that u ̸∈ X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So N(u) ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Assume wlog that u ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, we  get B ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Now to match the n vertices in B, we need at least n − 1 vertices from A in the  bag X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So mtw(Kn,n) ⩾ 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  It is easy to see that tw(Kn,n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, its matched treewidth is only 3 less than the  worst case 2n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We now use the Algorithm 2 and Definition 26 to count paths in cycles in sparse host graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Instead of analyzing all graphs in Spasm(P10), we completely characterize the matched  treewidth of graphs with treewidth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This will simplify the case analysis required for prov-  ing algorithmic upper bounds for many small pattern graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='1  Matched treewidth of partial 2-trees  In this section, we study the matched treewidth of partial 2-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A summary is given in  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' From Theorem 7, we have mtw(G) ⩽ 5 when tw(G) = 2 which yields the last row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The graph Y (See Figure 4) satisfies tw(Y) = 2 and mtw(Y) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' But Y is also an induced  subgraph of Z (See Figure 5) which satisfies tw(Z) = 2 and mtw(Z) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, a  forbidden induced minor subgraph characterization is not applicable for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We now  prove the remaining two characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  mtw ⩽ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Forbidden induced minor  2  C5  3  X  4  Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  5  None.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Table 2 Matched treewidth of partial 2-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any partial 2-tree G, the graph G is C5-induced-minor-free if and only if  mtw(G) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  23  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (Proof for the “if” direction) Suppose for contradiction that there is a partial 2-tree G  such that mtw(G) = 2 and G has a C5-induced-minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let T be a matched tree decomposition  of G with width 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since C5 is an induced minor, we can obtain C5 from G by deleting  vertices and contracting edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For all v that is deleted, delete v from all bags of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Similarly,  for all edges uv that are contracted, replace u and v consistently in all the bags of T by one  of u or v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We obtain a (not necessarily matched) tree decomposition T ′ of C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Assume wlog  that T ′ is a reduced tree decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since T has width at most 2, any bag in T ′ contains  at most 3 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If a bag in T ′ has 3 vertices, then they already form a P3 since it must  have been so in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' There cannot be a bag of one vertex in T ′ because it is reduced and  C5 is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We claim that a bag of size 2 in T ′ must contain some u and v such that  uv ∈ E(C5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▷ Claim 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let u and v be two vertices in C5 that are not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, the tree  decomposition T ′ cannot contain the bag {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose for contradiction that T ′ contains the bag B = {1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The other cases are  symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Root the tree T ′ at that bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We now analyze various cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (B has one child) Since 1 and 3 are adjacent to other vertices, the child of B must contain  both 1 and 3, contradicting the fact that T ′ is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (B has more than one child) We split this case sub-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (The edges 15 and 34 are covered in distinct subtrees (Say T1 and T2) of B) Since  45 ∈ E(C5), this edge must be covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The bag that covers 45 must have a path  containing 5 to the bag covering 15 in T1 due to the connectivity of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Similarly, this  bag must have a path containing 4 to the bag covering 34 in T2 due to the connectivity  of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' But, this is impossible since B contains only 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (The edges 15 and 34 are covered in the same subtree of B) This case is split into  further sub-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (The bag B, and the bags containing 15 and 34 occur on the same path in T ′)  Suppose the path is from B to the bag containing 3 and 4 via the bag B′ containing  1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The other case is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, the bag B′ must contain 3 to maintain the  connectivity of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Now, we have B′ ⊃ B, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (The bags containing 15 and 34 has a common ancestor that is a proper descendant  of B) Let B′ be this common ancestor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This bag B′ must contain 3 as it lies on the  path from B to the bag containing 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Also, this bag B′ must contain 1 as  this lies on the path from B to the bag containing 1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' But, then B′ ⊇ B, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  This completes the proof of the "if" direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (Proof of the “only if” direction) We use proof by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose G is a counter-  example on minimum number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▷ Claim 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The graph G is 2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  24  Finding and Counting Patterns in Sparse Graphs  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose G has a cut vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By deleting v, we obtain smaller graphs G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , Gm  for some m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Any cycle in G is also a cycle in Gi for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since G is minimal, each  Gi has a matched tree decomposition of width at most 2, say Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For all i, let Bi be a bag in  Ti that contains v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Add edges between B1 and Bi for all 1 < i ⩽ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is a matched tree  decomposition for G of width 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  We take a 2-tree G′ that is a super-graph of G and has the same vertex set as G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▷ Claim 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let uv be an edge in G′ but not in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then u and v have a common neighbor  in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since G is 2-connected, there are two internally disjoint paths P = uu1 · · · ukv and  P′ = uv1 · · · vℓv in G between u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We may assume that P and P′ are induced paths  in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If k or ℓ is 1, then u and v has a common neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So we assume k and ℓ are at  least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Now PP′ is a cycle of length at least 6 and it must have a chord (otherwise, this is a  C5-induced-minor in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, there is some i and j such that ui is adjacent to vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' the  edges on P, P′, together with this chord is a K4-minor in G′, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  Let T be a standard tree decomposition of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is a matched tree decomposition for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We show that this is also a matched tree decomposition for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It is enough to show that a  set {u, v, w} which forms a triangle in G′ induces either a P3 or a triangle in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Assuming  the contradiction, we get that v (say) is not adjacent to both u and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By Claim 34, we  obtain vertices x that is a common neighbor of u and v in G, and y that is a common  neighbor of v and w in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since G′ is K4-minor-free, we have x ̸= y, wx ̸∈ E(G), xy /∈ E(G),  and uy ̸∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If uw ∈ E(G), then uwyvxu is an induced C5 in G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So  uw ̸∈ E(G), and by Claim 34, we obtain a vertex z that is a common neighbor of u and  w in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Again, since G′ is K4-minor-free, we have z ̸= x, z ̸= y, zx ̸∈ E(G), zv /∈ E(G),  and zy ̸∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then uzwyvxu is an induced C6 in G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This completes the  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  We now show that the graph X (See Figure 1) is a partial 2-tree that has matched treewidth  more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' mtw(X) ⩾ 4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Consider a reduced, matched tree decomposition T of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose for contradiction  that X has width strictly less than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Consider a leaf bag Bl in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The bag Bl must contain  u0 or u2 or u4 as all edges in X are incident on one of these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' All cases are symmetric  so wlog, we can assume Bl is a super-set of {u0, u1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Consider the tree T to be rooted at Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  As neither u0 nor u1 is pendant and T is reduced, it follows that Bl must contain at least one  other vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let Bp be the bag adjacent to Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We claim that {u0, u2} or {u0, u4} is contained  in Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If u0 /∈ Bp, then {u1, u′  1, u5, u′  5} ⊆ Bl and therefore |Bl| ⩾ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose u1 ∈ Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since  T is reduced, this means that some vertex other than u0 and u1 was in Bl but not in Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If  this vertex is u2 or u4, then |Bl| ⩾ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If it is one of the other vertices, either u2 or u4 is in  Bl and Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If u1 /∈ Bp, then Bl must contain u2 and therefore so must Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Now, we assume  wlog that both u0 and u2 are in the bag Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If Bp also contains u4, then we are done as the  size of matched bag would be at least 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let Bc be a descendant bag of Bp in T such that Bc is the root of the subtree of T that  contains u4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Again, if {u0, u2, u4} ⊆ Bc, then |Bc| ⩾ 5 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So assume wlog  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  25  that u2 /∈ Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If u0 /∈ Bc, then Bc ⊇ {u5, u′  5, u3, u′  3} and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is because these  vertices are common neighbors of u4 with u0 or common neighbors of u4 with u2 and Bc is  the root bag of the sub-tree where u4 appears in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So u0 is also in Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' But Bc also contains  u3 and u′  3 and Bc must contain at least one more vertex to match u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  We now prove that forbidding X-induced-minor (for partial 2-trees) exactly gives us the class  of graphs with mtw(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=') ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Definition 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A minimum chordal completion ˜G of some graph G is said to be special if no  independent set of size 3 in G induces a clique in ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We write smcc instead of special minimum  chordal completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Note that treewidth of a graph is same as the treewidth of its smcc (if it exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For any partial 2-tree G, the graph G is X-induced-minor-free if and only if  mtw(G) ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (Proof for the “if” direction) Let G be a graph with a matched tree decomposition U  of width 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose for contradication that G has an X-induced-minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, we can also  obtain X by contracting some edges of an induced subgraph of X′ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let T ′ be the tree  decomposition of X′ obtained from U by deleting vertices in the set S from all bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since X  can be obtained from X′ by contracting some edges, it will have a set of three vertices, say  u0, u2, u4, that form an independent set and internally vertex-disjoint paths Qi+1, Q′  i+1 from  ui to ui+2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We define the map f : V(X′) �→ V(X) is such that f(ui) = ui for i ∈ {0, 2, 4}, f  maps all internal vertices in the path Qi+1 to ui+1 and Q′  i+1 to u′  i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The map f corresponds  to the edge contraction that we use to obtain X from X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let T be the tree-decomposition of  X obtained from T ′ by applying f to all vertices in all bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We can remove bags of size 0  and size 1 from T using standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Now we modify T to obtain a matched tree decomposition of width 3 for X thereby deriving  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Every bag B in T has a corresponding bag P in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that if uv is an edge  in X′, then f(u)f(v) is an edge in X, whenever f(u) ̸= f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If B has size 4, then since U is  a matched tree decomposition, the bag P must also be matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, if all bags in T ′  has size 4, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Otherwise, there is a bag B in T that has size less than three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We  root the tree T at B and modify T in a top-down fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We need the following claim to  prove the correctness of this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▷ Claim 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let B′ be some bag in T of size 3, then the vertices in B′ cannot form an  independent set of size 3 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If the corresponding bag P to B′ in U has size 4, then since U is a matched tree  decomposition, the bag P is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So B′ was obtained by the deletion of a vertex or the  contraction of an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Both of which will retain at least one edge in the bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  If the corresponding bag P has size 3, then since U is a matched tree composition, the  subgraph induced by B′ has P3 as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  So we can assume the following:  4 All indices i for ui and u′  i in this proof are modulo 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  26  Finding and Counting Patterns in Sparse Graphs  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Every bag in T of size 4 is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Every bag in T of size 3 has at least one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' There are no bags in T of size 0 or size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We now process a bag B, starting from the root, assuming that all ancestors of B are already  matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We split the procedure into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (|B| = 3) Let B = {u, v, w}, uv is an edge, and w is not adjacent to u or v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If w is present in  the parent of B, then it is matched to some w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We add w′ to B as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If w is not present  in the parent, then we choose a nearest descendant B′ of B in T that contains a neighbor w′  of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Every bag in the path from B to B′ contains w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We add w′ to every bag in this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Now, the bag B is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Also, every other bag that we modified (in the path from B to  B′) now has size 3 and has at least one edge or has size 4 and is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So all properties  are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (|B| = 2) Let B = {u, v} and uv is not an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If u is present in B’s parent in T, then it is  matched to some u′ there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We add u′ to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Otherwise, we choose a nearest descendant B′  of B in T that contains a neighbor u′ of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We add u′ to all bags in the path from B to B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B is now a bag of size 3 with at least one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Also, every other bag that we modified now  has size 3 and has at least one edge or has size 4 and is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Now, if B is matched we  are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Otherwise, we apply the previous case to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (Proof for the “only if” direction) We now prove a series of lemmas that will prove the  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Lemma 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph with tw(G) ⩽ 2 and ˜G is a smcc of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If there exists three  internally disjoint paths from a vertex u to v in G, then uv is an edge in ˜G  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose for contradiction u is not adjacent to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let Q1, Q2, Q3 be three internally  disjoint paths from u to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let Q′  1, Q′  2 be the shortest path from u to v in ˜G[V(Q1)] and  ˜G[V(Q2)], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that both of them are of length at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since ˜G is chordal,  the cycle obtained from Q′  1 and Q′  2 has a chord say xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that {x, y} ∩ {u, v} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Without  loss of generality, we may assume that x is in Q′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So y is in Q′  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, Q′  1, Q′  2 and Q3  gives a minor of K4 in ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  ▶ Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a X-induced-minor-free connected graph with tree-width 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, there  exists a smcc ˜G of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose for contradiction that there exists a minimal counter-example G with tw(G) ⩽  2 that is X-induced-minor-free but does not have a smcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since G is minimal, it has to be  biconnected and by the characterization of treewidth 2 graphs, we can assume that G is  series-parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We will use the series-parallel graph characterization to prove that G does  not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let s and t be the source and terminal of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since G is a counter example, it is not an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Suppose (G, s, t) is the series composition of (G1, s1, t1) with (G2, s2, t2) (t1 is identified with  s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since G is a minimal counter example, there exists smccs ˜  G1 of G1 and ˜  G2 of G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let ˜G  be the graph obtained from ˜  G1 and ˜  G2 by identifying t1 with s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The graph ˜G is a smcc of  G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Hence, the graph (G, s, t) is the parallel composition of strictly smaller  graphs (G1, s1, t1) and (G2, s2, t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  27  ▷ Claim 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let G be the parallel composition of two smaller graphs G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Further-  more, assume that G1 is the series composition of two or more graphs G′  i for 1 ⩽ i ⩽ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Then, for all 1 ⩽ i ⩽ m, if G′  i is not an edge, then s′  it′  i is also not an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose there exists 1 ⩽ i ⩽ m such that G′  i is not an edge but s′  it′  i is an edge in G′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  So G′  i is of order at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' There exists a smcc Gi of the graph obtained from G by deleting  all the internal vertices of G′  i (Since G is a minimum counter example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Again there exists  a smcc ˜  G′  i of G′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Define ˜G to be the graph obtained from Gi and ˜  G′  i by identifying the edge  s′  it′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The graph ˜G is a smcc of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  Note that at least one of G1 or G2 is not an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We now split the proof into two exhaustive  cases:  (Case 1) Either G1 or G2 is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Without loss of generality we may assume that G2 is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So s1t1 is not an edge in  G1 (as otherwise G1 = G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose (G1, s1, t1) is a parallel composition of (G3, s3, t3)  and (G4, s4, t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since s1t1 ̸∈ E(G1), we have s3t3 ̸∈ E(G3) and s4t4 ̸∈ E(G4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore,  both G3 and G4 has at least one internal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G′  3 be the graph obtained from  G3 by adding the edge s3t3 and G′  4 be the graph obtained from G4 by adding the edge  s4t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since G is a minimum counter example, there exists smccs ˜  G′  3 of G′  3 and ˜  G′  4 of  G′  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that G is obtained from G′  3 and G′  4 by identifying edges s3t3 and s4t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So the  graph ˜G obtained from ˜  G′  3 and ˜  G′  4 by identifying those edges is a smcc of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We may assume that (G1, s1, t1) is a series composition of smaller graphs (G′  1, s′  1, t′  1),  (G′  2, s′  2, t′  2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , (G′  m, s′  m, t′  m) for some m ⩾ 2 such that for all 2 ⩽ i ⩽ m, the vertex  t′  i−1 is identified with s′  i and for all 1 ⩽ i ⩽ m, (G′  j, s′  j, t′  j) is either an edge or a parallel  composition of two smaller graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Note that G is not a cycle since G is a counter example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So, there exists some i such that  the graph G′  i is not an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, by Claim 40, we have that s′  it′  i is not an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▷ Claim 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  There does not exist 1 ⩽ j(̸= i) < k(̸= i) ⩽ m such that s′  jt′  j, s′  kt′  k are  non-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Suppose s′  jt′  j, s′  kt′  k are non-edges, for some 1 < j(̸= i) < k(̸= i) ⩽ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Without  loss of generality, we may assume that i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So G′  i is a parallel composition of two  smaller graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Hence there exist two internally disjoints paths Q1, Q′  1 (each of length  at least 3), from s′  i to t′  i in G′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Similarly there exist two internally disjoints paths Q2, Q′  2  (each of length at least 3), from s′  j to t′  j in G′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Also there exist two internally disjoints  paths Q3, Q′  3 (each of length at least 3), from s′  k to t′  k in G′  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Again, there exists a  path from t′  i to s′  j, through G′  i+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' G′  j−1, a path from t′  j to s′  k through G′  j+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' G′  k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Again we have a path from t′  k to s′  i through G′  k+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' G′  m, s′  mt′  1, G′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' G′  i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This gives  a subdivision of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Hence, the above claim is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  ▷ Claim 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  There exists a smcc of G′  i such that s′  it′  i is an edge in the completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Since s′  i is not adjacent to t′  i, (G′  i, s′  i, t′  i) is parallel composition of two smaller  graphs (By Claim 40), say (G3, s3, t3) and (G4, s4, t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G′ be the graph obtained from  28  Finding and Counting Patterns in Sparse Graphs  G by contracting the edge st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is an X-induced-minor-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The minimality of  G says that G′ has an smcc, say ˜  G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By Lemma 38, s′  i is adjacent to t′  i in ˜  G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that  the graph induced by the vertices of G′  i in ˜  G′ is a smcc of G′  i in which s′  it′  i is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  By Claim 41, the following cases are now exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (Case 1a) s′  jt′  j is an edge, for all 1 ⩽ j(̸= i) ⩽ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  By the above claim, there exists a smcc ˜  G′  i of G′  i in which s′  it′  i is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We construct  an smcc of G as follows: It contains the smcc ˜  G′  k for all G′  k and the edge st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We then  add the edges ss′  i and s′  it′  j for all 1 ⩽ j(̸= i) ⩽ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' These edges ensure that there are  no chordless cycles through G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This smcc gives us a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (Case 1b) There exists 1 ⩽ j(̸= i) ⩽ m such that s′  i is not adjacent to t′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Without loss of generality we may assume that i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By the claim 42, there exists a  smcc ˜  G′  i, ˜  G′  j of G′  i and s′  j, respectively, such that s′  it′  i ∈ E( ˜  G′  i) and s′  jt′  j ∈ E( ˜  G′  j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We  construct an smcc of G as follows: It contains the smcc ˜  G′  k for all G′  k and the edge st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We then add edges t′  is′  k for all 1 ⩽ k < i, st′  k for all i < k < j, and s′  jt and s′  jt′  k for all  j < k ⩽ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This smcc gives us a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  The above constructions are illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The edges colored black are the  edges in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The red colored edges are those added to construct the smcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  case (1a)  case (1b)  Figure 3 Constructing smcc in case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (Case 2) Both G1 and G2 are not edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose G1 and G2 both are parallel composition  of smaller graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So there exists two internally disjoints paths P, P′ from s1 to t1 in G1  and two internally disjoints paths Q, Q′ from s2 to t2 in G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G′ be the graph obtained  from G1 by adding Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since, G is a minimum counter example, G′ has a smcc ˜  G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Again,  P, P′, Q are three mutually internally disjoint paths from s1 to t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By, lemma 38, s1t1 is  an edge in ˜  G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The graph ˜  G1 induced by V(G1) in ˜  G′ is a smcc such that s1t1 is an edge  in ˜  G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Similarly, we can show that there exists a smcc ˜  G2 of G2 such that s1t2 is an edge  in ˜  G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The graph obtained from ˜  G1 and ˜  G2 by identifying the edges s1 with s2 and t1  with t2 is a smcc of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  29  We may assume that at least one of G1 or G2 is a series composition of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We decom-  pose G1 and G2 into series composed graphs repeatedly, until we cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=', all the indi-  vidual components are parallel compositions or an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let (G1, s1, t1) be series compos-  itions of (G′  1, s′  1, t′  1), (G′  2, s′  2, t′  2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , (G′  m, s′  m, t′  m) and let (G2, s2, t2) be series composi-  tions of (G′  m+1, s′  m+1, t′  m+1), (G′  m+2, s′  m+2, t′  m+2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , (G′  m+ℓ, s′  m+ℓ, t′  m+ℓ) in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▷ Claim 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  There exists a 1 ⩽ i ⩽ m + ℓ such that s′  it′  i is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Suppose for contradiction that s′  i is not adjacent to t′  i, for all 1 ⩽ i ⩽ m + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So,  the graph (G′  1, s′  1, t′  1) is a parallel composition of two smaller graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Hence, there exists  two internally disjoint paths, each of length at least 3, from s′  1 to t′  1 in G′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Similarly,  there are two internally disjoint paths, each of length at least 3, from s′  i to t′  i in G′  i, for  all 1 ⩽ i ⩽ m+ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since m+ℓ > 2, we get an X-induced-minor in G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  Thus, there exists a 1 ⩽ i ⩽ m+ℓ such that s′  it′  i is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By Claim 40, we can conclude  that G′  i is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Without loss of generality, we may assume that 1 ⩽ i ⩽ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that  (G \\ {s′  it′  i}, s′  i, t′  i), is the graph obtained by the series composition of G′  i−1, G′  i−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , G′  1,  G2, G′  m, G′  m−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , G′  i+1 by identifying s′  i−k, s′  1, t2, t′  j with t′  i−k+1, s2, t′  m and s′  j, respect-  ively, for 1 ⩽ k < i and i < j ⩽ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Again, the graph (G, s′  i, t′  i) is the parallel composition  of (G\\{s′  it′  i}, s′  i, t′  i) with an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Now, we can use case (1) to get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Hence,  a minimum counter example does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  ▶ Lemma 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let G be a graph with treewidth k and T be a tree decomposition of G such that  no bag of size k + 1 of T is independent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, mtw(G) ⩽ 2tw(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Observe that in the proof of Theorem 7, the construction yields a matched tree  decomposition of width 2tw(G) − 1 if each bag of size tw(G) + 1 has at least one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  By Lemma 39, every G such that tw(G) = 2 and G is X-induced-minor-free has an smcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Now,  a standard tree decomposition T of ˜G is also a tree decomposition of G with the additional  property that every bag of size 3 has at least one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We apply Lemma 44 to construct a  matched tree decomposition of width 3 for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  To complete the characterization of matched treewidth of partial 2-trees, we prove the fol-  lowing lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Lemma 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' mtw(Y) ⩾ 5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The proof is similar to the proof of Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We additionally use the fact that  u1, u2, and u3 do not have any common neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose for contradiction that Y has  a reduced, matched tree decomposition T of width strictly less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let Bl be a leaf in  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We first argue that Bl must contain one of u1, u2, or u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If not, it contains only a subset  of the other edges, say, like u4u5 (other cases are symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since those vertices are not  pendant, the neighbor of Bl in T, which we call Bp, will be a superset of Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Suppose Bl  contains u1 and u4 (rest of the cases are symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since T is reduced and has width less  than 5, the bag Bp must contain both u1 and u2 (or u1 and u3, a symmetric case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Root T  at Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  30  Finding and Counting Patterns in Sparse Graphs  u1  u7  u5  u2  u3  u12  u14  u9  u11  u4  u6  u8  u10  u15  u13  Figure 4 The graph Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  If u3 ∈ Bp, we are done since |Bp| ⩾ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let Bc be the closest descendant of Bp that contains  u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Assume wlog that u2 /∈ Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Both u1 and u2 cannot be missing from Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In that case, all  neighbors of u3 must be in Bc and that will imply |Bc| ⩾ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So u1 and u3 are both in Bc (the  other case is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' It must also contain a vertex v that is a neighbor of u1 since it is  matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Consider the path P from Bp to Bc in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' If u8 (u10) does not appear in this path,  then u9 (u11 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=') must appear on all bags in this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, Bp will have to contain  u3 to match u9 (or u11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' So u8 and u10 must appear in this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Let B1 be the first bag in P where both u8 and u10 have appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We split the proof into  two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (Both u8 and u10 appear in B1) If u2 /∈ B1, then |B1| ⩾ 6 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Otherwise,  B1 = {u1, v, u2, u8, u10} for some neighbor v of u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let B2 be the first bag from B1 to Bc  where the edge u8u9 or u10u11 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, either B2 = {u1, v, u2, u10, u8, u9} (u10u11  has not appeared) or B2 = {u1, v, u10, u11, u8, u9} (both edges appear simultaneously) and  we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  (One of u8 or u10 is missing from B1) This means either u9 or u11 is in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We have either  B1 = {u1, v, u2, u10, u9, u3} or B1 = {u1, v, u2, u8, u11, u3} and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  Figure 5 is a super-graph of Y that has lower matched treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='2  Finding and counting Subgraphs and induced subgraphs using  matched treewidth  ▶ Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge graph H as input, we can count the number of Pk subgraphs,  where k ⩽ 10, in �O(m2)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' There are more than 300 graphs in Spasm(P10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We have verified that all of them  have mtw at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' To minimize the work, we can filter out all graphs in the spasm that  has tw(G′) = 1 or tw(G′) = 2 and G′ is X-induced-minor-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Observe that since X has  9 vertices 12 edges, it cannot be an induced minor in any of the graphs in Spasm(P10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Also, none of the forbidden minors for treewidth 4 can appear in Spasm(P10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  There-  fore, we only need to analyze graphs of treewidth 3 in Spasm(P10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  There are only 18  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  31  u1  u7  u5  u2  u3  u12  u14  u9  u11  u4  u6  u8  u10  u15  u13  v  u1u4u5u6u7  u1vu5u7u2  u1vu2u8u9  u1vu2u9u3  u1vu2u3u11  u1vu3u14u12  u1vu2u10u11  u1u13u12u14u15  Figure 5 The graph Z and its matched tree decomposition of width 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  such graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  They are listed in a pdf file in the repository associated with this paper  (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='com/anonymous1203/Spasm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Since Spasm(Pk) ⊆ Spasm(Pk+1) for all k ⩾ 2, we can make the same claim for all paths on  fewer than 10 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We now use Equation 1 to compute the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  ▶ Remark 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since K5 ∈ Spasm(P11), and treewidth of K5 is 4, the above method cannot  yield an �O(m2) algorithm for Pk where k ⩾ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We note that the above proof also yields �O(m2) time algorithms for counting any pattern  in Spasm(P10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is because if G ∈ Spasm(P10), then Spasm(G) ⊆ Spasm(P10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In the  following theorem, we point out an important class of graphs in Spasm(P10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge graph H as input, we can count the number of cycles of  length at most 9 in �O(m2)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We observe that Spasm(Ck) ⊂ Spasm(P10) for k ⩽ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  ▶ Remark 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Since K5 ∈ Spasm(C10), and treewidth of K5 is 4, the above method cannot  yield an �O(m2) algorithm for Ck where k ⩾ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We can also use our efficient construction of arithemetic circuits for homomorphism poly-  nomials for detecting small induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Brute-force search finds, or even counts  C6 as induced subgraphs in an m-edge host graph in O(m3) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Bläser, Komarath, and  Sreenivasaiah [2] showed that efficient computation of homomorphism polynomials can be  used to speed-up the detection of induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' For example, their techniques can be  used to show that detecting an induced C6 in an n-vertex host graph can be done in O(n4)  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In this section, we derive an �O(m2) algorithm for detecting induced C6 in an m-edge  host graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This algorithm is a natural analogue of the algorithm by Bla¨ser, Komarath and  Sreenivasaiah [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We now give some definitions that are necessary to understand how homomorphism poly-  nomials are used in induced subgraph detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  32  Finding and Counting Patterns in Sparse Graphs  ▶ Definition 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We define the induced subgraph isomorphism polynomial for pattern graph  G on n-vertex host graphs, denoted IndG, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The variables of the polynomial are yv  and x{u,v} for all u, v ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  IndG =  �  G′  �  v  yv  �  e  xe  �  f  (1 − xf)  where G′ ranges over all (not-necessarily induced) subgraphs of Kn isomorphic to G, v ranges  over the vertices of G′, e ranges over the edges of G′, and f ranges over the edges in Kn between  vertices in G′ but not in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We denote by IndG(E(H)), the polynomial obtained by substituting the adjacency in H for  the edge variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that the monomials of IndG(E(H)) are products of |V(G)| vertex  variables and correspond to the induced subgraphs of H isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In addition, these  monomials all have coefficient 1 because there can only be at most 1 induced subgraph  isomorphic to G on any given k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The induced subgraph polynomials and subgraph  polynomials are related via the equation:  IndG(E(H)) =  �  G′  (−1)|E(G′)−E(G)|#SubG[G′]SubG′[H](xe = 1)  (2)  where G′ ranges over all k-vertex supergraphs of G and #SubG[G′] denotes the number of  times G occurs as a subgraph in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We use the substitution (xe = 1) to denote that all edge  variables in the polynomial are substituted with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Variants of this equation have been used  by many authors for induced subgraph detection (See [23, 2, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We now briefly describe how to use homomorphism polynomials to detect induced subgraph  isomorphisms (See [2] for a more detailed description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Note that the IndG(E(H)) has the  monomial xv1 · · · xvk if and only if v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , vk induces a G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, to check whether G  occurs as an induced subgraph, we only have to test whether IndG(E(H)) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Fur-  thermore, the coefficient of every monomial is 1 because a k-vertex subgraph can contain at  most one induced subgraph isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, whether H contains an induced sub-  graph isomorphic to G can be reduced to whether IndG(E(H)) is non-zero modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The ad-  vantage of computing over a ring of characteristic 2 is that it eliminates all SubG′[H](xe = 1)  from the right-hand side of Equation 2 for which #SubG[G′] is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' However, we do not  have efficient computations for subgraph polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Here, we make use of the observation  that SubG′[H](xe = 1) is equal to the multilinear part of  1  #Aut(G′)HomG′[H](xe = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' There-  fore, to test whether SubG′[H](xe = 1) is non-zero modulo 2, we need only test whether  1  #Aut(G′)HomG′[H](xe = 1) contains a multilinear term with an odd coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  To check the presence of multilinear terms with odd coefficients, we can randomly substitute  elements that satisfy the equation x2 = 0 from group algebras over Z2 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We stress  that we only substitute these elements for the vertex variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The edge variables are all  always replaced by 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The use of a characteristic-2 ring introduces another issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We  now cannot compute  1  #Aut(G′)HomG′[H](xe = 1) for graphs that have an even number of  automorphisms, by finding the homomorphism polynomial and dividing by the number of  automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The solution is to compute a polynomial that avoids these automorphisms  in the multilinear part of HomG′[H](xe = 1) for each such G′ so that this division becomes  unnecessary, while being careful not to introduce additional multilinear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is the  crux of the following proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ▶ Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge host graph as input, we can find an induced C6 or report  that none exists in �O(m2)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  33  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We describe how to compute polynomials for which the multilinear part is the same  as HomG′[H](xe = 1) and the coefficient of all monomials are odd for all 6-vertex super-  graphs of C6 that contain C6 an odd number times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The complete list is given in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  These computations involve modifying Algorithm 2 slightly for each such G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We consider  the case of C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Each multilinear monomial in HomC6[H](xe = 1) has coefficient 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' By  ensuring that only homomorphisms σ where σ(2) = min(σ(2), σ(3), σ(5), σ(6)) are present  in the polynomial, we can ensure that all C6 subgraphs of H are counted exactly thrice, once  for each choice of {σ(1), σ(4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This check can be done when the algorithm processes bag  2365 (Figure 6) in Line 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Notice that we need to iterate only over the edges present in H  in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is because the other edge variables will be substituted with 0 anyway  and those monomials will definitely vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' This is crucial in ensuring that our algorithm  remains �O(m2) and not �O(n4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We consider one more case from our list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The graph in the first row and third column in  Figure 6 has four automorphisms (horizontal flip and vertical flip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We can ensure that  these subgraphs are counted exactly once in the polynomial by ensuring that Line 22 in  Algorithm 2 also checks that σ(3) < σ(6) (preventing horizontal flips) and σ(2) < σ(4)  (preventing vertical flips).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' These checks can be done when the algorithm processes the bag  1634 and 1234 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Figure 6 shows the matched tree decompositions and the constraints on σ that can be used  to apply Algorithm 2 to compute all these polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  ▶ Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Given an m-edge host graph as input, we can find an induced Pk or report  that none exists in �O(m(k−2)/2)-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We first prove that Pk has matched treewidth k−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We only consider the case of odd  k (the even case is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Let vertices in the path be 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , 2j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Then, our matched  tree decomposition will have three bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' A root bag that excludes only {j, j+2}, a left child of  the root bag that excludes {j, j+1}, and a right child of the root bag that excludes {j+1, j+2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  This is clearly a tree decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' To see that it is matched, at the root bag, we have the  matched matching (1, 2j + 1), (2, 2j), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' , (j − 1, j + 3), (j + 1, j + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' On the left child, we  can keep the rest of edges the same and match (j + 2, j − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Similarly, on the right child, we  match (j, j + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Since Pk has two automorphisms, we also need to show that we can avoid one automorph-  ism while computing the homomorphism polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' We note that the non-identity auto-  morphism τ must have τ(1) = k and τ(k) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Therefore, we can avoid this by always  ensuring that σ(1) < σ(k) when building the homomorphism polynomial in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  We know that IndPk(E(H)) = SubPk[H](xe = 1) (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' The theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  ◀  5  Acknowledgement:  The research work of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra is partially funded by Fondecyt Postdoctoral grant 3220618  of Agencia National de Investigatión y Desarrollo (ANID), Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  34  Finding and Counting Patterns in Sparse Graphs  1  2  3  4  5  6  126  2365  345  σ(2) = min(σ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ(5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ(6))  1  2  3  4  5  6  σ(2) < σ(6)  1  2  3  4  5  6  126  2365  345  1234  1634  1654  σ(3) < σ(6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ(2) < σ(4)  1  2  3  4  5  6  1236  3645  1  2  3  4  5  6  6123  6234  654  σ(4) < σ(6)  2  4  6  1  2345  1245  1456  σ(3) < σ(5)  3  5  1  2  3  4  5  6  1256  2356  3456  1  2  3  4  5  6  1264  3264  5264  σ(2) < σ(4) < σ(6)  1  2  3  4  5  6  6123  6234  6254  σ(1) < σ(3)  1  2  3  4  5  6  126  2365  345  σ(2) = min(σ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ(5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ(6))  1  2  3  4  5  6  126  2365  345  σ(2) < σ(5)  1  2  3  4  5  6  1256  2356  3456  σ(4) < σ(6)  1  2  3  4  5  6  1263  6134  6534  σ(5) < σ(6)  1  2  3  4  6  5  σ(2) < σ(6)  1265  2564  2534  1  2  3  4  5  6  1426  2645  2345  σ(1) < σ(4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' σ(1) < σ(2) < σ(6)  1  2  3  4  5  6  1426  2645  2345  σ(2) < σ(4)  1  2  3  5  6  4  σ(2) < σ(6)  1265  2564  2534  1  2  3  5  6  4  σ(2) < σ(6)  1246  2564  2534  Figure 6 Detecting Induced C6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Komarath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Mishra, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sethia  35  References  1  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' Yuster, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Zwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Finding and counting given length cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Algorithmica,  17(3):209–223, Mar 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='1007/BF02523189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 2, 3, 8, 14)  2  Markus Bläser, Balagopal Komarath, and Karteek Sreenivasaiah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Graph pattern polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In  Sumit Ganguly and Paritosh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Pandya, editors, 38th IARCS Annual Conference on Foundations  of Software Technology and Theoretical Computer Science, FSTTCS 2018, December 11-13, 2018,  Ahmedabad, India, volume 122 of LIPIcs, pages 18:1–18:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Schloss Dagstuhl - Leibniz-Zentrum  für Informatik, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 4, 8, 31, 32)  3  Marco Bressan and Marc Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Exact and approximate pattern counting in degenerate  graphs:  New algorithms, hardness results, and complexity dichotomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  In 2021 IEEE  62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 276–285, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' (↑ 8)  4  Radu Curticapean, Holger Dell, and Dániel Marx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Homomorphisms are a good basis for count-  ing small subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of  Computing, STOC 2017, page 210–223, New York, NY, USA, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Association for Computing  Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='3055502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 2, 3, 8, 14)  5  Radu Curticapean and Dániel Marx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Complexity of counting subgraphs: Only the boundedness  of the vertex-cover number counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In 55th IEEE Annual Symposium on Foundations of Computer  Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 130–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' IEEE Computer  Society, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='  In Proceedings of the 7th Annual International Conference on Computing and Combinatorics, CO-  COON ’01, page 298–307, Berlin, Heidelberg, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Springer-Verlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 4, 6)  7  Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Counting induced sub-  graphs:  An algebraic approach to #w[1]-hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' (↑ 8)  8  Friedrich  Eisenbrand  and  Fabrizio  Grandoni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  On  the  complexity  of  fixed  para-  meter clique and dominating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Theoretical Computer Science, 326:57–67, 10 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' Induced subgraph  isomorphism: Are some patterns substantially easier than others?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' In Joachim Gudmundsson, Ju-  lián Mestre, and Taso Viglas, editors, Computing and Combinatorics, pages 37–48, Berlin, Heidel-  berg, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' (↑ 7, 8)  10  Jacob Focke and Marc Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Counting small induced subgraphs with hereditary properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  In Stefano Leonardi and Anupam Gupta, editors, STOC ’22: 54th Annual ACM SIGACT Sym-  posium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1543–1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' ACM, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' (↑ 8)  11  Fedor V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Fomin, Daniel Lokshtanov, Venkatesh Raman, Saket Saurabh, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Raghavendra  Rao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Faster algorithms for finding and counting subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='jcss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 3)  12  Archontia  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Giannopoulou  and  Dimitrios  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Thilikos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Obstructions  for  tree-  depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='com/science/article/pii/S1571065309000821,  doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='org/doc/17394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 2)  18  Marc Roth,  Johannes Schmitt,  and Philip Wellnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Detecting and Counting Small  Subgraphs, and Evaluating a Parameterized Tutte Polynomial:  Lower Bounds via Tor-  oidal Grids and Cayley Graph Expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  In Nikhil Bansal,  Emanuela Merelli,  and  James Worrell, editors, 48th International Colloquium on Automata, Languages, and Pro-  gramming (ICALP 2021),  volume 198 of Leibniz International Proceedings in Informat-  ics (LIPIcs), pages 108:1–108:16, Dagstuhl, Germany, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Schloss Dagstuhl – Leibniz-  Zentrum für Informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='dagstuhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='de/opus/volltexte/2021/14177,  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='ICALP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 8)  19  Virginia Vassilevska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  Efficient Algorithms for Path Problems in Weighted Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  PhD thesis,  Carnegie Mellon University, Pittsburgh, PA 15213, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 8)  20  Douglas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' (↑ 9)  21  Ryan Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Maximum two-satisfiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Encyclopedia of Algorithms, pages 507–510, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='1007/978-0-387-30162-4_227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 2)  22  Ryan Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Finding paths of length k in o*(2k) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='ipl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
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+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 2, 3)  23  Virginia Vassilevska Williams, Joshua R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Wang, Ryan Williams, and Huacheng Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Finding four-  node subgraphs in triangle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content='  In Proceedings of the Twenty-Sixth Annual ACM-SIAM Sym-  posium on Discrete Algorithms, SODA ’15, page 1671–1680, USA, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' Society for Industrial  and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
+page_content=' (↑ 4, 32)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfrwEw/content/2301.02569v1.pdf'}
diff --git a/sdAzT4oBgHgl3EQfcvxN/content/tmp_files/2301.01407v1.pdf.txt b/sdAzT4oBgHgl3EQfcvxN/content/tmp_files/2301.01407v1.pdf.txt
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+Dynamics of correlation spreading in low-dimensional transverse-field Ising models
+Ryui Kaneko∗ and Ippei Danshita†
+Department of Physics, Kindai University, Higashi-Osaka, Osaka 577-8502, Japan
+(Dated: January 5, 2023)
+We investigate the dynamical spreading of spatial correlations after a quantum quench starting
+from a magnetically disordered state in the transverse-field Ising model at one (1D) and two spa-
+tial dimensions (2D). We analyze specifically the longitudinal and transverse spin-spin correlation
+functions at equal time with use of several methods. From the comparison of the results in 1D
+obtained by the linear spin-wave approximation (LSWA) and those obtained by the rigorous ana-
+lytical approach, we show that the LSWA can asymptotically reproduce the exact group velocity
+in the limit of strong transverse fields while it fails to capture the detailed time dependence of the
+correlation functions. By applying the LSWA to the 2D case, in which the rigorous analytical ap-
+proach is unavailable, we estimate the propagation velocity to be Ja/(2ℏ) at the strong-field limit,
+where J is the Ising interaction and a is the lattice spacing. We also utilize the tensor-network
+method based on the projected-entangled pair states for 2D and quantitatively compute the time
+evolution of the correlation functions for a relatively short time. Our findings provide useful bench-
+marks for quantum simulation experiments of correlation spreading and theoretical refinement of
+the Lieb-Robinson bound in the future.
+I.
+INTRODUCTION
+Neutral atoms trapped in optical-tweezer arrays are
+promising platforms for analog quantum simulations [1–
+3].
+The controllability of individual atoms with laser
+pulses and of interatomic interactions via Rydberg ex-
+citations enables one to realize fast and high-fidelity
+quantum operations. Recent rapid technological devel-
+opments allow for manipulating many Rydberg atoms
+in large arrays [4–6] and investigating the ground state
+of quantum lattice systems experimentally [7–15]. Such
+experiments have also stimulated theoretical research on
+fundamental quantum many-body systems. For instance,
+the ground-state phase diagrams of the transverse-field
+Ising model and those of its large Ising interaction limit,
+the PXP model, have been intensively examined using
+the quantum Monte Carlo method [16–19].
+Rydberg-atom arrays have also given the opportunity
+to study the nonequilibrium dynamics of isolated quan-
+tum many-body systems, which are hard to simulate nu-
+merically with classical computers.
+In particular, the
+correlation-spreading dynamics of quantum Ising mod-
+els [16, 20] is one of the intriguing topics that is likely
+to be further addressed. At present, experiments with
+more than 200 Rydberg atoms are feasible [4–6], allow-
+ing one to study unprecedentedly large lattice systems in
+one (1D) and two spatial dimensions (2D).
+These recent experiments on long-time dynamics in
+quantum many-body systems have motivated us to quan-
+titatively calculate the velocity of the correlation prop-
+agation, which will serve as useful references for future
+experiments. In general, there are two kinds of propa-
+gation velocities for correlation spreading dynamics; one
+is the phase velocity and the other is the group velocity.
+∗ rkaneko@phys.kindai.ac.jp
+† danshita@phys.kindai.ac.jp
+The former can be captured by the first peak of the wave
+packet, whereas the latter can be extracted by the enve-
+lope of the wave packet. The group velocity is bounded
+from above in non-relativistic quantum systems, and this
+upper limit is known as the Lieb-Robinson bound [21, 22].
+While significant progress has been made concerning
+rigorous inequalities related to the Lieb-Robinson bound,
+such inequalities do not necessarily offer practical refer-
+ence values for experiments. Usually, the Lieb-Robinson
+bound is intended to provide general conditions for ar-
+bitrary correlations.
+Consequently, the bound can be
+too loose and sometimes meaningless when examining
+the propagation velocity of particular correlation func-
+tions that are measurable in experiments. With this in
+mind, the Lieb-Robinson bound has been improved very
+recently [23]; however, their method still gives a looser
+bound than the exact solution if it is available.
+In some cases, direct numerical simulations on classi-
+cal computers would give much more detailed informa-
+tion about correlation spreading than rigorous inequali-
+ties for the Lieb-Robinson bound. Such numerical data
+would also strengthen the validity of experimental find-
+ings through cross-checking experimental and theoreti-
+cal results.
+Indeed, many numerical efforts have been
+made to calculate the quench or sweep dynamics in 1D
+and 2D. These attempts include the time-dependent vari-
+ational Monte Carlo method with the Slater-Jastrow
+wave function [24] and with more sophisticated neural-
+network wave functions [25–30], the form factor expan-
+sions [31], the numerical linked-cluster expansion [32–
+34], the tensor-network method based on matrix prod-
+uct states (MPS) [35, 36], and that based on projected
+entangled pair states (PEPS) [37–39].
+In this paper, we study quench dynamics in the
+transverse-field Ising model on a chain in 1D and that
+on a square lattice in 2D by using several methods, in-
+cluding the tensor-network method based on PEPS and
+the linear spin-wave approximation (LSWA). We take
+arXiv:2301.01407v1  [cond-mat.quant-gas]  4 Jan 2023
+
+2
+the initial state to be the magnetically disordered prod-
+uct state, which is the ground state in the strong-field
+limit, and calculate time evolution of spin-spin correla-
+tions at equal time after a sudden quench of the trans-
+verse field. We focus on the quench within a parameter
+region where the ground state is magnetically disordered.
+We extract the group velocity of the correlation propa-
+gation from the spin-spin correlations for several values
+of the transverse field. In the 1D case, we show that the
+group velocity extracted from the LSWA results asymp-
+totically approaches that extracted from the rigorous an-
+alytical results with increasing the transverse field, while
+the agreement in the time dependence of the correlation
+functions is limited to a short time before the first peak
+appears. Our results indicate that the LSWA can quan-
+titatively predict the propagation velocity as long as the
+final transverse field is sufficiently strong. In the 2D case,
+using the LSWA, we estimate the group velocity to be
+Ja/(2ℏ), where J is the Ising interaction and a is the
+lattice spacing. We use the PEPS method in a comple-
+mentary way to perform more quantitative calculations
+on the time evolution of the correlation functions for a
+relatively short time.
+This paper is organized as follows: In Sec. II, we in-
+troduce the model and all the analytical and numerical
+methods used in the present study. In Secs. III and IV,
+we present the time-dependent spin-spin correlation func-
+tions and extract the corresponding group velocity in 1D
+and 2D, respectively. We discuss the relation between
+our propagation velocity and the Lieb-Robinson bound
+proposed recently, and draw our conclusions in Sec. V.
+For simplicity, we set ℏ = 1 throughout this paper.
+II.
+MODEL AND METHODS
+We consider the transverse-field Ising model with the
+periodic boundary condition defined as
+ˆH = −J
+�
+⟨i,j⟩
+ˆSz
+i ˆSz
+j − Γ
+�
+i
+ˆSx
+i ,
+(1)
+where ˆSz
+i and ˆSx
+i correspond to the z and x components
+of the S = 1/2 Pauli spin, J represents the strength of the
+spin exchange interaction, and Γ represents the strength
+of the transverse field. The symbol ⟨i, j⟩ means that the
+sum is taken over nearest-neighbor sites. We focus on
+the ferromagnetic spin exchange interaction (J > 0) on a
+chain in 1D and that on a square lattice in 2D. Both fer-
+romagnetic and antiferromagnetic models are equivalent
+under appropriate unitary transformations for bipartite
+lattices.
+The ground state is ordered (disordered) for
+Γ < Γc (Γ > Γc), where Γc is the transition point given
+as Γc/J = 1/2 [40] in 1D and Γc/J ≈ 1.522 [17, 41, 42]
+in 2D. Hereafter we take J as the unit of energy. We
+also take the lattice constant to be unity throughout this
+paper.
+We investigate the quench dynamic starting from the
+disordered state |ψ0⟩ = ⊗i |→⟩i at Γ → ∞ to the dis-
+ordered parameter region Γ ∈ (Γc, ∞).
+We study the
+equal-time longitudinal and connected transverse corre-
+lation functions at distance r, which are defined as
+Czz(r, t) = ⟨ψ(t)| ˆSz
+r ˆSz
+0|ψ(t)⟩,
+(2)
+Cxx
+connected(r, t) = ⟨ψ(t)| ˆSx
+r ˆSx
+0|ψ(t)⟩
+− ⟨ψ(t)| ˆSx
+r|ψ(t)⟩⟨ψ(t)| ˆSx
+0|ψ(t)⟩
+(3)
+with |ψ(t)⟩ = e−i ˆ
+Ht|ψ0⟩, respectively. In 1D, we obtain
+them by the exact calculations via the Jordan-Wigner
+transformation and by the LSWA via the Holstein-
+Primakoff transformation.
+In 2D, we use the tensor-
+network method, the exact diagonalization (ED) method,
+and the LSWA. We will summarize each method below.
+We extract the group velocity from the envelope of the
+wave packet in the spin-spin correlation functions. Let
+us first discuss how the correlation spreading is related to
+the Lieb-Robinson bound. In a system with short-range
+interaction, a commutator of any operators ˆOA and ˆOB
+in regions A and B satisfies the relation
+��[ ˆOA(t), ˆOB]
+�� ≤ const. × exp
+�
+−L − vt
+χ
+�
+,
+(4)
+where ˆOA(t) = exp
+�
+i ˆHt
+� ˆOA exp
+�
+−i ˆHt
+�
+, L is the dis-
+tance between the regions A and B, and χ is con-
+stant [21, 22]. The velocity v corresponds to the Lieb-
+Robinson bound. This relation means that the informa-
+tion from the region A is transmitted to the region B up
+to a time t ≈ L/v.
+Then, the inequality of the Lieb-
+Robinson bound ensures that, for any operators ˆOA and
+ˆOB in regions A and B having the distance L, the ex-
+pectation value for a state |ψ⟩ with a finite correlation
+length χ satisfies [43]
+|⟨ψ(t)| ˆOA ˆOB|ψ(t)⟩ − ⟨ψ(t)| ˆOA|ψ(t)⟩⟨ψ(t)| ˆOB|ψ(t)⟩|
+≤ const. × e− L−2vt
+χ′
+,
+(5)
+where χ′ is a constant that depends on χ. This velocity
+2v on the right-hand side corresponds to twice the Lieb-
+Robinson bound. When the correlation spreading is well
+described by the quasiparticle, the group velocity (vgroup)
+of the fastest quasiparticle is often regarded as the Lieb-
+Robinson bound [44–47].
+Under these circumstances, the slope obtained from
+the peak-time dependence of the distance gives twice the
+group velocity (2vgroup) of a certain quasiparticle.
+In-
+tuitively, the factor two originates from pairs of quasi-
+particles moving to the left or right from a given point.
+This quasiparticle picture has been discussed intensively
+in the dynamics of the Bose-Hubbard model [45, 48–50].
+In the present analysis, we have presented the group ve-
+locity vgroup of a certain single quasiparticle estimated
+from one half of the slope obtained from the peak-time
+dependence of the distance.
+
+3
+A.
+Exact calculations in 1D
+The analytical form of the time-dependent correla-
+tion functions can be obtained rigorously for the 1D
+transverse-field Ising model [40, 51–56]. We briefly re-
+view the detailed derivation of the time-dependent cor-
+relation functions in Appendix A and present the final
+results below.
+The longitudinal correlation function is represented as
+a Pfaffian of a 2r × 2r skew symmetric matrix:
+Czz(r, t) = 1
+4pf
+� S
+G
+−GT Q
+�
+.
+(6)
+Here elements of the matrices S, Q, and G are defined as
+si,j =
+�
+�
+�
+�
+�
+Si−1,j−1
+if i < j
+−Sj−1,i−1
+if i > j
+0
+if i = j
+,
+(7)
+qi,j =
+�
+�
+�
+�
+�
+Qi−1,j−1
+if i < j
+−Qj−1,i−1
+if i > j
+0
+if i = j
+,
+(8)
+gi,j = Gi−1,j,
+(9)
+where the time-dependent correlation functions Si,j, Qi,j,
+and Gi,j are given as
+Si,j = − 2
+L
+�
+k>0
+�
+cos[k(ri − rj)][|˜uk(t)|2 + |˜vk(t)|2]
+− i sin[k(ri − rj)][˜uk(t)˜v∗
+k(t) + ˜vk(t)˜u∗
+k(t)]
+�
+, (10)
+Qi,j = + 2
+L
+�
+k>0
+�
+cos[k(ri − rj)][|˜uk(t)|2 + |˜vk(t)|2]
++ i sin[k(ri − rj)][˜uk(t)˜v∗
+k(t) + ˜vk(t)˜u∗
+k(t)]
+�
+, (11)
+Gi,j = − 2
+L
+�
+k>0
+�
+cos[−k(ri − rj)][|˜uk(t)|2 − |˜vk(t)|2]
+− i sin[−k(ri − rj)][˜uk(t)˜v∗
+k(t) − ˜vk(t)˜u∗
+k(t)]
+�
+.
+(12)
+The symbol �
+k>0 means the sum taken over all k =
+2πn/L with n = 1/2, 3/2, . . . , (L−3)/2, (L−1)/2 for even
+L or n = 1, 2, . . . , (L − 3)/2, (L − 1)/2 for odd L. For the
+quench starting from the disordered state (Γ → ∞), the
+parameters ˜uk(t) and ˜vk(t) for 0 < k < π are described
+as
+˜uk(t) = i
+˜b′
+k
+ω′
+k
+sin
+�
+2ω′
+k × tJ
+4
+�
+,
+(13)
+˜vk(t) = −i cos
+�
+2ω′
+k × tJ
+4
+�
+− ˜a′
+k
+ω′
+k
+sin
+�
+2ω′
+k × tJ
+4
+�
+(14)
+with
+˜a′
+k = 2Γ
+J + cos k,
+(15)
+˜b′
+k = sin k,
+(16)
+ω′
+k =
+�
+4Γ2
+J2 + 4Γ
+J cos k + 1,
+(17)
+respectively.
+Parameters with prime symbols indicate
+physical quantities after the quench. On the other hand,
+the transverse correlation function is given as
+Cxx
+connected(r, t) = −1
+4(Q0,rS0,r + Gr,0G0,r).
+(18)
+We numerically evaluate each correlation function for suf-
+ficiently large systems. We use the library for Pfaffian
+computations [57] in the case of the longitudinal correla-
+tion function.
+B.
+Spin-wave approximation
+We investigate a small quench starting from the com-
+pletely disordered point (Γ → ∞) to the parameter
+within a disordered phase (Γclassical
+c
+≪ Γ < ∞, where
+Γclassical
+c
+= JD with D being the spatial dimension). We
+focus on small quantum fluctuations around the disor-
+dered state and map quantum Ising spins to bosons using
+the linearized Holstein-Primakoff transformation [58–61].
+The equal-time correlation functions for quantum spins
+can be obtained by calculating those for bosons. They
+serve as a good approximation as long as the transverse
+magnetization is large enough (⟨Sx
+i ⟩ ≈ 1/2). We give the
+detailed derivation in Appendix B and show the obtained
+spin-spin correlation functions below.
+The longitudinal correlation function at distance r
+(1 ≤ rν ≤ L/2 with ν = 1, 2, . . . , D) is given as
+Czz(r, t) =
+S
+2LD
+�
+k
+eik·r
+B′
+k
+A′
+k + B′
+k
+(cos 2Ω′
+kt − 1), (19)
+where S(= 1/2) is the size of spin and other parameters
+are defined as
+Ω′
+k = sgn(A′
+k)
+�
+A′
+k
+2 − B′
+k
+2,
+(20)
+A′
+k = −z
+2JSγk + Γ,
+(21)
+B′
+k = −z
+2JSγk,
+(22)
+γk = 1
+D
+D
+�
+ν=1
+cos kν
+(23)
+with z = 2D being the coordination number. Parame-
+ters with prime symbols correspond to physical quanti-
+ties after the quench. On the other hand, the transverse
+correlation function at distance r (1 ≤ rν ≤ L/2 with
+ν = 1, 2, . . . , D) is given as
+Cxx
+connected(r, t)
+=
+�����
+1
+LD
+�
+k
+eik·r B′
+k
+2Ω′
+k
+�A′
+k
+Ω′
+k
+(cos 2Ω′
+kt − 1) + i sin 2Ω′
+kt
+������
+2
+
+4
+Dphys
+Dvirt
+A
+B
+FIG. 1.
+Schematic picture of the iPEPS having a two-site
+unit-cell structure. The two sublattice sites are represented
+by A and B. Each ball corresponds to a rank-five tensor, which
+is located at a lattice site and has four thin sticks and a thick
+stick.
+The thin and thick sticks represent the virtual and
+physical degrees of freedom, and the bond dimensions of the
+former and the latter are defined as Dvirt and Dphys, respec-
+tively.
++
+�����
+1
+LD
+�
+k
+eik·r B′
+k
+2
+2Ω′
+k
+2 (cos 2Ω′
+kt − 1)
+�����
+2
+.
+(24)
+We numerically calculate each correlation function for
+sufficiently large systems.
+C.
+2D tensor-network method
+We use the infinite projected entangled pair state
+(iPEPS) [62–69] or the infinite tensor product state [70–
+74] to investigate short-time dynamics in the infinite sys-
+tem. We choose translationally invariant iPEPS consist-
+ing of a two-site unit-cell structure as shown in Fig. 1.
+The dimension of the local Hilbert space is Dphys = 2
+for spin S = 1/2. The initial state |ψ0⟩ = ⊗i| →⟩i is
+the ground state in the limit of Γ/J → ∞, which can be
+prepared by the virtual bond dimension Dvirt = 1.
+We apply the simple update algorithm [66, 75] to
+simulate the real-time dynamics of the transverse-field
+Ising model.
+In this algorithm, we approximate the
+real-time evolution operator in a very short-time step
+dt using the Suzuki-Trotter decomposition [76–78] and
+obtain the two-site gate e−idt ˆ
+H ≈ �
+⟨i,j⟩ e−idt ˆ
+Hij with
+ˆHij = −J ˆSz
+i ˆSz
+j − Γ( ˆSx
+i + ˆSx
+j )/z (z = 4) satisfying
+ˆH = �
+⟨i,j⟩ ˆHij. The gate acts on two neighboring ten-
+sors and increases the virtual bond dimensions. We trun-
+cate the bond dimensions of the local tensors using the
+singular value decomposition so that the bond dimen-
+sions of iPEPS remains Dvirt. In the actual calculations,
+the second-order Suzuki-Trotter decomposition is used,
+and the time step is typically chosen as dtJ = 0.001 for
+a quench to a strong field Γ. Simulations using doubled
+or halved dt show no significant change in the short-time
+dynamics as for the present model.
+We improve the accuracy of time-evolved wave func-
+tions by increasing the dimension of the virtual bond
+Dvirt and confirm the convergence of physical quantities.
+Previous studies [38, 79] suggest that results for bond
+dimensions Dvirt ≳ 6 already show good convergence
+within a short-time frame tJ ≲ 4 even in one of the most
+difficult cases, i.e., the quench to the critical point. (Con-
+cerning the unit of time, the energy scale is four times
+larger in previous studies [38, 79] because they used the
+Pauli spin σz = 2Sz.) In the present iPEPS simulations,
+we adopt the tensor-network library TeNeS [80–82] and
+increase the virtual bond dimensions up to Dvirt = 8 for
+safety.
+The corner transfer matrix renormalization group
+method [67–69, 71, 83–89] is used to calculate physi-
+cal quantities in the thermodynamic limit. We take the
+bond dimensions of the environment tensors as χvirt =
+2(Dvirt)2 so that physical quantities are well converged.
+D.
+Exact diagonalization method
+The ED method is often used to get insight into the
+dynamics of small quantum many-body systems [90–94].
+We use the QuSpin library [95, 96] for ED calculations.
+We consider the system sizes up to 28 sites under the pe-
+riodic boundary condition. In the present setup, both the
+Hamiltonian and the initial state are translationally in-
+variant, and the total momentum of the initial and time-
+evolved states remains zero. We restrict ourselves to the
+zero-momentum sector [97] and follow the dynamics of
+the state. Instead of generating matrix elements on the
+fly to reduce the memory cost, we keep all the elements
+of sparse matrices in the compressed sparse row format
+to accelerate calculations.
+To compute the matrix ex-
+ponential applied to a vector, we use the Taylor series
+expansion with error analysis proposed by Al-Mohy and
+Higham [98, 99].
+We confirm that the ED results (up to 28 sites) re-
+produce the exact analytical results in 1D (not shown).
+We mainly show the ED results in 2D (for 5 × 5 sites)
+hereafter.
+III.
+RESULTS IN 1D
+We first present the time dependence of spin-spin cor-
+relations in 1D using the exact analytical approach and
+the LSWA. We extract the group velocity after a sudden
+quench to a strong field from these data.
+
+5
+0.0
+5.0
+Czz(r = 1, t)
+×10−2
+Γ/J = 3.0
+0.0
+2.0
+Czz(r = 2, t)
+L = 256
+−1.0
+0.0
+1.0
+Czz(r = 3, t)
+−1.0
+0.0
+1.0
+Czz(r = 4, t)
+−1.0
+0.0
+1.0
+Czz(r = 5, t)
+−1.0
+0.0
+1.0
+Czz(r = 6, t)
+−1.0
+0.0
+1.0
+Czz(r = 7, t)
+−1.0
+0.0
+1.0
+Czz(r = 8, t)
+−1.0
+0.0
+1.0
+Czz(r = 9, t)
+0
+5
+10
+15
+20
+time tJ
+−1.0
+0.0
+1.0
+Czz(r = 10, t)
+FIG. 2. Exact equal-time longitudinal correlation functions
+in 1D. We consider the quench to Γ/J = 3 for a system size
+L = 256 and show the short-time dynamics for distances r =
+1, 2, . . . , and 10. The envelope of each correlation function is
+a guide to the eye.
+A.
+Exact results
+We show the exact equal-time longitudinal correlation
+functions in Fig. 2. At an early time (tJ ≪ r), the in-
+tensity of correlation is nearly zero. On the other hand,
+when tJ ≳ r, the correlation starts to develop and ex-
+hibits rapid oscillations. For each distance, the earliest
+peak in the envelope of the wave packet has the largest
+intensity.
+The peak time of the largest envelope peak
+moves almost linearly with the distance, suggesting the
+light-cone-like spreading of correlations.
+We also show the exact equal-time transverse correla-
+tion functions in Fig. 3. In contrast to the longitudinal
+correlations, the rapid oscillations appear only for short
+0.0
+5.0
+Cxx(r = 1, t)
+×10−3
+Γ/J = 3.0
+0.0
+1.0
+Cxx(r = 2, t)
+L = 256
+0.0
+0.5
+Cxx(r = 3, t)
+0.0
+0.5
+Cxx(r = 4, t)
+0.0
+0.5
+Cxx(r = 5, t)
+0.0
+0.5
+Cxx(r = 6, t)
+0.0
+0.5
+Cxx(r = 7, t)
+0.0
+0.5
+Cxx(r = 8, t)
+0.0
+0.5
+Cxx(r = 9, t)
+0
+5
+10
+15
+20
+time tJ
+0.0
+0.5
+Cxx(r = 10, t)
+FIG. 3. Exact equal-time transverse correlation functions in
+1D. The parameters are the same as those in Fig. 2.
+distances (r ≲ 3) and are negligibly small for most of
+the distances.
+Besides, the intensity of the transverse
+correlation is much smaller than that of the longitudinal
+one. On the other hand, the peak time of the transverse
+correlation almost coincides with that of the largest enve-
+lope peak in the longitudinal correlation. The transverse
+correlation decays rapidly just before and after the peak
+time.
+To estimate the propagation velocity, we first extract
+the peak time of the envelope of correlations as a function
+of distance. We show the corresponding time and dis-
+tance in Fig. 4. The data for longitudinal and transverse
+correlations overlap very well.
+The distance is nearly
+proportional to the peak time for both correlations. For
+each field Γ/J and size L, we estimate the group velocity
+vgroup from one half of the slope so that it corresponds
+directly to the speed of one of quasiparticle pairs moving
+
+6
+0
+25
+50
+75
+100
+125
+first peak time tJ
+0
+50
+100
+distance r
+Γ/J = 3.0
+L = 256
+longitudinal
+transverse
+FIG. 4.
+First-peak time dependence of distance for exact
+correlations in 1D. We show data points when the distance is
+a multiple of four. The group velocity is estimated from one
+half of the slope.
+0
+2
+4
+6
+8
+10
+Γ/J
+0.0
+0.2
+0.4
+0.6
+0.8
+vgroup/J
+L = 256
+Lieb-Robinson bound
+longitudinal
+transverse
+FIG. 5. Field dependence of the group velocity in 1D. The
+exact Lieb-Robinson bound vLR/J = 1/2 is shown as a refer-
+ence. For Γ/J < 3, we only show the group velocity estimated
+from the transverse correlations because the envelopes of lon-
+gitudinal correlations become unclear for a weaker field.
+to the left or right. Since the data points are slightly
+out of the straight line at the very short and long dis-
+tances, we discard those for r ≤ 5 and r ≥ L/2 − 5 when
+extracting the velocity.
+To see how the group velocity behaves as a function of
+the transverse field, we first examine a sufficiently large
+system (L = 256) as shown in Fig. 5. Both velocities
+estimated from longitudinal and transverse correlations
+are nearly 0.5J for all fields Γ/J ≥ 3. The group velocity
+of the spin-spin correlations agrees with the exact Lieb-
+Robinson bound in the 1D transverse-field Ising model
+(see Appendix A 5 for the derivation of the exact value).
+This fact suggests that the quasiparticles with the fastest
+propagation velocity among the various correlation func-
+tions are directly responsible for the spreading of spin-
+spin correlations.
+Although the estimated velocity is very close to 0.5J,
+it is slightly smaller than the exact value in finite-size
+systems. To check the size dependence and confirm the
+convergence, we perform the finite-size scaling of the es-
+timated velocity.
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+1/L2/3
+0.40
+0.45
+0.50
+vgroup/J
+Γ/J = 3.0
+Lieb-Robinson bound
+longitudinal
+transverse
+FIG. 6. Size scaling of the group velocity in 1D. The group
+velocity is well fitted by 1/L2/3 with L being the length of
+a chain and is extrapolated to the value of the exact Lieb-
+Robinson bound vLR/J = 1/2.
+For this purpose, let us first discuss how the finite-
+size effect appears. The spin-spin correlation functions in
+the 1D transverse-field Ising model are described by the
+single-particle correlation functions of fermionic quasi-
+particles.
+In the thermodynamic limit, they are writ-
+ten by the Bessel functions [53].
+The size dependence
+of the Bessel functions has been carefully investigated in
+the case of long-time dynamics of the 1D Bose-Hubbard
+model [48], as well as in that of the 1D transverse-field
+Ising model [100]. The distance r dependence of the peak
+time t is given as
+t ≈
+1
+v∞
+(r + ϵr1/3),
+(25)
+where v∞ is the velocity at large distances, and ϵ is a
+constant related to the peak position of the Bessel func-
+tion [48]. The instantaneous velocity v(r) = [t(r + 1) −
+t(r)]−1 at each time t is independent of distances and
+becomes v∞ if ϵ = 0, but it is slightly modified in the
+presence of finite ϵ. For ϵ ̸= 0, the instantaneous velocity
+is obtained as
+v(r) ≈ v∞
+�
+1 − ϵ
+3r−2/3�
+.
+(26)
+Since the farthest distance for a chain of length L is
+r = L/2 (∝ L), we may safely assume that the devi-
+ation between the finite-size and infinite-size velocities
+∆v(L) follows the relation
+∆v(L) :=
+����v
+�L
+2
+�
+− lim
+L→∞ v
+�L
+2
+����� ∝ L−2/3
+(27)
+for L ≫ 1.
+We then extrapolate the finite-size group velocities to
+the thermodynamic limit using Eq. (27). As shown in
+Fig. 6, all the data points lie on an expected straight line
+for both correlations. The extrapolated group velocity at
+Γ/J = 3 is vgroup/J = 0.5005(4) [vgroup/J = 0.4999(3)]
+for the longitudinal (transverse) correlations and almost
+converges to the exact Lieb-Robinson bound (vLR/J =
+
+7
+0.0
+5.0
+Czz(r = 1, t)
+×10−2
+Γ/J = 3.0
+0.0
+2.0
+Czz(r = 2, t)
+L = 256
+−1.0
+0.0
+1.0
+Czz(r = 3, t)
+−1.0
+0.0
+1.0
+Czz(r = 4, t)
+−1.0
+0.0
+1.0
+Czz(r = 5, t)
+−1.0
+0.0
+1.0
+Czz(r = 6, t)
+−1.0
+0.0
+1.0
+Czz(r = 7, t)
+−1.0
+0.0
+1.0
+Czz(r = 8, t)
+−1.0
+0.0
+1.0
+Czz(r = 9, t)
+0
+5
+10
+15
+20
+time tJ
+−1.0
+0.0
+1.0
+Czz(r = 10, t)
+exact
+LSWA
+FIG. 7.
+Equal-time longitudinal correlation functions ob-
+tained by the LSWA in 1D. The parameters are the same
+as those in Fig. 2. We show the exact correlations (dashed
+line) for comparison.
+0.5) of the 1D transverse field Ising model within the
+error bar of the extrapolation. We have also confirmed
+that the estimated velocity converges to the exact one
+for all the other transverse fields that we have studied
+(Γ/J ≥ 1). Therefore, the fastest correlation spreading
+can be measured by the spin-spin correlations in the 1D
+transverse-field Ising model.
+B.
+Results by the LSWA
+To examine how good the LSWA is as for the correla-
+tion spreading, we calculate the equal-time spin-spin cor-
+relation functions by the LSWA and compare the results
+with those of the exact analysis. In general, the LSWA
+0.0
+5.0
+Cxx(r = 1, t)
+×10−3
+Γ/J = 3.0
+0.0
+1.0
+Cxx(r = 2, t)
+L = 256
+0.0
+0.5
+Cxx(r = 3, t)
+0.0
+0.5
+Cxx(r = 4, t)
+0.0
+0.5
+Cxx(r = 5, t)
+0.0
+0.5
+Cxx(r = 6, t)
+0.0
+0.5
+Cxx(r = 7, t)
+0.0
+0.5
+Cxx(r = 8, t)
+0.0
+0.5
+Cxx(r = 9, t)
+0
+5
+10
+15
+20
+time tJ
+0.0
+0.5
+Cxx(r = 10, t)
+exact
+LSWA
+FIG. 8. Equal-time transverse correlation functions obtained
+by the LSWA in 1D. The parameters are the same as those
+in Fig. 2. We show the exact correlations (dashed line) for
+comparison.
+gets better with increasing spatial dimensions. Here we
+will demonstrate that the group velocity of the correla-
+tion propagation obtained by the LSWA agrees well with
+the exact one even in the lowest 1D.
+We show the longitudinal correlation functions in
+Fig. 7. As in the case of the exact analysis, the correla-
+tions are suppressed for tJ ≲ r and begin to develop for
+tJ ≳ r at a given distance r. The LSWA quantitatively
+reproduces the period of oscillations and the intensity
+of exact correlations up to about tJ ≈ r. In the short
+time (tJ ≲ r), a very small number of quasiparticle ex-
+citations would come into play, and the LSWA becomes
+more accurate in this dilute regime.
+On the other hand, the transverse correlation functions
+appear to be accurate up to the point where they begin
+
+8
+0
+25
+50
+75
+100
+125
+first peak time tJ
+0
+50
+100
+distance r
+Γ/J = 3.0
+L = 256
+longitudinal
+transverse
+FIG. 9.
+First-peak time dependence of distance for corre-
+lations obtained by the LSWA in 1D. We show data points
+when the distance is a multiple of four.
+0
+2
+4
+6
+8
+10
+Γ/J
+0.0
+0.2
+0.4
+0.6
+0.8
+vgroup/J
+L = 256
+from spin-wave dispersion
+Lieb-Robinson bound
+longitudinal
+transverse
+FIG. 10.
+Field dependence of the group velocity ob-
+tained by the LSWA in 1D. The exact Lieb-Robinson bound
+vLR/J = 1/2 and the maximum group velocity vSW/J =
+[1 +
+�
+1 − (J/Γ)2]−1/2/
+√
+2 estimated from the spin-wave dis-
+persion (see Appendix B 4) are shown as references.
+For
+Γ/J < 3, we only show the group velocity estimated from
+the transverse correlations because the envelopes of longitu-
+dinal correlations become unclear for a weaker field.
+to increase (see Fig. 8). In contrast to the exact analyti-
+cal result, where the earliest peak has the largest inten-
+sity, the LSWA predicts that the second earliest peak has
+the largest intensity. Nevertheless, the time of maximum
+intensity does not differ significantly between the exact
+and approximate results. The first-peak time is typically
+about 2tJ early, while the time of maximum intensity is
+typically about 2tJ late for all distances in the case of
+the LSWA. These effects do not change the propagation
+velocity significantly. Therefore, the group velocity esti-
+mated by the LSWA is expected to be close to the exact
+one.
+As in the case of exact analysis, we observe the suppres-
+sion of rapid oscillations in the transverse correlations
+using the LSWA. This phenomenon can be easily un-
+derstood in the magnon picture. The original transverse
+correlation corresponds to the density-density correlation
+of magnons. The density operator is less susceptible to
+the effects of phases.
+On the other hand, the original
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+1/L2/3
+0.40
+0.45
+0.50
+vgroup/J
+Γ/J = 3.0
+from spin-wave dispersion
+Lieb-Robinson bound
+longitudinal
+transverse
+FIG. 11. Size dependence of the group velocity estimated by
+the LSWA in 1D. The size dependence is smaller than the
+exact case (see Fig. 6).
+longitudinal correlation corresponds to the single-particle
+correlation of magnons, which directly feels the effects of
+phases. Therefore, the transverse (longitudinal) correla-
+tion tends to exhibit less (more) oscillations. Such effects
+have been intensively examined in the correlation spread-
+ing of the Bose-Hubbard model [45, 48, 49, 101, 102].
+Likewise, the LSWA also predicts that the intensity
+of the transverse correlation is smaller than that of
+the longitudinal one. They are approximately given as
+|Czz(r, t)| = O(J/Γ) and |Cxx
+connected(r, t)| = O(J2/Γ2),
+respectively (see Appendices B 2 and B 3).
+Having confirmed the accuracy of the LSWA, we ex-
+tract the group velocity from the 1D correlations. We
+first investigate the distance dependence of peak time
+for a sufficiently large system (L = 256), as shown in
+Fig. 9. Again, both correlations show almost the same
+result, and the distance is nearly proportional to the
+peak time. We estimate the velocity using the data for
+5 < r < L/2 − 5.
+We summarize the field dependence of the group ve-
+locity in Fig. 10. Both group velocities estimated from
+the longitudinal and transverse correlations are nearly
+0.5J irrespective of the choice of the transverse field for
+Γ/J ≳ 3.
+Note that the LSWA group velocity is ex-
+pected to deviate from the exact one at Γ/J ≲ 2 because
+too many quasiparticles are created due to such a large
+quench. The LSWA also suggests that, in the case of the
+1D transverse-field Ising model with a sufficiently strong
+field, the spin-spin correlations can be used to extract the
+Lieb-Robinson bound, the fastest group velocity among
+any correlation functions.
+Finally, we have confirmed the size dependence of the
+estimated group velocity. As shown in Fig. 11, the LSWA
+shows much smaller size dependence than the exact anal-
+ysis in Fig. 6. The velocity is nearly converged for L ≥ 48
+and is extrapolated to 0.5J, corresponding to the Lieb-
+Robinson bound.
+
+9
+0.0
+2.0
+Czz(r = 1, t)
+×10−2
+Γ/J = 9.0
+0.0
+0.5
+Czz(r = 2, t)
+−0.2
+0.0
+0.2
+Czz(r = 3, t)
+0.0
+0.2
+Czz(r = 4, t)
+−0.1
+0.0
+0.1
+Czz(r = 5, t)
+0
+2
+4
+6
+time tJ
+−0.5
+0.0
+0.5
+Czz(r =
+√
+2, t)
+PEPS, Dvirt = 8
+ED, Ns = 5 × 5
+LSWA, Ns = 128 × 128
+FIG. 12.
+Equal-time longitudinal correlation functions ob-
+tained by the LSWA, the ED method, and the tensor-network
+method in 2D. We consider the quench to Γ/J = 9 for a finite
+system of Ns = L2, L = 128 by the LSWA (solid line), for
+a finite system of Ns = L2, L = 5 by ED simulations (small
+circles), and for the infinite system with the bond dimensions
+Dvirt = 8 by iPEPS simulations (dashed line).
+IV.
+RESULTS IN 2D
+Next, we examine the time-dependent correlations in
+2D using the ED method, the tensor-network method
+based on iPEPS, and the LSWA. As in the case of 1D,
+we estimate the group velocity after a sudden quench to
+a strong field.
+A.
+Results by the LSWA
+We apply the LSWA to calculate the spin-spin cor-
+relation functions and to extract the group velocity.
+In the case of the 1D transverse-field Ising model, the
+LSWA reproduces the exact results to the extent that
+the group velocity of the correlation propagation quan-
+titatively agrees at a sufficiently strong field.
+We will
+demonstrate that it reproduces the 2D correlations ob-
+tained by the nearly exact simulations much better than
+in 1D. It also allows us to estimate the group velocity
+from the correlations at farther distances than the ED
+and iPEPS simulations, as we will demonstrate below.
+We compare the longitudinal correlation functions ob-
+tained by the ED method and the LSWA in Fig. 12. The
+LSWA well reproduces the correlations obtained by the
+ED method up to the point where the second peak of
+0.0
+100.0
+Cxx(r = 1, t)
+×10−5
+Γ/J = 9.0
+0.0
+10.0
+Cxx(r = 2, t)
+0.0
+2.5
+Cxx(r = 3, t)
+0.0
+1.0
+Cxx(r = 4, t)
+0.0
+0.5
+Cxx(r = 5, t)
+0
+2
+4
+6
+time tJ
+0.0
+25.0
+Cxx(r =
+√
+2, t)
+PEPS, Dvirt = 8
+ED, Ns = 5 × 5
+LSWA, Ns = 128 × 128
+FIG. 13.
+Equal-time transverse correlation functions ob-
+tained by the LSWA, the ED method, and the tensor-network
+method in 2D. The parameters are the same as those in
+Fig. 12.
+0
+10
+20
+30
+time tJ
+10
+20
+30
+distance r
+0.00
+0.25
+0.50
+0.75
+1.00
+FIG. 14.
+Contour plot of normalized intensity of equal-
+time transverse correlation functions as a function of time
+and distance obtained by the LSWA in 2D. The param-
+eters L
+=
+128 and Γ/J
+=
+9 are the same as those
+in Fig. 12.
+We show the normalized correlation function
+Cxx
+connected(r, t)/ maxt∈[0,L/(2J)] Cxx
+connected(r, t) (∈ [0, 1]) along
+the horizontal axis [r = (r, 0)] for each distance up to r = 32.
+the envelope appears [see, e.g., Czz(r = 2, t) in Fig. 12].
+The period of oscillations almost coincides between the
+ED method and the LSWA. As expected in the LSWA
+in higher spatial dimensions, the agreement in 2D looks
+much better than in 1D (compare Fig. 7 and Fig. 12).
+The longitudinal correlations exhibit rapid oscillations
+as in the case of 1D. On the other hand, in contrast to
+the 1D case, where the earliest envelope peak has the
+
+10
+0
+20
+40
+60
+first peak time tJ
+0
+20
+40
+60
+distance r
+Γ/J = 9.0
+L = 128
+longitudinal
+transverse
+FIG. 15. Dominant-peak time dependence of distance for cor-
+relations obtained by the LSWA in 2D.
+0
+2
+4
+6
+8
+10
+Γ/J
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+vgroup/J
+L = 128
+from spin-wave dispersion
+longitudinal
+transverse
+FIG. 16.
+Field dependence of the group velocity obtained
+by the LSWA in 2D. The maximum group velocity vSW/J =
+(1−J/Γ+
+�
+1 − 2J/Γ)−1/2/
+√
+2 estimated from the spin-wave
+dispersion (see Appendix B 4) is shown as a reference. For
+Γ/J < 3, we only show the group velocity estimated from the
+transverse correlations because the envelopes of longitudinal
+correlations become unclear for a weaker field.
+largest intensity, it does not always exhibit the largest
+intensity in 2D. The order of the envelope peaks with the
+largest intensity varies with distance in 2D, which would
+make it more difficult to extract the group velocity. This
+observation may be ascribed to the complex interference
+effects in 2D.
+The transverse correlation function obtained by the
+LSWA also qualitatively reproduces the ED result (see
+Fig. 13). In contrast to the longitudinal correlations, the
+rapid oscillations are much weaker for r ≳ 3.
+To clarify how the correlation develops for a longer
+time and to examine the complex interference effects in
+2D, we depict the normalized intensity of the transverse
+correlations as a function of time and distance in Fig. 14.
+We observe a stronger intensity near the line satisfying
+r ≈ tJ. However, areas of high intensity are not continu-
+ously connected and are rather separated in small pieces.
+Such pieces are bundled together forming the boundary
+of the light cone. When we focus on the short-time and
+short-distance region, we can only look at the first small
+area of high intensity. If we use such data, we would in-
+0.00
+0.01
+0.02
+0.03
+0.04
+1/L
+0.00
+0.25
+0.50
+0.75
+1.00
+vgroup/J
+Γ/J = 9.0
+from spin-wave dispersion
+longitudinal
+transverse
+FIG. 17. Size dependence of the group velocity estimated by
+the LSWA in 2D.
+correctly estimate the group velocity. Indeed, as we will
+see later in Sec. IV B, the velocity obtained by the iPEPS
+method for a relatively short time shows a rather large
+error bar.
+To estimate the group velocity in 2D, we collect the
+peak times and distances in Fig. 15. Both correlations
+exhibit the consistent results. Although the jagged be-
+havior caused by the complex interference effects is ob-
+served in the data points, the distance becomes nearly
+proportional to the peak time for sufficiently large sys-
+tems. We extract the group velocity from one half of the
+slope so that it corresponds directly to the velocity of one
+quasiparticle.
+We show the field dependence of the group velocity
+along the horizontal axis for a large system (Ns = L2,
+L = 128) in Fig. 16. At a very strong transverse field,
+the velocity turns out to be nearly 0.5J. The velocity
+is likely to increase with decreasing the transverse field
+although we do not know to what extent the LSWA is
+reliable for a weaker field, corresponding to a relatively
+larger quench. The velocity estimated from correlations
+is basically on the curve represented by vSW/J = (1 −
+J/Γ +
+�
+1 − 2J/Γ)−1/2/
+√
+2, which is determined by the
+derivative of the spin-wave dispersion (see Appendix B 4).
+We finally check the size dependence of the estimated
+group velocity in Fig. 17. As in the case of 1D, the veloc-
+ity does not depend on the size significantly for L ≳ 48
+and converges to the value close to 0.5J. Therefore, the
+LSWA predicts that the speed of spin-spin correlation
+spreading is vgroup ≈ 0.5J for a small quench to Γ ≫ J
+in the 2D transverse-field Ising model.
+B.
+Tensor-network results
+As a complementary method to the LSWA, we use the
+tensor-network method based on the iPEPS to calculate
+the spin-spin correlation functions. We will see that the
+tensor-network method has an advantage in calculating
+the time dependence of correlations more accurately than
+the LSWA.
+Before presenting the correlations obtained by the
+
+11
+0.00
+0.05
+0.10
+0.15
+Γ/J = 5.0
+PEPS, Dvirt = 5, 6, 7, 8
+0.00
+0.05
+0.10
+0.15
+Γ/J = 6.0
+0.00
+0.05
+0.10
+0.15
+[e(tJ) − e(0)]/J
+Γ/J = 7.0
+0.00
+0.05
+0.10
+0.15
+Γ/J = 8.0
+0
+2
+4
+6
+8
+time tJ
+0.00
+0.05
+0.10
+0.15
+Γ/J = 9.0
+FIG. 18. Bond-dimension dependence of energy density ob-
+tained by iPEPS simulations.
+We consider the quench to
+Γ/J = 5, 6, 7, 8, and 9 and subtract the energy density
+at t = 0.
+The lines correspond to the time-dependent en-
+ergy density for the bond dimensions Dvirt = 5, 6, 7, and 8
+(from lighter to darker). The energy is nearly conserved for
+Dvirt ≥ 6 in a time frame tJ ∈ [0, 4].
+iPEPS simulations, let us examine the time range of the
+applicability. In general, as for numerical simulations of
+a quench dynamics, the obtained correlations would be
+reliable for short times until the energy is conserved. We
+investigate the time dependence of the energy density in
+the unit of Ising interaction J for different fields Γ/J
+with increasing the bond dimensions Dvirt (see Fig. 18).
+The energy density is nearly conserved for a short time
+(tJ ≲ 4) when Dvirt ≥ 6 regardless of the choice of the
+transverse field. Therefore, we will present the correla-
+tions within this time frame hereafter.
+We show the longitudinal correlation functions ob-
+tained by the ED and iPEPS simulations in Fig. 19. The
+ED method can deal with small systems in 2D and gives
+the correlations up to r ≈ 2 at the farthest. For these
+distances (r ≲ 2) and short times (tJ ≲ 4), the data by
+the ED and iPEPS methods completely overlap. Since
+the iPEPS method directly handles the infinite system,
+the ED method appears to provide the correlations that
+can almost be regarded as those at the thermodynamic
+limit in this regime. The iPEPS method can predict the
+peak positions of correlations at slightly farther distances
+until the energy is conserved (tJ ≲ 4). The peak in the
+envelope of correlation hits tJ ≈ 4 when r = 5, and thus
+the correlations up to r = 5 would be reliable for the
+velocity estimation.
+As in the case of the LSWA, the longitudinal corre-
+lations exhibit rapid oscillations. Moreover, in 2D, the
+tensor-network method also predicts that the earliest en-
+0.0
+2.0
+Czz(r = 1, t)
+×10−2
+Γ/J = 9.0
+0.0
+0.5
+Czz(r = 2, t)
+−0.2
+0.0
+0.2
+Czz(r = 3, t)
+0.0
+0.2
+Czz(r = 4, t)
+−0.1
+0.0
+0.1
+Czz(r = 5, t)
+0
+2
+4
+6
+time tJ
+−0.5
+0.0
+0.5
+Czz(r =
+√
+2, t)
+PEPS, Dvirt = 5, 6, 7, 8
+ED, Ns = 5 × 5
+FIG. 19. Equal-time longitudinal correlation functions in 2D.
+We consider the quench to Γ/J = 9 for the infinite system
+with the bond dimensions Dvirt = 5, 6, 7, and 8 (solid lines
+from lighter to darker) by iPEPS simulations and for a finite
+system of Ns = L2, L = 5 (small circles) by ED simulations.
+We show the short-time dynamics for distances r = 1, 2, . . . ,
+5, and
+√
+2. Both data agree very well for t/J ≲ 4.
+velope peak does not always correspond to the peak hav-
+ing the largest intensity (see Fig. 19). This observation
+suggests that the complex interference effects in 2D are
+not the artifact of the LSWA.
+We also examine the transverse correlation functions
+in Fig. 20. The ED and iPEPS methods provide almost
+the same correlations for r ≲ 2 and tJ ≲ 4. Again, the
+iPEPS method is applicable to farther distances up to
+r = 5. The rapid oscillations are quickly suppressed for
+r ≳ 3, as in the case of 1D and also as in the LSWA
+for 2D. The peak positions of the transverse correlations
+are nearly the same as those of the envelope peak in the
+longitudinal correlations.
+The qualitative behavior of correlations obtained by
+the tensor-network method and the LSWA is similar (see
+Figs. 12 and 13). The peak time of the correlations does
+not differ significantly between the two methods.
+Al-
+though the first-peak time is a little ahead in the LSWA,
+the peak-time difference is typically 0.5tJ in 2D, which
+is smaller than 2tJ in 1D. Because the LSWA is appli-
+cable to a much longer time, it is more suitable for esti-
+mating the group velocity. On the other hand, the time
+dependences of correlations agree well between the ED
+and tensor-network methods, whereas they slightly dif-
+fer between the ED method and the LSWA. Therefore,
+the tensor-network method is more appropriate to obtain
+
+12
+0.0
+100.0
+Cxx(r = 1, t)
+×10−5
+Γ/J = 9.0
+0.0
+10.0
+Cxx(r = 2, t)
+0.0
+2.5
+Cxx(r = 3, t)
+0.0
+1.0
+Cxx(r = 4, t)
+0.0
+0.5
+Cxx(r = 5, t)
+0
+2
+4
+6
+time tJ
+0.0
+25.0
+Cxx(r =
+√
+2, t)
+PEPS, Dvirt = 5, 6, 7, 8
+ED, Ns = 5 × 5
+FIG. 20. Equal-time transverse correlation functions in 2D.
+The parameters are the same as those in Fig. 19.
+0
+1
+2
+3
+4
+5
+6
+peak time tJ
+0
+2
+4
+6
+distance r
+Γ/J = 9.0
+Dvirt = 8
+longitudinal
+transverse
+FIG. 21. Dominant-peak time dependence of distance for cor-
+relations obtained by iPEPS simulations (Dvirt = 8) in 2D.
+The group velocity is estimated from the value r/[2t(r)] for
+distances r = 3, 4, and 5.
+quantitative data.
+To estimate the group velocity, we pick up the peak
+time for each distance from these correlations obtained
+by iPEPS simulations, as shown in Fig. 21. Since the data
+obtained by the bond dimensions Dvirt = 6, 7, and 8 are
+well converged, we present the result for the largest bond
+dimension Dvirt = 8. Within the range of time where
+the iPEPS simulations are considered to be reliable, it
+is hard to tell whether the light-cone-like spreading of
+correlations exists or not in 2D. However, as we have
+shown by the LSWA in Sec. IV A, such behavior is caused
+by the complex interference effects in 2D; it is highly
+0
+2
+4
+6
+8
+10
+Γ/J
+0.0
+0.2
+0.4
+0.6
+0.8
+vgroup(r)/J
+Dvirt = 8
+longitudinal, r = 5
+longitudinal, r = 4
+longitudinal, r = 3
+transverse, r = 5
+transverse, r = 4
+transverse, r = 3
+FIG. 22. Field dependence of the group velocity estimated
+from iPEPS simulations (Dvirt = 8) in 2D. For Γ/J < 5, we
+only show the group velocity estimated from the transverse
+correlations because the envelopes of longitudinal correlations
+become unclear for a weaker field.
+probable that the light cone exists. Therefore, we may
+assume that the distance eventually grows linearly with
+the peak time also in the iPEPS results. We then extract
+the group velocity as vgroup = r/[2t(r)] for each distance
+r.
+We mainly focus on the data for farther distances
+(r = 3, 4, and 5) because data for short distances tend
+to be off the light-cone behavior in general.
+The field dependence of the group velocities along
+the horizontal axis for distances r = 3, 4, and 5 are
+given in Fig. 22.
+They do not vary significantly for
+Γ/J ∈ [2, 9]. Since the velocity increases with increas-
+ing the distance, we estimate the group velocity as the
+average of the smallest and largest values with the error
+bar given by one half of their difference. It is given as
+vgroup/J = 0.54(11) for all transverse fields that we have
+studied using the iPEPS method. As we have discussed
+in Sec. IV A, the LSWA also predicts the similar velocity
+vgroup/J ≈ 0.5. The velocities obtained by the LSWA
+and those obtained by the tensor-network method agree
+within the error bar.
+V.
+DISCUSSION AND SUMMARY
+Let us compare our group velocity estimated from
+the spin-spin correlations with the recent Lieb-Robinson
+bound.
+In 1D, our estimate of the group velocity is
+vgroup = J/2.
+This is the same as the exact Lieb-
+Robinson bound vLR = J/2 in the 1D transverse-field
+Ising model, indicating that the spin-spin correlations
+propagate at the speed of fastest quasiparticles. On the
+other hand, the recent Lieb-Robinson bound for general
+lattice systems provides the speed vrecent = 1.51J [23].
+As was already pointed out in Ref. [23], it is approxi-
+mately three times as large as the exact Lieb-Robinson
+bound.
+In 2D, the group velocity along the horizontal axis is
+estimated to be vhorizontal ≈ J/2 as well. We do not know
+
+13
+the exact Lieb-Robinson bound in the 2D transverse-
+field Ising model so far.
+However, for a small quench
+within a disorder phase, we might expect that the fastest
+quasiparticles are responsible for spin-correlation spread-
+ing also in 2D. We come to this conclusion because the
+dispersion corresponding to the fastest quasiparticles ob-
+tained in perturbation theory [103] turns out to be the
+same as the dispersion estimated in the LSWA (see Ap-
+pendix B 4), and the LSWA reproduces the spin-spin cor-
+relations obtained by the exact analysis in 1D and those
+obtained by the nearly exact simulations in 2D fairly well
+(see Secs. III and IV). Therefore, even in 2D, it is natural
+to regard the group velocity of the spin-spin correlations
+as the tightest Lieb-Robinson bound. From the compar-
+ison between this value (the horizontal vhorizontal ≈ J/2
+or the diagonal vdiagonal ≈ J/
+√
+2 velocity) and the best
+currently available estimate (vrecent = 3.775J) [23], it
+is likely that there is still much room for improving the
+Lieb-Robinson bound in 2D.
+In
+conclusion,
+we
+have
+studied
+the
+correlation-
+spreading dynamics in the transverse-field Ising model
+on a chain and that on a square lattice. We have calcu-
+lated the longitudinal and transverse spin-spin correla-
+tion functions after a sudden quench starting from the
+disordered state to a strong field within a disordered
+phase. We have applied several analytical and numer-
+ical methods and crossvalidated all data.
+In 1D, we have compared the time-dependent correla-
+tions using the exact analytical formulae and the LSWA.
+We have found that the group velocity of the correlation
+propagation extracted from the LSWA results asymptot-
+ically approaches that from the exact analytical formulae
+as the transverse field increases. Moreover, the 1D spin-
+spin correlations are found to propagate at the speed of
+fastest quasiparticles corresponding to the exact Lieb-
+Robinson bound.
+In 2D, we have calculated the correlations using the
+ED method, the tensor-network method based on iPEPS,
+and the LSWA. As in the case of 1D, we have confirmed
+that the three methods reproduce nearly the same cor-
+relations within a short-time frame. The tensor-network
+method and the LSWA allow us to calculate the corre-
+lations for much farther distances than the ED method
+can deal with. In particular, the LSWA is convenient for
+estimating the propagation velocity, whereas the tensor-
+network method is advantageous in calculating the time
+dependence of correlations accurately. We have extracted
+the group velocity by these two methods and obtained the
+value which is nearly equal to one half of the magnitude of
+the Ising interaction. The group velocity of the spin-spin
+correlations in 2D turns out to be much smaller than the
+best currently available estimate for the Lieb-Robinson
+bound [23].
+Our findings on the group velocity would be helpful for
+future analog quantum simulations of Rydberg-atom ar-
+rays and stimulate further research on the Lieb-Robinson
+bound. The present tensor-network method, which can
+accurately calculate the dynamics in one of the most fun-
+damental two-dimensional quantum many-body systems,
+opens the possibility of future applications to other sys-
+tems.
+ACKNOWLEDGMENTS
+The authors acknowledge fruitful discussions with
+Shimpei
+Goto,
+Daichi
+Kagamihara,
+and
+Mathias
+Mikkelsen. This work was financially supported by JSPS
+KAKENHI (Grants Nos. JP18H05228, JP21H01014, and
+JP21K13855),
+by MEXT Q-LEAP (Grant No. JP-
+MXS0118069021), and by JST FOREST (Grant No.
+JPMJFR202T). The numerical computations were per-
+formed on computers at the Yukawa Institute Computer
+Facility and on computers at the Supercomputer Center,
+the Institute for Solid State Physics, the University of
+Tokyo.
+Appendix A: Details of exact calculations in 1D
+1.
+Hamiltonian
+We review the derivation of the exact form of two-body
+correlation functions after a sudden quench [40, 51–56].
+For simplicity, we consider the Hamiltonian
+ˆH = −
+�
+i
+ˆσz
+i ˆσz
+i+1 − ˜g
+�
+i
+ˆσx
+i ,
+(A1)
+which corresponds to the Hamiltonian in Eq. (1) with
+J = 4 and Γ = ˜gJ/2. To get the correlation functions for
+the original Hamiltonian, we have to use ˜g = 2Γ/J and
+replace time 4t with tJ.
+After the Jordan-Wigner transformation
+ˆσx
+i = 2ˆc†
+i ˆci − 1,
+(A2)
+ˆσz
+i =
+i−1
+�
+j=1
+(1 − 2ˆc†
+jˆcj)(ˆci + ˆc†
+i)
+(A3)
+and the Fourier transformation
+ˆcj =
+1
+√
+L
+�
+k
+e−ikrj ˆck,
+(A4)
+k = 2πn
+L ,
+(A5)
+where n = −(L − 1)/2, −(L − 3)/2, . . . , −1/2, 1/2, . . . ,
+(L − 3)/2, (L − 1)/2 for even L or n = −(L − 1)/2,
+(L − 3)/2, . . . , −2, −1, 0, 1, 2, . . . , (L − 3)/2, (L − 1)/2
+for odd L, we obtain
+ˆH =
+�
+k
+(ˆc†
+k ˆc−k)
+�
+˜ak −i˜bk
+i˜bk −˜ak
+� � ˆck
+ˆc†
+−k
+�
+,
+(A6)
+˜ak = ˜g + cos k,
+(A7)
+˜bk = sin k.
+(A8)
+
+14
+Using the Bogoliubov transformation
+� ˆck
+ˆc†
+−k
+�
+=
+�
+uk ivk
+ivk uk
+� � ˆγk
+ˆγ†
+−k
+�
+(A9)
+uk = cos θk
+2 ,
+(A10)
+vk = sin θk
+2 ,
+(A11)
+tan θk =
+sin k
+g + cos k
+(A12)
+satisfying u−k = uk and v−k = −vk, we get
+ˆH = 2
+�
+k
+ωk
+�
+ˆγ†
+kˆγk − 1
+2
+�
+,
+(A13)
+ωk =
+�
+˜g2 + 2˜g cos k + 1.
+(A14)
+The coefficients uk and vk can be described by ˜ak, ˜bk,
+and ωk as
+uk =
+˜ak − ωk
+�
+2ωk(ωk − ˜ak)
+= (˜ak − ωk)√ωk + ˜ak
+√2ωk|˜bk|
+,
+(A15)
+vk =
+˜bk
+�
+2ωk(ωk − ˜ak)
+= sgn(˜bk)√ωk + ˜ak
+√2ωk
+.
+(A16)
+In the present paper, we mainly consider the quantum
+quench from ˜g = g0 to ˜g = g < ∞ within the disor-
+dered phase.
+We write the Hamiltonian before (after)
+the quench as ˆH ( ˆH′). For ˜g → ∞, we have uk → 0 and
+vk → sgn(sin k).
+2.
+Longitudinal correlation functions
+We evaluate the time-dependent longitudinal correla-
+tion functions defined as
+¯Czz(r, t) = ⟨ψ0|ei ˆ
+H′tˆσz
+i ˆσz
+i+re−i ˆ
+H′t|ψ0⟩
+(A17)
+= ⟨ψ0|ei ˆ
+H′t(ˆc†
+i + ˆci)
+�
+�
+i+r−1
+�
+j=i
+(1 − 2ˆc†
+jˆcj)
+�
+�
+· (ˆc†
+i+r + ˆci+r)e−i ˆ
+H′t|ψ0⟩.
+(A18)
+Using the equality 1 − 2ˆc†
+jˆcj = (ˆc†
+j + ˆcj)(ˆc†
+j − ˆcj) and
+defining the operators
+ˆAi = ˆc†
+i + ˆci,
+ˆBi = ˆc†
+i − ˆci,
+(A19)
+ˆAi(t) = ei ˆ
+H′t ˆAie−i ˆ
+H′t,
+ˆBi(t) = ei ˆ
+H′t ˆBie−i ˆ
+H′t,
+(A20)
+we obtain the correlation function
+¯Czz(r, t) = ⟨ψ0| ˆBi(t) ˆAi+1(t) ˆBi+1(t) ˆAi+2(t) ˆBi+2(t) · · ·
+· ˆAi+r−1(t) ˆBi+r−1(t) ˆBi+r(t)|ψ0⟩.
+(A21)
+It can be evaluated by the Pfaffian of a 2r × 2r skew
+symmetric matrix A using the Wick’s theorem:
+¯Czz(r, t) = pfA.
+(A22)
+The matrix A is given as
+A =
+� S
+G
+−GT Q
+�
+(A23)
+with matrices
+S =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+S0,1
+S0,2
+· · ·
+S0,r−2
+S0,r−1
+−S0,1
+0
+S1,2
+· · ·
+S1,r−2
+S1,r−1
+−S0,2
+−S1,2
+0
+· · ·
+S2,r−2
+S2,r−1
+...
+...
+...
+...
+...
+...
+−S0,r−2
+−S1,r−2
+−S2,r−2
+· · ·
+0
+Sr−2,r−1
+−S0,r−1
+−S1,r−1
+−S2,r−1
+· · ·
+−Sr−2,r−1
+0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(A24)
+Q =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+Q0,1
+Q0,2
+· · ·
+Q0,r−2
+Q0,r−1
+−Q0,1
+0
+Q1,2
+· · ·
+Q1,r−2
+Q1,r−1
+−Q0,2
+−Q1,2
+0
+· · ·
+Q2,r−2
+Q2,r−1
+...
+...
+...
+...
+...
+...
+−Q0,r−2
+−Q1,r−2
+−Q2,r−2
+· · ·
+0
+Qr−2,r−1
+−Q0,r−1
+−Q1,r−1
+−Q2,r−1
+· · ·
+−Qr−2,r−1
+0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(A25)
+G =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+G0,1
+G0,2
+G0,3
+· · ·
+G0,r−1
+G0,r
+G1,1
+G1,2
+G1,3
+· · ·
+G1,r−1
+G1,r
+G2,1
+G2,2
+G2,3
+· · ·
+G2,r−1
+G2,r
+...
+...
+...
+...
+...
+...
+Gr−2,1
+Gr−2,2
+Gr−2,3
+· · ·
+Gr−2,r−1
+Gr−2,r
+Gr−1,1
+Gr−1,2
+Gr−1,3
+· · ·
+Gr−1,r−1
+Gr−1,r
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+. (A26)
+Here we define time-dependent correlation functions
+Si,j = ⟨ ˆBi(t) ˆBj(t)⟩,
+(A27)
+Qi,j = ⟨ ˆAi(t) ˆAj(t)⟩,
+(A28)
+Gi,j = ⟨ ˆBi(t) ˆAj(t)⟩ = −⟨ ˆAj(t) ˆBi(t)⟩
+(A29)
+and use that they are translational invariant. We will
+obtain the explicit form of evaluating Si,j, Qi,j, and Gi,j
+in Appendix A 4.
+3.
+Transverse correlation functions
+We evaluate the time-dependent transverse correlation
+functions defined as
+¯Cxx(r, t) = ⟨ψ0|ei ˆ
+H′tˆσx
+i ˆσx
+i+re−i ˆ
+H′t|ψ0⟩
+(A30)
+= ⟨ψ0|ei ˆ
+H′t(2ˆc†
+i ˆci − 1)(2ˆc†
+i+rˆci+r − 1)e−i ˆ
+H′t|ψ0⟩.
+(A31)
+Using the equality 1 − 2ˆc†
+jˆcj = (ˆc†
+j + ˆcj)(ˆc†
+j − ˆcj) and the
+expressions for ˆAi and ˆBi, we obtain
+¯Cxx(r, t) = ⟨ψ0| ˆAi(t) ˆBi(t) ˆAi+r(t) ˆBi+r(t)|ψ0⟩
+(A32)
+
+15
+= G2
+i,i − Qi,i+rSi,i+r + (−Gi+r,i)Gi,i+r.
+(A33)
+Subtracting
+the
+correlation
+of
+the
+local
+trans-
+verse magnetization,
+which is given as ⟨ˆσx
+i (t)⟩
+=
+⟨ψ0|ei ˆ
+H′tˆσx
+i e−i ˆ
+H′t|ψ0⟩ = −⟨ψ0| ˆAi(t) ˆBi(t)|ψ0⟩ = −Gi,i =
+−G0,0, we get the connected correlation function
+¯Cxx
+connected(r, t) := ¯Cxx(r, t) − ⟨ˆσx
+i (t)⟩⟨ˆσx
+i+r(t)⟩
+(A34)
+= −Qi,i+rSi,i+r − Gi+r,iGi,i+r.
+(A35)
+The explicit form of evaluating Si,j, Qi,j, and Gi,j will
+be given in Appendix A 4.
+4.
+Single-particle correlation functions for fermions
+Let us focus on t = 0 operators. Using the Fourier
+transformation ˆcj =
+1
+√
+L
+�
+k e−ikrj ˆck and the Bogoli-
+ubov transformation ˆck = ukˆγk + ivkˆγ−k, and then split-
+ting the sum �
+k into the positive and negative parts
+�
+k>0 + �
+k<0, we rewrite ˆAi and ˆBi as
+ˆAi = ˆa†
+i + ˆai,
+(A36)
+ˆBi = ˆb†
+i − ˆbi
+(A37)
+with
+ˆai =
+1
+√
+L
+�
+k>0
+�
+eikrj(uk − ivk)ˆγk + e−ikrj(uk + ivk)ˆγ−k
+�
+,
+(A38)
+ˆbi =
+1
+√
+L
+�
+k>0
+�
+eikrj(uk + ivk)ˆγk + e−ikrj(uk − ivk)ˆγ−k
+�
+.
+(A39)
+The operators ˆai and ˆbi satisfy
+{ˆai, ˆaj} = {ˆbi,ˆbj} = {ˆai,ˆbj} = 0,
+(A40)
+{ˆai, ˆa†
+j} = {ˆbi,ˆb†
+j} = δij,
+(A41)
+{ˆai,ˆb†
+j} = {ˆa†
+i,ˆbj} =: −Gab
+i−j
+(A42)
+with
+Gab
+i−j = − 1
+√
+L
+�
+k>0
+�
+eik(ri−rj)(uk − ivk)2
++ e−ik(ri−rj)(uk + ivk)2�
+.
+(A43)
+After the quench, the Heisenberg equation for ˆck(t) is
+written as
+i d
+dtˆck(t) = −2˜a′
+kˆck(t) − 2i˜b′
+kˆc†
+−k(t),
+(A44)
+˜a′
+k = g + cos k,
+(A45)
+˜b′
+k = sin k,
+(A46)
+where the prime symbols indicate the parameters after
+the quench. As in the static case, we can introduce ˜uk(t)
+and ˜vk(t) for the Bogoliubov transformation at time t
+� ˆck(t)
+ˆc†
+−k(t)
+�
+=
+�
+˜uk(t) −˜v∗
+k(t)
+˜vk(t)
+˜u∗
+k(t)
+� � ˆγk
+ˆγ†
+−k
+�
+(A47)
+satisfying
+˜u−k(t) = ˜uk(t),
+(A48)
+˜v−k(t) = −˜vk(t),
+(A49)
+|˜uk(t)|2 + |˜vk(t)|2 = 1.
+(A50)
+Here ˆγk corresponds to the Bogoliubov excitations before
+the quench. From Eqs. (A44) and (A47), ˜uk(t) and ˜vk(t)
+should satisfy
+i d
+dt
+�
+˜uk(t)
+˜vk(t)
+�
+=
+�
+−2˜a′
+k −2i˜b′
+k
+2i˜b′
+k
+2˜a′
+k
+� �
+˜uk(t)
+˜vk(t)
+�
+.
+(A51)
+Then, for the sudden quench (g0 → g), ˜uk(t) and ˜vk(t)
+are explicitly given as
+�
+˜uk(t)
+˜vk(t)
+�
+=
+�
+� uk cos 2ω′
+kt + i ˜a′
+kuk+˜b′
+kvk
+ω′
+k
+sin 2ω′
+kt
+−ivk cos 2ω′
+kt +
+˜b′
+kuk−˜a′
+kvk
+ω′
+k
+sin 2ω′
+kt
+�
+� ,
+(A52)
+where each variable is represented as
+˜ak = g0 + cos k,
+(A53)
+˜bk = sin k,
+(A54)
+ωk =
+�
+g2
+0 + 2g0 cos k + 1,
+(A55)
+ω′
+k =
+�
+g2 + 2g cos k + 1,
+(A56)
+and Eqs.(A45) and (A46). The parameters uk and vk are
+defined in Eqs. (A15) and (A16).
+As in the case of t = 0, the Heisenberg representation
+of each operator satisfies
+ˆAi(t) = ˆa†
+i(t) + ˆai(t),
+(A57)
+ˆBi(t) = ˆb†
+i(t) − ˆbi(t)
+(A58)
+with
+ˆai(t) =
+1
+√
+L
+�
+k>0
+�
+eikrj[˜uk(t) + ˜vk(t)]ˆγk
++ e−ikrj[˜uk(t) − ˜vk(t)]ˆγ−k
+�
+,
+(A59)
+ˆbi(t) =
+1
+√
+L
+�
+k>0
+�
+eikrj[˜uk(t) − ˜vk(t)]ˆγk
++ e−ikrj[˜uk(t) + ˜vk(t)]ˆγ−k
+�
+.
+(A60)
+Note that ˆai(t) ̸= ei ˆ
+H′tˆaie−i ˆ
+H′t and ˆbi(t) ̸= ei ˆ
+H′tˆbie−i ˆ
+H′t
+in our notation. Then, the commutation relations for the
+operators are
+{ˆai(t), ˆaj(t)} = {ˆbi(t),ˆbj(t)} = {ˆai(t),ˆbj(t)} = 0,
+(A61)
+
+16
+{ˆai(t), ˆa†
+j(t)} =: −Gaa
+i−j(t),
+(A62)
+{ˆbi(t),ˆb†
+j(t)} =: −Gbb
+i−j(t),
+(A63)
+{ˆai(t),ˆb†
+j(t)} = {ˆa†
+i(t),ˆbj(t)} =: −Gab
+i−j(t)
+(A64)
+with
+Gaa
+i−j(t) = − 2
+L
+�
+k>0
+�
+cos[k(ri − rj)][|˜uk(t)|2 + |˜vk(t)|2]
++ i sin[k(ri − rj)][˜uk(t)˜v∗
+k(t) + ˜vk(t)˜u∗
+k(t)]
+�
+,
+(A65)
+Gbb
+i−j(t) = − 2
+L
+�
+k>0
+�
+cos[k(ri − rj)][|˜uk(t)|2 + |˜vk(t)|2]
+− i sin[k(ri − rj)][˜uk(t)˜v∗
+k(t) + ˜vk(t)˜u∗
+k(t)]
+�
+,
+(A66)
+Gab
+i−j(t) = − 2
+L
+�
+k>0
+�
+cos[k(ri − rj)][|˜uk(t)|2 − |˜vk(t)|2]
+− i sin[k(ri − rj)][˜uk(t)˜v∗
+k(t) − ˜vk(t)˜u∗
+k(t)]
+�
+.
+(A67)
+Using these results, we obtain
+S0,r = ⟨ ˆB0(t) ˆBr(t)⟩ = −⟨{ˆb0(t),ˆb†
+r(t)}⟩ = +Gbb
+−r(t),
+(A68)
+Q0,r = ⟨ ˆA0(t) ˆAr(t)⟩ = +⟨{ˆa0(t), ˆa†
+r(t)}⟩ = −Gaa
+−r(t),
+(A69)
+G0,r = ⟨ ˆB0(t) ˆAr(t)⟩ = −⟨{ˆb0(t), ˆa†
+r(t)}⟩ = +Gab
++r(t).
+(A70)
+5.
+Maximum group velocity
+In the 1D transverse-field Ising model, the Lieb-
+Robinson bound is obtained as the maximum group ve-
+locity determined from the derivative of the band disper-
+sion [44–47]. It is given as
+vLR = max
+k
+����
+dωk
+dk
+���� =
+�
+2˜g
+if ˜g ≤ 1
+2
+if ˜g ≥ 1
+(A71)
+for the Hamiltonian defined in Eq. (A1), and it is ob-
+tained as
+vLR =
+�
+Γ
+if Γ ≤ J/2
+J/2
+if Γ ≥ J/2
+(A72)
+for the original Hamiltonian given in Eq. (1).
+Appendix B: Details of the LSWA
+1.
+Bosonic quadratic Hamiltonian
+We consider a sudden quench within the disordered
+phase for the Hamiltonian in Eq. (1).
+We investigate
+the effect of small quantum fluctuations around the com-
+pletely disordered state at Γ → ∞ using a linear spin-
+wave expansion [58–61]. As long as we consider a quench
+to a strong transverse field so that the transverse mag-
+netization is large enough (⟨Sx
+i ⟩ ≈ 1/2), this approach
+should be a good approximation. We specifically study
+the parameter region Γ ∈ (Γclassical
+c
+, ∞), where the clas-
+sical transition point obtained by the mean-field approx-
+imation [17, 104, 105] is Γclassical
+c
+= JD with D being
+the spatial dimension. We review the derivation of the
+longitudinal correlation functions [59] and then calculate
+the transverse correlation functions.
+We apply the linearized Holstein-Primakoff transfor-
+mation, which is given as
+ˆSx
+i = S − ˆb†
+iˆbi,
+ˆSz
+i =
+√
+2S
+2
+(ˆb†
+i + ˆbi)
+(B1)
+before the quench and is represented as
+ˆSx
+i
+′ = S − ˆa†
+iˆai,
+ˆSz
+i
+′ =
+√
+2S
+2
+(ˆa†
+i + ˆai)
+(B2)
+after the quench.
+The prime symbols indicate opera-
+tors after the quench. After the Fourier transformation
+(ˆbi =
+1
+√
+LD
+�
+k e−ik·riˆbk, ˆai =
+1
+√
+LD
+�
+k e−ik·riˆak) and
+the Bogoliubov transformation, we obtain the Hamilto-
+nian for free bosons before the quench (up to constant
+terms) as
+ˆH =
+�
+k
+Ωk ˆβ†
+k ˆβk,
+(B3)
+ˆbk = sk ˆβk + tk ˆβ†
+−k,
+(B4)
+and that after the quench (up to constant terms) as
+ˆH′ =
+�
+k
+Ω′
+k ˆα†
+k ˆαk,
+(B5)
+ˆak = s′
+k ˆαk + t′
+k ˆα†
+−k.
+(B6)
+Here the corresponding dispersions and coefficients are
+defined as
+Ωk = sgn(Ak)
+�
+A2
+k − B2
+k,
+(B7)
+sk = sgn(Ak)
+�
+1
+2
+�|Ak|
+|Ωk| + 1
+�
+,
+(B8)
+tk = −sgn(Bk)
+�
+1
+2
+�|Bk|
+|Ωk| − 1
+�
+,
+(B9)
+Ω′
+k = sgn(A′
+k)
+�
+A′
+k
+2 − B′
+k
+2,
+(B10)
+s′
+k = sgn(A′
+k)
+�
+1
+2
+�|A′
+k|
+|Ω′
+k| + 1
+�
+,
+(B11)
+t′
+k = −sgn(B′
+k)
+�
+1
+2
+�|B′
+k|
+|Ω′
+k| − 1
+�
+,
+(B12)
+where
+Ak = −z
+2JSγk + Γ,
+Bk = −z
+2JSγk,
+(B13)
+
+17
+A′
+k = −z
+2J′Sγk + Γ′,
+B′
+k = −z
+2J′Sγk,
+(B14)
+γk = 1
+D
+D
+�
+ν=1
+cos kν
+(B15)
+with z = 2D being the coordination number. We add
+the prime symbols to distinguish parameters after the
+quench.
+At t = 0, the vacuum of both Hamiltonians are the
+same, and bosons before the Holstein-Primakoff trans-
+formation satisfy ˆbk = ˆak. Then, these operators should
+fulfill
+� ˆbk
+ˆb†
+−k
+�
+=
+�
+sk tk
+tk sk
+� � ˆβk
+ˆβ†
+−k
+�
+=
+�
+s′
+k t′
+k
+t′
+k s′
+k
+� � ˆαk
+ˆα†
+−k
+�
+=
+� ˆak
+ˆa†
+−k
+�
+.
+(B16)
+This means that Bogoliubov excitations before and after
+the quench are connected by
+� ˆαk
+ˆα†
+−k
+�
+=
+�
+s′
+ksk − t′
+ktk s′
+ktk − skt′
+k
+s′
+ktk − skt′
+k s′
+ksk − t′
+ktk
+� � ˆβk
+ˆβ†
+−k
+�
+=:
+�
+uk vk
+vk uk
+� � ˆβk
+ˆβ†
+−k
+�
+,
+(B17)
+where the coefficients satisfy
+u2
+k − v2
+k = s′
+k
+2 − t′
+k
+2 = s2
+k − t2
+k = 1.
+(B18)
+2.
+Longitudinal correlation functions
+We evaluate the time-dependent longitudinal correla-
+tion functions defined as
+Czz(r, t) = ⟨ψ0|ei ˆ
+H′t ˆSz
+r ˆSz
+0e−i ˆ
+H′t|ψ0⟩
+(B19)
+= S
+2 ⟨ψ0|ei ˆ
+H′t(ˆb†
+r + ˆbr)(ˆb†
+0 + ˆb0)e−i ˆ
+H′t|ψ0⟩.
+(B20)
+Writing
+them
+in
+the
+Fourier
+space
+(ˆbi
+=
+1
+√
+LD
+�
+k e−ik·riˆbk)
+and
+in
+the
+Heisenberg
+picture,
+we obtain
+Czz(r, t) =
+S
+2LD
+�
+k
+eik·r⟨ψ0|[ˆb†
+k(t)ˆb†
+−k(t) + ˆb†
+k(t)ˆbk(t)
++ ˆb−k(t)ˆb†
+−k(t) + ˆb−k(t)ˆbk(t)]|ψ0⟩.
+(B21)
+We then replace all operators ˆbk(t) by ˆβk.
+Because
+ˆbk(t) = s′
+k ˆαk(t) + t′
+k ˆα†
+−k(t), ˆαk(t) = e−iΩ′
+ktˆαk, and
+ˆαk = uk ˆβk + vk ˆβ†
+−k, the following relation holds:
+ˆbk(t) = (e−iΩ′
+kts′
+kuk + e+iΩ′
+ktt′
+kvk)ˆβk
++ (e−iΩ′
+kts′
+kvk + e+iΩ′
+ktt′
+kuk)ˆβ†
+−k.
+(B22)
+After straightforward calculations using ˆβk|ψ0⟩ = 0, we
+obtain
+Czz(r, t) =
+S
+2LD
+�
+k
+eik·r(s′
+k + t′
+k)2
+· (u2
+k + v2
+k + 2ukvk cos 2Ω′
+kt).
+(B23)
+Substituting uk and vk with sk, s′
+k, tk, and t′
+k using
+Eq. (B17), we get
+Czz(r, 0) =
+S
+2LD
+�
+k
+eik·r(s′
+k + t′
+k)2(uk + vk)2
+(B24)
+=
+S
+2LD
+�
+k
+eik·r(sk + tk)2,
+(B25)
+˜Czz(r, t) := Czz(r, t) − Czz(r, 0)
+(B26)
+=
+S
+2LD
+�
+k
+eik·r(s′
+k + t′
+k)22ukvk(cos 2Ω′
+kt − 1)
+(B27)
+=
+S
+2LD
+�
+k
+eik·r(−2)[(s2
+k + t2
+k)s′
+kt′
+k
+− (s′
+k
+2 + t′
+k
+2)sktk](s′
+k + t′
+k)2(cos 2Ω′
+kt − 1).
+(B28)
+Using the relations defined in Eqs. (B7-B15), we finally
+get [59]
+Czz(r, 0) =
+S
+2LD
+�
+k
+eik·r
+Ωk
+Ak + Bk
+,
+(B29)
+˜Czz(r, t) =
+S
+2LD
+�
+k
+eik·r AkB′
+k − A′
+kBk
+Ωk(A′
+k + B′
+k) (cos 2Ω′
+kt − 1).
+(B30)
+For Γ → ∞ before the quench, Ωk/(Ak + Bk) = 1 is
+satisfied, and hence, Czz(r, 0) = S/2 × δr,Lm (mν: inte-
+ger, ν = 1, 2, . . . , D) holds. This means that Czz(r, t) =
+˜Czz(r, t) for 1 ≤ rν ≤ L/2 (ν = 1, 2, . . . , D).
+Be-
+sides, when J = 0 and J′ ≪ Γ′ < Γ, the intensity
+of the correlation would be approximately |Czz(r, t)| =
+O[S/LD × �
+k B′
+k/(A′
+k + B′
+k)] = O(zS2J′/Γ′).
+3.
+Transverse correlation functions
+We evaluate the time-dependent transverse correlation
+functions defined as
+Cxx(r, t) = ⟨ψ0|ei ˆ
+H′t ˆSx
+r ˆSx
+0e−i ˆ
+H′t|ψ0⟩
+(B31)
+= ⟨ψ0|ei ˆ
+H′t(S − ˆb†
+rˆbr)(S − ˆb†
+0ˆb0)e−i ˆ
+H′t|ψ0⟩
+(B32)
+and the connected one defined as
+Cxx
+connected(r, t) = Cxx(r, t) − ⟨ψ0|ei ˆ
+H′t ˆSx
+re−i ˆ
+H′t|ψ0⟩
+· ⟨ψ0|ei ˆ
+H′t ˆSx
+0e−i ˆ
+H′t|ψ0⟩.
+(B33)
+
+18
+Writing
+them
+in
+the
+Fourier
+space
+(ˆbi
+=
+1
+√
+LD
+�
+k e−ik·riˆbk)
+and
+in
+the
+Heisenberg
+picture,
+we obtain
+Cxx(r, t) = S2 − 2S
+LD
+�
+k
+⟨ψ0|ˆb†
+k(t)ˆbk(t)|ψ0⟩
++
+1
+L2D
+�
+k,l,p
+ei(k−l)·r⟨ψ0|ˆb†
+k(t)ˆbl(t)ˆb†
+p(t)ˆbk−l+p(t)|ψ0⟩
+(B34)
+and
+Cxx
+connected(r, t) = −
+�
+S
+LD
+�
+k
+⟨ψ0|ˆb†
+k(t)ˆbk(t)|ψ0⟩
+�2
++
+1
+L2D
+�
+k,l,p
+ei(k−l)·r⟨ψ0|ˆb†
+k(t)ˆbl(t)ˆb†
+p(t)ˆbk−l+p(t)|ψ0⟩.
+(B35)
+As in the case of longitudinal correlation functions,
+we replace ˆbk(t) by ˆβk and use ˆβk|ψ0⟩ = 0.
+Non-
+vanishing terms contain ⟨ψ0|ˆβ−k ˆβ†
+−l ˆβ−p ˆβ†
+−(k−l+p)|ψ0⟩ =
+δk,l, ⟨ψ0|ˆβ−k ˆβl ˆβ†
+p ˆβ†
+−(k−l+p)|ψ0⟩
+=
+δk,−p + δl,p, and
+⟨ψ0|ˆβ−k ˆβ†
+−k|ψ0⟩ = 1. After straightforward calculations,
+we get
+1
+LD
+�
+k
+⟨ψ0|ˆb†
+k(t)ˆbk(t)|ψ0⟩
+=
+1
+LD
+�
+k
+(s′
+k
+2v2
+k + t′
+k
+2u2
+k + 2s′
+kt′
+kukvk cos 2Ω′
+kt) (B36)
+and
+Cxx
+connected(r, t) =
+�����
+1
+LD
+�
+k
+eik·r �
+[(s′
+k
+2 + t′
+k
+2) cos 2Ω′
+kt + i sin 2Ω′
+kt]ukvk + s′
+kt′
+k(u2
+k + v2
+k)
+������
+2
++
+1
+L2D
+�
+k,l
+ei(k−l)·r �
+t2
+k + 2s′
+kt′
+kukvk(cos 2Ω′
+kt − 1)
+� �
+s2
+l + 2s′
+lt′
+lulvl(cos 2Ω′
+lt − 1)
+�
+(B37)
+=
+�����
+1
+LD
+�
+k
+eik·r �
+[(s′
+k
+2 + t′
+k
+2) cos 2Ω′
+kt + i sin 2Ω′
+kt]ukvk + s′
+kt′
+k(u2
+k + v2
+k)
+������
+2
++
+�����
+1
+LD
+�
+k
+eik·r �
+t2
+k + 2s′
+kt′
+kukvk(cos 2Ω′
+kt − 1)
+�
+�����
+2
++ 1
+LD
+�
+k
+�
+t2
+k + 2s′
+kt′
+kukvk(cos 2Ω′
+kt − 1)
+�
+× δr,Lm
+(B38)
+with mν (ν = 1, 2, . . . , D) being integer. Substituting the
+parameters uk, vk, sk, tk, s′
+k, and t′
+k with the parameters
+Ak, Bk, Ωk, A′
+k, B′
+k, and Ω′
+k, we finally get
+Cxx
+connected(r, t) =
+�����
+1
+LD
+�
+k
+eik·r
+�
+−B′
+k
+2
+AkA′
+k − BkB′
+k
+ΩkΩ′
+k
+2
++ AkB′
+k − A′
+kBk
+2ΩkΩ′
+k
+�A′
+k
+Ω′
+k
+cos 2Ω′
+kt + i sin 2Ω′
+kt
+�������
+2
++
+�����
+1
+LD
+�
+k
+eik·r
+��
+A′
+k
+2
+AkA′
+k − BkB′
+k
+ΩkΩ′
+k
+2
+− 1
+2
+�
+− B′
+k
+2
+AkB′
+k − A′
+kBk
+ΩkΩ′
+k
+2
+cos 2Ω′
+kt
+������
+2
++ 1
+LD
+�
+k
+��
+A′
+k
+2
+AkA′
+k − BkB′
+k
+ΩkΩ′
+k
+2
+− 1
+2
+�
+− B′
+k
+2
+AkB′
+k − A′
+kBk
+ΩkΩ′
+k
+2
+cos 2Ω′
+kt
+�
+× δr,Lm
+(B39)
+=
+�����
+1
+LD
+�
+k
+eik·r
+�
+− Bk
+2Ωk
++ AkB′
+k − A′
+kBk
+2ΩkΩ′
+k
+�A′
+k
+Ω′
+k
+(cos 2Ω′
+kt − 1) + i sin 2Ω′
+kt
+�������
+2
++
+�����
+1
+LD
+�
+k
+eik·r
+�
+1
+2
+�Ak
+Ωk
+− 1
+�
+− B′
+k
+2
+AkB′
+k − A′
+kBk
+ΩkΩ′
+k
+2
+(cos 2Ω′
+kt − 1)
+������
+2
+
+19
++ 1
+LD
+�
+k
+�
+1
+2
+�Ak
+Ωk
+− 1
+�
+− B′
+k
+2
+AkB′
+k − A′
+kBk
+ΩkΩ′
+k
+2
+(cos 2Ω′
+kt − 1)
+�
+× δr,Lm
+(B40)
+with mν (ν = 1, 2, . . . , D) being integer.
+When J = 0 and J′ ≪ Γ′ < Γ, the intensity of the
+correlation would be approximately |Cxx
+connected(r, t)| =
+O[|1/LD × �
+k B′
+kA′
+k/(Ω′
+k)2|2] = O(z2S2J′2/Γ′2).
+4.
+Dispersion relation and maximum group velocity
+Within the LSWA, the dispersion relation for S = 1/2
+is expressed as
+Ωk =
+�
+Γ2 − z
+2ΓJγk,
+(B41)
+= Γ
+�
+1 − 1
+2
+zJ
+2Γγk − 1
+8
+�zJ
+2Γ
+�2
+γ2
+k + O
+��zJ
+2Γ
+�3��
+.
+(B42)
+This result is consistent with the dispersion relation
+Ωperturb
+k
+= Γ
+�
+1 − 1
+2
+zJ
+2Γγk − 1
+8
+�zJ
+2Γ
+�2 �
+γ2
+k − 2
+9
+�
+− 1
+16
+�zJ
+2Γ
+�3 �
+γ3
+k − γk
+2 − 3z
+z2
+�
++ O
+��zJ
+2Γ
+�4��
+(B43)
+obtained by the perturbation calculation [103] up to
+O{[(zJ)/(2Γ)]2} terms.
+It is widely believed that the Lieb-Robinson bound
+should be the maximum group velocity determined from
+the derivative of band dispersion [44–47]. Although the
+dispersion obtained by the LSWA does not necessarily
+offer the exact Lieb-Robinson bound, we calculate the
+reference value using the dispersion. In 1D, the maxi-
+mum group velocity of the spin-wave dispersion is given
+as
+vSW = max
+k
+����
+dΩk
+dk
+���� = J
+√
+2
+�
+�1 +
+�
+1 −
+�J
+Γ
+�2
+�
+�
+−1/2
+.
+(B44)
+For Γ → ∞, the group velocity satisfies vSW → J/2 =
+vLR, reproducing the exact maximum group velocity. On
+the other hand, for Γ ∈ (Γclassical
+c
+, ∞), the LSWA always
+gives vSW > J/2 = vLR.
+Its worst (largest) estimate
+vSW = J/
+√
+2 ≈ 0.707J at Γ = Γclassical,1D
+c
+(= J) is still
+tighter than the recent bound 1.51J obtained by the gen-
+eral formula for the Lieb-Robinson bound [23].
+In the same manner, we can extract the group velocity
+as vSW = maxk ∇k Ωk from the spin-wave dispersion in
+higher dimensions. In 2D, the horizontal and diagonal
+velocities are given as
+vSW,horizontal = J
+√
+2
+�
+1 − J
+Γ +
+�
+1 − 2J
+Γ
+�−1/2
+,
+(B45)
+vSW,diagonal = J
+�
+�1 +
+�
+1 −
+�2J
+Γ
+�2
+�
+�
+−1/2
+,
+(B46)
+respectively. The maximum velocity along the horizon-
+tal (diagonal) axis is estimated to be vSW,horizontal →
+J/2 (vSW,diagonal
+→ J/
+√
+2) for Γ → ∞.
+On the
+other hand, for both axes, it approaches the value J
+(vSW,horizontal, vSW,diagonal → J) for Γ → Γclassical,2D
+c
+(=
+2J).
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diff --git a/sdAzT4oBgHgl3EQfcvxN/content/tmp_files/load_file.txt b/sdAzT4oBgHgl3EQfcvxN/content/tmp_files/load_file.txt
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@@ -0,0 +1,1754 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf,len=1753
+page_content='Dynamics of correlation spreading in low-dimensional transverse-field Ising models  Ryui Kaneko∗ and Ippei Danshita†  Department of Physics, Kindai University, Higashi-Osaka, Osaka 577-8502, Japan  (Dated: January 5, 2023)  We investigate the dynamical spreading of spatial correlations after a quantum quench starting  from a magnetically disordered state in the transverse-field Ising model at one (1D) and two spa-  tial dimensions (2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We analyze specifically the longitudinal and transverse spin-spin correlation  functions at equal time with use of several methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' From the comparison of the results in 1D  obtained by the linear spin-wave approximation (LSWA) and those obtained by the rigorous ana-  lytical approach, we show that the LSWA can asymptotically reproduce the exact group velocity  in the limit of strong transverse fields while it fails to capture the detailed time dependence of the  correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' By applying the LSWA to the 2D case, in which the rigorous analytical ap-  proach is unavailable, we estimate the propagation velocity to be Ja/(2ℏ) at the strong-field limit,  where J is the Ising interaction and a is the lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We also utilize the tensor-network  method based on the projected-entangled pair states for 2D and quantitatively compute the time  evolution of the correlation functions for a relatively short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Our findings provide useful bench-  marks for quantum simulation experiments of correlation spreading and theoretical refinement of  the Lieb-Robinson bound in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  INTRODUCTION  Neutral atoms trapped in optical-tweezer arrays are  promising platforms for analog quantum simulations [1–  3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The controllability of individual atoms with laser  pulses and of interatomic interactions via Rydberg ex-  citations enables one to realize fast and high-fidelity  quantum operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Recent rapid technological devel-  opments allow for manipulating many Rydberg atoms  in large arrays [4–6] and investigating the ground state  of quantum lattice systems experimentally [7–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Such  experiments have also stimulated theoretical research on  fundamental quantum many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For instance,  the ground-state phase diagrams of the transverse-field  Ising model and those of its large Ising interaction limit,  the PXP model, have been intensively examined using  the quantum Monte Carlo method [16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Rydberg-atom arrays have also given the opportunity  to study the nonequilibrium dynamics of isolated quan-  tum many-body systems, which are hard to simulate nu-  merically with classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In particular, the  correlation-spreading dynamics of quantum Ising mod-  els [16, 20] is one of the intriguing topics that is likely  to be further addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' At present, experiments with  more than 200 Rydberg atoms are feasible [4–6], allow-  ing one to study unprecedentedly large lattice systems in  one (1D) and two spatial dimensions (2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  These recent experiments on long-time dynamics in  quantum many-body systems have motivated us to quan-  titatively calculate the velocity of the correlation prop-  agation, which will serve as useful references for future  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In general, there are two kinds of propa-  gation velocities for correlation spreading dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' one  is the phase velocity and the other is the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  ∗ rkaneko@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='kindai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='jp  † danshita@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='kindai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='jp  The former can be captured by the first peak of the wave  packet, whereas the latter can be extracted by the enve-  lope of the wave packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The group velocity is bounded  from above in non-relativistic quantum systems, and this  upper limit is known as the Lieb-Robinson bound [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  While significant progress has been made concerning  rigorous inequalities related to the Lieb-Robinson bound,  such inequalities do not necessarily offer practical refer-  ence values for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Usually, the Lieb-Robinson  bound is intended to provide general conditions for ar-  bitrary correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Consequently, the bound can be  too loose and sometimes meaningless when examining  the propagation velocity of particular correlation func-  tions that are measurable in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' With this in  mind, the Lieb-Robinson bound has been improved very  recently [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' however, their method still gives a looser  bound than the exact solution if it is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In some cases, direct numerical simulations on classi-  cal computers would give much more detailed informa-  tion about correlation spreading than rigorous inequali-  ties for the Lieb-Robinson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Such numerical data  would also strengthen the validity of experimental find-  ings through cross-checking experimental and theoreti-  cal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Indeed, many numerical efforts have been  made to calculate the quench or sweep dynamics in 1D  and 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' These attempts include the time-dependent vari-  ational Monte Carlo method with the Slater-Jastrow  wave function [24] and with more sophisticated neural-  network wave functions [25–30], the form factor expan-  sions [31], the numerical linked-cluster expansion [32–  34], the tensor-network method based on matrix prod-  uct states (MPS) [35, 36], and that based on projected  entangled pair states (PEPS) [37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In this paper, we study quench dynamics in the  transverse-field Ising model on a chain in 1D and that  on a square lattice in 2D by using several methods, in-  cluding the tensor-network method based on PEPS and  the linear spin-wave approximation (LSWA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We take  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='01407v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='quant-gas]  4 Jan 2023  2  the initial state to be the magnetically disordered prod-  uct state, which is the ground state in the strong-field  limit, and calculate time evolution of spin-spin correla-  tions at equal time after a sudden quench of the trans-  verse field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We focus on the quench within a parameter  region where the ground state is magnetically disordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We extract the group velocity of the correlation propa-  gation from the spin-spin correlations for several values  of the transverse field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In the 1D case, we show that the  group velocity extracted from the LSWA results asymp-  totically approaches that extracted from the rigorous an-  alytical results with increasing the transverse field, while  the agreement in the time dependence of the correlation  functions is limited to a short time before the first peak  appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Our results indicate that the LSWA can quan-  titatively predict the propagation velocity as long as the  final transverse field is sufficiently strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In the 2D case,  using the LSWA, we estimate the group velocity to be  Ja/(2ℏ), where J is the Ising interaction and a is the  lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We use the PEPS method in a comple-  mentary way to perform more quantitative calculations  on the time evolution of the correlation functions for a  relatively short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  This paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' II, we in-  troduce the model and all the analytical and numerical  methods used in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' III and IV,  we present the time-dependent spin-spin correlation func-  tions and extract the corresponding group velocity in 1D  and 2D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We discuss the relation between  our propagation velocity and the Lieb-Robinson bound  proposed recently, and draw our conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  For simplicity, we set ℏ = 1 throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  MODEL AND METHODS  We consider the transverse-field Ising model with the  periodic boundary condition defined as  ˆH = −J  �  ⟨i,j⟩  ˆSz  i ˆSz  j − Γ  �  i  ˆSx  i ,  (1)  where ˆSz  i and ˆSx  i correspond to the z and x components  of the S = 1/2 Pauli spin, J represents the strength of the  spin exchange interaction, and Γ represents the strength  of the transverse field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The symbol ⟨i, j⟩ means that the  sum is taken over nearest-neighbor sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We focus on  the ferromagnetic spin exchange interaction (J > 0) on a  chain in 1D and that on a square lattice in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Both fer-  romagnetic and antiferromagnetic models are equivalent  under appropriate unitary transformations for bipartite  lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The ground state is ordered (disordered) for  Γ < Γc (Γ > Γc), where Γc is the transition point given  as Γc/J = 1/2 [40] in 1D and Γc/J ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='522 [17, 41, 42]  in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Hereafter we take J as the unit of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We  also take the lattice constant to be unity throughout this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We investigate the quench dynamic starting from the  disordered state |ψ0⟩ = ⊗i |→⟩i at Γ → ∞ to the dis-  ordered parameter region Γ ∈ (Γc, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We study the  equal-time longitudinal and connected transverse corre-  lation functions at distance r, which are defined as  Czz(r, t) = ⟨ψ(t)| ˆSz  r ˆSz  0|ψ(t)⟩,  (2)  Cxx  connected(r, t) = ⟨ψ(t)| ˆSx  r ˆSx  0|ψ(t)⟩  − ⟨ψ(t)| ˆSx  r|ψ(t)⟩⟨ψ(t)| ˆSx  0|ψ(t)⟩  (3)  with |ψ(t)⟩ = e−i ˆ  Ht|ψ0⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In 1D, we obtain  them by the exact calculations via the Jordan-Wigner  transformation and by the LSWA via the Holstein-  Primakoff transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In 2D, we use the tensor-  network method, the exact diagonalization (ED) method,  and the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We will summarize each method below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We extract the group velocity from the envelope of the  wave packet in the spin-spin correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Let  us first discuss how the correlation spreading is related to  the Lieb-Robinson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In a system with short-range  interaction, a commutator of any operators ˆOA and ˆOB  in regions A and B satisfies the relation  ��[ ˆOA(t), ˆOB]  �� ≤ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' × exp  �  −L − vt  χ  �  ,  (4)  where ˆOA(t) = exp  �  i ˆHt  � ˆOA exp  �  −i ˆHt  �  , L is the dis-  tance between the regions A and B, and χ is con-  stant [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The velocity v corresponds to the Lieb-  Robinson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' This relation means that the informa-  tion from the region A is transmitted to the region B up  to a time t ≈ L/v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Then, the inequality of the Lieb-  Robinson bound ensures that, for any operators ˆOA and  ˆOB in regions A and B having the distance L, the ex-  pectation value for a state |ψ⟩ with a finite correlation  length χ satisfies [43]  |⟨ψ(t)| ˆOA ˆOB|ψ(t)⟩ − ⟨ψ(t)| ˆOA|ψ(t)⟩⟨ψ(t)| ˆOB|ψ(t)⟩|  ≤ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' × e− L−2vt  χ′  ,  (5)  where χ′ is a constant that depends on χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' This velocity  2v on the right-hand side corresponds to twice the Lieb-  Robinson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' When the correlation spreading is well  described by the quasiparticle, the group velocity (vgroup)  of the fastest quasiparticle is often regarded as the Lieb-  Robinson bound [44–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Under these circumstances, the slope obtained from  the peak-time dependence of the distance gives twice the  group velocity (2vgroup) of a certain quasiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In-  tuitively, the factor two originates from pairs of quasi-  particles moving to the left or right from a given point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  This quasiparticle picture has been discussed intensively  in the dynamics of the Bose-Hubbard model [45, 48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In the present analysis, we have presented the group ve-  locity vgroup of a certain single quasiparticle estimated  from one half of the slope obtained from the peak-time  dependence of the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  3  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Exact calculations in 1D  The analytical form of the time-dependent correla-  tion functions can be obtained rigorously for the 1D  transverse-field Ising model [40, 51–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We briefly re-  view the detailed derivation of the time-dependent cor-  relation functions in Appendix A and present the final  results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The longitudinal correlation function is represented as  a Pfaffian of a 2r × 2r skew symmetric matrix:  Czz(r, t) = 1  4pf  � S  G  −GT Q  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (6)  Here elements of the matrices S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' and G are defined as  si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j =  �  �  �  �  �  Si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j−1  if i < j  −Sj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='i−1  if i > j  0  if i = j  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (7)  qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j =  �  �  �  �  �  Qi−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j−1  if i < j  −Qj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='i−1  if i > j  0  if i = j  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (8)  gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j = Gi−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (9)  where the time-dependent correlation functions Si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  and Gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j are given as  Si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j = − 2  L  �  k>0  �  cos[k(ri − rj)][|˜uk(t)|2 + |˜vk(t)|2]  − i sin[k(ri − rj)][˜uk(t)˜v∗  k(t) + ˜vk(t)˜u∗  k(t)]  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (10)  Qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j = + 2  L  �  k>0  �  cos[k(ri − rj)][|˜uk(t)|2 + |˜vk(t)|2]  + i sin[k(ri − rj)][˜uk(t)˜v∗  k(t) + ˜vk(t)˜u∗  k(t)]  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (11)  Gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='j = − 2  L  �  k>0  �  cos[−k(ri − rj)][|˜uk(t)|2 − |˜vk(t)|2]  − i sin[−k(ri − rj)][˜uk(t)˜v∗  k(t) − ˜vk(t)˜u∗  k(t)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (12)  The symbol �  k>0 means the sum taken over all k =  2πn/L with n = 1/2, 3/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , (L−3)/2, (L−1)/2 for even  L or n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , (L − 3)/2, (L − 1)/2 for odd L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For the  quench starting from the disordered state (Γ → ∞), the  parameters ˜uk(t) and ˜vk(t) for 0 < k < π are described  as  ˜uk(t) = i  ˜b′  k  ω′  k  sin  �  2ω′  k × tJ  4  �  ,  (13)  ˜vk(t) = −i cos  �  2ω′  k × tJ  4  �  − ˜a′  k  ω′  k  sin  �  2ω′  k × tJ  4  �  (14)  with  ˜a′  k = 2Γ  J + cos k,  (15)  ˜b′  k = sin k,  (16)  ω′  k =  �  4Γ2  J2 + 4Γ  J cos k + 1,  (17)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Parameters with prime symbols indicate  physical quantities after the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' On the other hand,  the transverse correlation function is given as  Cxx  connected(r, t) = −1  4(Q0,rS0,r + Gr,0G0,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (18)  We numerically evaluate each correlation function for suf-  ficiently large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We use the library for Pfaffian  computations [57] in the case of the longitudinal correla-  tion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Spin-wave approximation  We investigate a small quench starting from the com-  pletely disordered point (Γ → ∞) to the parameter  within a disordered phase (Γclassical  c  ≪ Γ < ∞, where  Γclassical  c  = JD with D being the spatial dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We  focus on small quantum fluctuations around the disor-  dered state and map quantum Ising spins to bosons using  the linearized Holstein-Primakoff transformation [58–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The equal-time correlation functions for quantum spins  can be obtained by calculating those for bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' They  serve as a good approximation as long as the transverse  magnetization is large enough (⟨Sx  i ⟩ ≈ 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We give the  detailed derivation in Appendix B and show the obtained  spin-spin correlation functions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The longitudinal correlation function at distance r  (1 ≤ rν ≤ L/2 with ν = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , D) is given as  Czz(r, t) =  S  2LD  �  k  eik·r  B′  k  A′  k + B′  k  (cos 2Ω′  kt − 1), (19)  where S(= 1/2) is the size of spin and other parameters  are defined as  Ω′  k = sgn(A′  k)  �  A′  k  2 − B′  k  2,  (20)  A′  k = −z  2JSγk + Γ,  (21)  B′  k = −z  2JSγk,  (22)  γk = 1  D  D  �  ν=1  cos kν  (23)  with z = 2D being the coordination number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Parame-  ters with prime symbols correspond to physical quanti-  ties after the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' On the other hand, the transverse  correlation function at distance r (1 ≤ rν ≤ L/2 with  ν = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , D) is given as  Cxx  connected(r, t)  =  �����  1  LD  �  k  eik·r B′  k  2Ω′  k  �A′  k  Ω′  k  (cos 2Ω′  kt − 1) + i sin 2Ω′  kt  ������  2  4  Dphys  Dvirt  A  B  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Schematic picture of the iPEPS having a two-site  unit-cell structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The two sublattice sites are represented  by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Each ball corresponds to a rank-five tensor, which  is located at a lattice site and has four thin sticks and a thick  stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The thin and thick sticks represent the virtual and  physical degrees of freedom, and the bond dimensions of the  former and the latter are defined as Dvirt and Dphys, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  +  �����  1  LD  �  k  eik·r B′  k  2  2Ω′  k  2 (cos 2Ω′  kt − 1)  �����  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (24)  We numerically calculate each correlation function for  sufficiently large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  2D tensor-network method  We use the infinite projected entangled pair state  (iPEPS) [62–69] or the infinite tensor product state [70–  74] to investigate short-time dynamics in the infinite sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We choose translationally invariant iPEPS consist-  ing of a two-site unit-cell structure as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The dimension of the local Hilbert space is Dphys = 2  for spin S = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The initial state |ψ0⟩ = ⊗i| →⟩i is  the ground state in the limit of Γ/J → ∞, which can be  prepared by the virtual bond dimension Dvirt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We apply the simple update algorithm [66, 75] to  simulate the real-time dynamics of the transverse-field  Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In this algorithm, we approximate the  real-time evolution operator in a very short-time step  dt using the Suzuki-Trotter decomposition [76–78] and  obtain the two-site gate e−idt ˆ  H ≈ �  ⟨i,j⟩ e−idt ˆ  Hij with  ˆHij = −J ˆSz  i ˆSz  j − Γ( ˆSx  i + ˆSx  j )/z (z = 4) satisfying  ˆH = �  ⟨i,j⟩ ˆHij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The gate acts on two neighboring ten-  sors and increases the virtual bond dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We trun-  cate the bond dimensions of the local tensors using the  singular value decomposition so that the bond dimen-  sions of iPEPS remains Dvirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In the actual calculations,  the second-order Suzuki-Trotter decomposition is used,  and the time step is typically chosen as dtJ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='001 for  a quench to a strong field Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Simulations using doubled  or halved dt show no significant change in the short-time  dynamics as for the present model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We improve the accuracy of time-evolved wave func-  tions by increasing the dimension of the virtual bond  Dvirt and confirm the convergence of physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Previous studies [38, 79] suggest that results for bond  dimensions Dvirt ≳ 6 already show good convergence  within a short-time frame tJ ≲ 4 even in one of the most  difficult cases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=', the quench to the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (Con-  cerning the unit of time, the energy scale is four times  larger in previous studies [38, 79] because they used the  Pauli spin σz = 2Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=') In the present iPEPS simulations,  we adopt the tensor-network library TeNeS [80–82] and  increase the virtual bond dimensions up to Dvirt = 8 for  safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The corner transfer matrix renormalization group  method [67–69, 71, 83–89] is used to calculate physi-  cal quantities in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We take the  bond dimensions of the environment tensors as χvirt =  2(Dvirt)2 so that physical quantities are well converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Exact diagonalization method  The ED method is often used to get insight into the  dynamics of small quantum many-body systems [90–94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We use the QuSpin library [95, 96] for ED calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We consider the system sizes up to 28 sites under the pe-  riodic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In the present setup, both the  Hamiltonian and the initial state are translationally in-  variant, and the total momentum of the initial and time-  evolved states remains zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We restrict ourselves to the  zero-momentum sector [97] and follow the dynamics of  the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Instead of generating matrix elements on the  fly to reduce the memory cost, we keep all the elements  of sparse matrices in the compressed sparse row format  to accelerate calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  To compute the matrix ex-  ponential applied to a vector, we use the Taylor series  expansion with error analysis proposed by Al-Mohy and  Higham [98, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We confirm that the ED results (up to 28 sites) re-  produce the exact analytical results in 1D (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We mainly show the ED results in 2D (for 5 × 5 sites)  hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  RESULTS IN 1D  We first present the time dependence of spin-spin cor-  relations in 1D using the exact analytical approach and  the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We extract the group velocity after a sudden  quench to a strong field from these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 1, t)  ×10−2  Γ/J = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 2, t)  L = 256  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 3, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 4, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 5, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 6, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 7, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 8, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 9, t)  0  5  10  15  20  time tJ  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 10, t)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Exact equal-time longitudinal correlation functions  in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We consider the quench to Γ/J = 3 for a system size  L = 256 and show the short-time dynamics for distances r =  1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The envelope of each correlation function is  a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Exact results  We show the exact equal-time longitudinal correlation  functions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' At an early time (tJ ≪ r), the in-  tensity of correlation is nearly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' On the other hand,  when tJ ≳ r, the correlation starts to develop and ex-  hibits rapid oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For each distance, the earliest  peak in the envelope of the wave packet has the largest  intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The peak time of the largest envelope peak  moves almost linearly with the distance, suggesting the  light-cone-like spreading of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We also show the exact equal-time transverse correla-  tion functions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In contrast to the longitudinal  correlations, the rapid oscillations appear only for short  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 1, t)  ×10−3  Γ/J = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 2, t)  L = 256  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 3, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 4, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 5, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 6, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 7, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 8, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 9, t)  0  5  10  15  20  time tJ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 10, t)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Exact equal-time transverse correlation functions in  1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The parameters are the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  distances (r ≲ 3) and are negligibly small for most of  the distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Besides, the intensity of the transverse  correlation is much smaller than that of the longitudinal  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' On the other hand, the peak time of the transverse  correlation almost coincides with that of the largest enve-  lope peak in the longitudinal correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The transverse  correlation decays rapidly just before and after the peak  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  To estimate the propagation velocity, we first extract  the peak time of the envelope of correlations as a function  of distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We show the corresponding time and dis-  tance in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The data for longitudinal and transverse  correlations overlap very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The distance is nearly  proportional to the peak time for both correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For  each field Γ/J and size L, we estimate the group velocity  vgroup from one half of the slope so that it corresponds  directly to the speed of one of quasiparticle pairs moving  6  0  25  50  75  100  125  first peak time tJ  0  50  100  distance r  Γ/J = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  L = 256  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  First-peak time dependence of distance for exact  correlations in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We show data points when the distance is  a multiple of four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The group velocity is estimated from one  half of the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  0  2  4  6  8  10  Γ/J  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='8  vgroup/J  L = 256  Lieb-Robinson bound  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Field dependence of the group velocity in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The  exact Lieb-Robinson bound vLR/J = 1/2 is shown as a refer-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For Γ/J < 3, we only show the group velocity estimated  from the transverse correlations because the envelopes of lon-  gitudinal correlations become unclear for a weaker field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  to the left or right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Since the data points are slightly  out of the straight line at the very short and long dis-  tances, we discard those for r ≤ 5 and r ≥ L/2 − 5 when  extracting the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  To see how the group velocity behaves as a function of  the transverse field, we first examine a sufficiently large  system (L = 256) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Both velocities  estimated from longitudinal and transverse correlations  are nearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5J for all fields Γ/J ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The group velocity  of the spin-spin correlations agrees with the exact Lieb-  Robinson bound in the 1D transverse-field Ising model  (see Appendix A 5 for the derivation of the exact value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  This fact suggests that the quasiparticles with the fastest  propagation velocity among the various correlation func-  tions are directly responsible for the spreading of spin-  spin correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Although the estimated velocity is very close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5J,  it is slightly smaller than the exact value in finite-size  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' To check the size dependence and confirm the  convergence, we perform the finite-size scaling of the es-  timated velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='10  1/L2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='50  vgroup/J  Γ/J = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Lieb-Robinson bound  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Size scaling of the group velocity in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The group  velocity is well fitted by 1/L2/3 with L being the length of  a chain and is extrapolated to the value of the exact Lieb-  Robinson bound vLR/J = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  For this purpose, let us first discuss how the finite-  size effect appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The spin-spin correlation functions in  the 1D transverse-field Ising model are described by the  single-particle correlation functions of fermionic quasi-  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In the thermodynamic limit, they are writ-  ten by the Bessel functions [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The size dependence  of the Bessel functions has been carefully investigated in  the case of long-time dynamics of the 1D Bose-Hubbard  model [48], as well as in that of the 1D transverse-field  Ising model [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The distance r dependence of the peak  time t is given as  t ≈  1  v∞  (r + ϵr1/3),  (25)  where v∞ is the velocity at large distances, and ϵ is a  constant related to the peak position of the Bessel func-  tion [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The instantaneous velocity v(r) = [t(r + 1) −  t(r)]−1 at each time t is independent of distances and  becomes v∞ if ϵ = 0, but it is slightly modified in the  presence of finite ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For ϵ ̸= 0, the instantaneous velocity  is obtained as  v(r) ≈ v∞  �  1 − ϵ  3r−2/3�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (26)  Since the farthest distance for a chain of length L is  r = L/2 (∝ L), we may safely assume that the devi-  ation between the finite-size and infinite-size velocities  ∆v(L) follows the relation  ∆v(L) :=  ����v  �L  2  �  − lim  L→∞ v  �L  2  ����� ∝ L−2/3  (27)  for L ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We then extrapolate the finite-size group velocities to  the thermodynamic limit using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 6, all the data points lie on an expected straight line  for both correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The extrapolated group velocity at  Γ/J = 3 is vgroup/J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5005(4) [vgroup/J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='4999(3)]  for the longitudinal (transverse) correlations and almost  converges to the exact Lieb-Robinson bound (vLR/J =  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 1, t)  ×10−2  Γ/J = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 2, t)  L = 256  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 3, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 4, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 5, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 6, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 7, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 8, t)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 9, t)  0  5  10  15  20  time tJ  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 10, t)  exact  LSWA  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Equal-time longitudinal correlation functions ob-  tained by the LSWA in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The parameters are the same  as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We show the exact correlations (dashed  line) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5) of the 1D transverse field Ising model within the  error bar of the extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We have also confirmed  that the estimated velocity converges to the exact one  for all the other transverse fields that we have studied  (Γ/J ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Therefore, the fastest correlation spreading  can be measured by the spin-spin correlations in the 1D  transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Results by the LSWA  To examine how good the LSWA is as for the correla-  tion spreading, we calculate the equal-time spin-spin cor-  relation functions by the LSWA and compare the results  with those of the exact analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In general, the LSWA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 1, t)  ×10−3  Γ/J = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 2, t)  L = 256  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 3, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 4, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 5, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 6, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 7, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 8, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 9, t)  0  5  10  15  20  time tJ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 10, t)  exact  LSWA  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Equal-time transverse correlation functions obtained  by the LSWA in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The parameters are the same as those  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We show the exact correlations (dashed line) for  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  gets better with increasing spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Here we  will demonstrate that the group velocity of the correla-  tion propagation obtained by the LSWA agrees well with  the exact one even in the lowest 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We show the longitudinal correlation functions in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As in the case of the exact analysis, the correla-  tions are suppressed for tJ ≲ r and begin to develop for  tJ ≳ r at a given distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The LSWA quantitatively  reproduces the period of oscillations and the intensity  of exact correlations up to about tJ ≈ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In the short  time (tJ ≲ r), a very small number of quasiparticle ex-  citations would come into play, and the LSWA becomes  more accurate in this dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  On the other hand, the transverse correlation functions  appear to be accurate up to the point where they begin  8  0  25  50  75  100  125  first peak time tJ  0  50  100  distance r  Γ/J = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  L = 256  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  First-peak time dependence of distance for corre-  lations obtained by the LSWA in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We show data points  when the distance is a multiple of four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  0  2  4  6  8  10  Γ/J  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='8  vgroup/J  L = 256  from spin-wave dispersion  Lieb-Robinson bound  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Field dependence of the group velocity ob-  tained by the LSWA in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The exact Lieb-Robinson bound  vLR/J = 1/2 and the maximum group velocity vSW/J =  [1 +  �  1 − (J/Γ)2]−1/2/  √  2 estimated from the spin-wave dis-  persion (see Appendix B 4) are shown as references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  For  Γ/J < 3, we only show the group velocity estimated from  the transverse correlations because the envelopes of longitu-  dinal correlations become unclear for a weaker field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  to increase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In contrast to the exact analyti-  cal result, where the earliest peak has the largest inten-  sity, the LSWA predicts that the second earliest peak has  the largest intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Nevertheless, the time of maximum  intensity does not differ significantly between the exact  and approximate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The first-peak time is typically  about 2tJ early, while the time of maximum intensity is  typically about 2tJ late for all distances in the case of  the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' These effects do not change the propagation  velocity significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Therefore, the group velocity esti-  mated by the LSWA is expected to be close to the exact  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  As in the case of exact analysis, we observe the suppres-  sion of rapid oscillations in the transverse correlations  using the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' This phenomenon can be easily un-  derstood in the magnon picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The original transverse  correlation corresponds to the density-density correlation  of magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The density operator is less susceptible to  the effects of phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  On the other hand, the original  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='10  1/L2/3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='50  vgroup/J  Γ/J = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  from spin-wave dispersion  Lieb-Robinson bound  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Size dependence of the group velocity estimated by  the LSWA in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The size dependence is smaller than the  exact case (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  longitudinal correlation corresponds to the single-particle  correlation of magnons, which directly feels the effects of  phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Therefore, the transverse (longitudinal) correla-  tion tends to exhibit less (more) oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Such effects  have been intensively examined in the correlation spread-  ing of the Bose-Hubbard model [45, 48, 49, 101, 102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Likewise, the LSWA also predicts that the intensity  of the transverse correlation is smaller than that of  the longitudinal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' They are approximately given as  |Czz(r, t)| = O(J/Γ) and |Cxx  connected(r, t)| = O(J2/Γ2),  respectively (see Appendices B 2 and B 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Having confirmed the accuracy of the LSWA, we ex-  tract the group velocity from the 1D correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We  first investigate the distance dependence of peak time  for a sufficiently large system (L = 256), as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Again, both correlations show almost the same  result, and the distance is nearly proportional to the  peak time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We estimate the velocity using the data for  5 < r < L/2 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We summarize the field dependence of the group ve-  locity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Both group velocities estimated from  the longitudinal and transverse correlations are nearly  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5J irrespective of the choice of the transverse field for  Γ/J ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Note that the LSWA group velocity is ex-  pected to deviate from the exact one at Γ/J ≲ 2 because  too many quasiparticles are created due to such a large  quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The LSWA also suggests that, in the case of the  1D transverse-field Ising model with a sufficiently strong  field, the spin-spin correlations can be used to extract the  Lieb-Robinson bound, the fastest group velocity among  any correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Finally, we have confirmed the size dependence of the  estimated group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 11, the LSWA  shows much smaller size dependence than the exact anal-  ysis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The velocity is nearly converged for L ≥ 48  and is extrapolated to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5J, corresponding to the Lieb-  Robinson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 1, t)  ×10−2  Γ/J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Czz(r = 2, t)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='2  Czz(r = 3, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='2  Czz(r = 4, t)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='1  Czz(r = 5, t)  0  2  4  6  time tJ  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Czz(r =  √  2, t)  PEPS, Dvirt = 8  ED, Ns = 5 × 5  LSWA, Ns = 128 × 128  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Equal-time longitudinal correlation functions ob-  tained by the LSWA, the ED method, and the tensor-network  method in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We consider the quench to Γ/J = 9 for a finite  system of Ns = L2, L = 128 by the LSWA (solid line), for  a finite system of Ns = L2, L = 5 by ED simulations (small  circles), and for the infinite system with the bond dimensions  Dvirt = 8 by iPEPS simulations (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  RESULTS IN 2D  Next, we examine the time-dependent correlations in  2D using the ED method, the tensor-network method  based on iPEPS, and the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As in the case of 1D,  we estimate the group velocity after a sudden quench to  a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Results by the LSWA  We apply the LSWA to calculate the spin-spin cor-  relation functions and to extract the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In the case of the 1D transverse-field Ising model, the  LSWA reproduces the exact results to the extent that  the group velocity of the correlation propagation quan-  titatively agrees at a sufficiently strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We will  demonstrate that it reproduces the 2D correlations ob-  tained by the nearly exact simulations much better than  in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' It also allows us to estimate the group velocity  from the correlations at farther distances than the ED  and iPEPS simulations, as we will demonstrate below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We compare the longitudinal correlation functions ob-  tained by the ED method and the LSWA in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The  LSWA well reproduces the correlations obtained by the  ED method up to the point where the second peak of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 1, t)  ×10−5  Γ/J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 2, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 3, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 4, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 5, t)  0  2  4  6  time tJ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r =  √  2, t)  PEPS, Dvirt = 8  ED, Ns = 5 × 5  LSWA, Ns = 128 × 128  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Equal-time transverse correlation functions ob-  tained by the LSWA, the ED method, and the tensor-network  method in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The parameters are the same as those in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  0  10  20  30  time tJ  10  20  30  distance r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Contour plot of normalized intensity of equal-  time transverse correlation functions as a function of time  and distance obtained by the LSWA in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The param-  eters L  =  128 and Γ/J  =  9 are the same as those  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We show the normalized correlation function  Cxx  connected(r, t)/ maxt∈[0,L/(2J)] Cxx  connected(r, t) (∈ [0, 1]) along  the horizontal axis [r = (r, 0)] for each distance up to r = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  the envelope appears [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=', Czz(r = 2, t) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The period of oscillations almost coincides between the  ED method and the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As expected in the LSWA  in higher spatial dimensions, the agreement in 2D looks  much better than in 1D (compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 7 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The longitudinal correlations exhibit rapid oscillations  as in the case of 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' On the other hand, in contrast to  the 1D case, where the earliest envelope peak has the  10  0  20  40  60  first peak time tJ  0  20  40  60  distance r  Γ/J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  L = 128  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Dominant-peak time dependence of distance for cor-  relations obtained by the LSWA in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  0  2  4  6  8  10  Γ/J  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='25  vgroup/J  L = 128  from spin-wave dispersion  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Field dependence of the group velocity obtained  by the LSWA in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The maximum group velocity vSW/J =  (1−J/Γ+  �  1 − 2J/Γ)−1/2/  √  2 estimated from the spin-wave  dispersion (see Appendix B 4) is shown as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For  Γ/J < 3, we only show the group velocity estimated from the  transverse correlations because the envelopes of longitudinal  correlations become unclear for a weaker field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  largest intensity, it does not always exhibit the largest  intensity in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The order of the envelope peaks with the  largest intensity varies with distance in 2D, which would  make it more difficult to extract the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' This  observation may be ascribed to the complex interference  effects in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The transverse correlation function obtained by the  LSWA also qualitatively reproduces the ED result (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In contrast to the longitudinal correlations, the  rapid oscillations are much weaker for r ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  To clarify how the correlation develops for a longer  time and to examine the complex interference effects in  2D, we depict the normalized intensity of the transverse  correlations as a function of time and distance in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We observe a stronger intensity near the line satisfying  r ≈ tJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' However, areas of high intensity are not continu-  ously connected and are rather separated in small pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Such pieces are bundled together forming the boundary  of the light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' When we focus on the short-time and  short-distance region, we can only look at the first small  area of high intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' If we use such data, we would in-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='04  1/L  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  vgroup/J  Γ/J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  from spin-wave dispersion  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Size dependence of the group velocity estimated by  the LSWA in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  correctly estimate the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Indeed, as we will  see later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' IV B, the velocity obtained by the iPEPS  method for a relatively short time shows a rather large  error bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  To estimate the group velocity in 2D, we collect the  peak times and distances in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Both correlations  exhibit the consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Although the jagged be-  havior caused by the complex interference effects is ob-  served in the data points, the distance becomes nearly  proportional to the peak time for sufficiently large sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We extract the group velocity from one half of the  slope so that it corresponds directly to the velocity of one  quasiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We show the field dependence of the group velocity  along the horizontal axis for a large system (Ns = L2,  L = 128) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' At a very strong transverse field,  the velocity turns out to be nearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The velocity  is likely to increase with decreasing the transverse field  although we do not know to what extent the LSWA is  reliable for a weaker field, corresponding to a relatively  larger quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The velocity estimated from correlations  is basically on the curve represented by vSW/J = (1 −  J/Γ +  �  1 − 2J/Γ)−1/2/  √  2, which is determined by the  derivative of the spin-wave dispersion (see Appendix B 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We finally check the size dependence of the estimated  group velocity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As in the case of 1D, the veloc-  ity does not depend on the size significantly for L ≳ 48  and converges to the value close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Therefore, the  LSWA predicts that the speed of spin-spin correlation  spreading is vgroup ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5J for a small quench to Γ ≫ J  in the 2D transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Tensor-network results  As a complementary method to the LSWA, we use the  tensor-network method based on the iPEPS to calculate  the spin-spin correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We will see that the  tensor-network method has an advantage in calculating  the time dependence of correlations more accurately than  the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Before presenting the correlations obtained by the  11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='15  Γ/J = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  PEPS, Dvirt = 5, 6, 7, 8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='15  Γ/J = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='15  [e(tJ) − e(0)]/J  Γ/J = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='15  Γ/J = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0  2  4  6  8  time tJ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='15  Γ/J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Bond-dimension dependence of energy density ob-  tained by iPEPS simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We consider the quench to  Γ/J = 5, 6, 7, 8, and 9 and subtract the energy density  at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The lines correspond to the time-dependent en-  ergy density for the bond dimensions Dvirt = 5, 6, 7, and 8  (from lighter to darker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The energy is nearly conserved for  Dvirt ≥ 6 in a time frame tJ ∈ [0, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  iPEPS simulations, let us examine the time range of the  applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In general, as for numerical simulations of  a quench dynamics, the obtained correlations would be  reliable for short times until the energy is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We  investigate the time dependence of the energy density in  the unit of Ising interaction J for different fields Γ/J  with increasing the bond dimensions Dvirt (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The energy density is nearly conserved for a short time  (tJ ≲ 4) when Dvirt ≥ 6 regardless of the choice of the  transverse field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Therefore, we will present the correla-  tions within this time frame hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We show the longitudinal correlation functions ob-  tained by the ED and iPEPS simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The  ED method can deal with small systems in 2D and gives  the correlations up to r ≈ 2 at the farthest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For these  distances (r ≲ 2) and short times (tJ ≲ 4), the data by  the ED and iPEPS methods completely overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Since  the iPEPS method directly handles the infinite system,  the ED method appears to provide the correlations that  can almost be regarded as those at the thermodynamic  limit in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The iPEPS method can predict the  peak positions of correlations at slightly farther distances  until the energy is conserved (tJ ≲ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The peak in the  envelope of correlation hits tJ ≈ 4 when r = 5, and thus  the correlations up to r = 5 would be reliable for the  velocity estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  As in the case of the LSWA, the longitudinal corre-  lations exhibit rapid oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Moreover, in 2D, the  tensor-network method also predicts that the earliest en-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Czz(r = 1, t)  ×10−2  Γ/J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Czz(r = 2, t)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='2  Czz(r = 3, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='2  Czz(r = 4, t)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='1  Czz(r = 5, t)  0  2  4  6  time tJ  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Czz(r =  √  2, t)  PEPS, Dvirt = 5, 6, 7, 8  ED, Ns = 5 × 5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Equal-time longitudinal correlation functions in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We consider the quench to Γ/J = 9 for the infinite system  with the bond dimensions Dvirt = 5, 6, 7, and 8 (solid lines  from lighter to darker) by iPEPS simulations and for a finite  system of Ns = L2, L = 5 (small circles) by ED simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We show the short-time dynamics for distances r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' ,  5, and  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Both data agree very well for t/J ≲ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  velope peak does not always correspond to the peak hav-  ing the largest intensity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' This observation  suggests that the complex interference effects in 2D are  not the artifact of the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We also examine the transverse correlation functions  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The ED and iPEPS methods provide almost  the same correlations for r ≲ 2 and tJ ≲ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Again, the  iPEPS method is applicable to farther distances up to  r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The rapid oscillations are quickly suppressed for  r ≳ 3, as in the case of 1D and also as in the LSWA  for 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The peak positions of the transverse correlations  are nearly the same as those of the envelope peak in the  longitudinal correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The qualitative behavior of correlations obtained by  the tensor-network method and the LSWA is similar (see  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 12 and 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The peak time of the correlations does  not differ significantly between the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Al-  though the first-peak time is a little ahead in the LSWA,  the peak-time difference is typically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5tJ in 2D, which  is smaller than 2tJ in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Because the LSWA is appli-  cable to a much longer time, it is more suitable for esti-  mating the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' On the other hand, the time  dependences of correlations agree well between the ED  and tensor-network methods, whereas they slightly dif-  fer between the ED method and the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Therefore,  the tensor-network method is more appropriate to obtain  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 1, t)  ×10−5  Γ/J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 2, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 3, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r = 4, t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5  Cxx(r = 5, t)  0  2  4  6  time tJ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Cxx(r =  √  2, t)  PEPS, Dvirt = 5, 6, 7, 8  ED, Ns = 5 × 5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Equal-time transverse correlation functions in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The parameters are the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  0  1  2  3  4  5  6  peak time tJ  0  2  4  6  distance r  Γ/J = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  Dvirt = 8  longitudinal  transverse  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Dominant-peak time dependence of distance for cor-  relations obtained by iPEPS simulations (Dvirt = 8) in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The group velocity is estimated from the value r/[2t(r)] for  distances r = 3, 4, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  quantitative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  To estimate the group velocity, we pick up the peak  time for each distance from these correlations obtained  by iPEPS simulations, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Since the data  obtained by the bond dimensions Dvirt = 6, 7, and 8 are  well converged, we present the result for the largest bond  dimension Dvirt = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Within the range of time where  the iPEPS simulations are considered to be reliable, it  is hard to tell whether the light-cone-like spreading of  correlations exists or not in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' However, as we have  shown by the LSWA in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' IV A, such behavior is caused  by the complex interference effects in 2D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' it is highly  0  2  4  6  8  10  Γ/J  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='8  vgroup(r)/J  Dvirt = 8  longitudinal, r = 5  longitudinal, r = 4  longitudinal, r = 3  transverse, r = 5  transverse, r = 4  transverse, r = 3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Field dependence of the group velocity estimated  from iPEPS simulations (Dvirt = 8) in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For Γ/J < 5, we  only show the group velocity estimated from the transverse  correlations because the envelopes of longitudinal correlations  become unclear for a weaker field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  probable that the light cone exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Therefore, we may  assume that the distance eventually grows linearly with  the peak time also in the iPEPS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We then extract  the group velocity as vgroup = r/[2t(r)] for each distance  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We mainly focus on the data for farther distances  (r = 3, 4, and 5) because data for short distances tend  to be off the light-cone behavior in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The field dependence of the group velocities along  the horizontal axis for distances r = 3, 4, and 5 are  given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  They do not vary significantly for  Γ/J ∈ [2, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Since the velocity increases with increas-  ing the distance, we estimate the group velocity as the  average of the smallest and largest values with the error  bar given by one half of their difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' It is given as  vgroup/J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='54(11) for all transverse fields that we have  studied using the iPEPS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As we have discussed  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' IV A, the LSWA also predicts the similar velocity  vgroup/J ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The velocities obtained by the LSWA  and those obtained by the tensor-network method agree  within the error bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  DISCUSSION AND SUMMARY  Let us compare our group velocity estimated from  the spin-spin correlations with the recent Lieb-Robinson  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In 1D, our estimate of the group velocity is  vgroup = J/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  This is the same as the exact Lieb-  Robinson bound vLR = J/2 in the 1D transverse-field  Ising model, indicating that the spin-spin correlations  propagate at the speed of fastest quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' On the  other hand, the recent Lieb-Robinson bound for general  lattice systems provides the speed vrecent = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='51J [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  As was already pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' [23], it is approxi-  mately three times as large as the exact Lieb-Robinson  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In 2D, the group velocity along the horizontal axis is  estimated to be vhorizontal ≈ J/2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We do not know  13  the exact Lieb-Robinson bound in the 2D transverse-  field Ising model so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  However, for a small quench  within a disorder phase, we might expect that the fastest  quasiparticles are responsible for spin-correlation spread-  ing also in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We come to this conclusion because the  dispersion corresponding to the fastest quasiparticles ob-  tained in perturbation theory [103] turns out to be the  same as the dispersion estimated in the LSWA (see Ap-  pendix B 4), and the LSWA reproduces the spin-spin cor-  relations obtained by the exact analysis in 1D and those  obtained by the nearly exact simulations in 2D fairly well  (see Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' III and IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Therefore, even in 2D, it is natural  to regard the group velocity of the spin-spin correlations  as the tightest Lieb-Robinson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' From the compar-  ison between this value (the horizontal vhorizontal ≈ J/2  or the diagonal vdiagonal ≈ J/  √  2 velocity) and the best  currently available estimate (vrecent = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='775J) [23], it  is likely that there is still much room for improving the  Lieb-Robinson bound in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In  conclusion,  we  have  studied  the  correlation-  spreading dynamics in the transverse-field Ising model  on a chain and that on a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We have calcu-  lated the longitudinal and transverse spin-spin correla-  tion functions after a sudden quench starting from the  disordered state to a strong field within a disordered  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We have applied several analytical and numer-  ical methods and crossvalidated all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In 1D, we have compared the time-dependent correla-  tions using the exact analytical formulae and the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We have found that the group velocity of the correlation  propagation extracted from the LSWA results asymptot-  ically approaches that from the exact analytical formulae  as the transverse field increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Moreover, the 1D spin-  spin correlations are found to propagate at the speed of  fastest quasiparticles corresponding to the exact Lieb-  Robinson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In 2D, we have calculated the correlations using the  ED method, the tensor-network method based on iPEPS,  and the LSWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As in the case of 1D, we have confirmed  that the three methods reproduce nearly the same cor-  relations within a short-time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The tensor-network  method and the LSWA allow us to calculate the corre-  lations for much farther distances than the ED method  can deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In particular, the LSWA is convenient for  estimating the propagation velocity, whereas the tensor-  network method is advantageous in calculating the time  dependence of correlations accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We have extracted  the group velocity by these two methods and obtained the  value which is nearly equal to one half of the magnitude of  the Ising interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The group velocity of the spin-spin  correlations in 2D turns out to be much smaller than the  best currently available estimate for the Lieb-Robinson  bound [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Our findings on the group velocity would be helpful for  future analog quantum simulations of Rydberg-atom ar-  rays and stimulate further research on the Lieb-Robinson  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The present tensor-network method, which can  accurately calculate the dynamics in one of the most fun-  damental two-dimensional quantum many-body systems,  opens the possibility of future applications to other sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors acknowledge fruitful discussions with  Shimpei  Goto,  Daichi  Kagamihara,  and  Mathias  Mikkelsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' This work was financially supported by JSPS  KAKENHI (Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' JP18H05228, JP21H01014, and  JP21K13855),  by MEXT Q-LEAP (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' JP-  MXS0118069021), and by JST FOREST (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  JPMJFR202T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The numerical computations were per-  formed on computers at the Yukawa Institute Computer  Facility and on computers at the Supercomputer Center,  the Institute for Solid State Physics, the University of  Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Appendix A: Details of exact calculations in 1D  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Hamiltonian  We review the derivation of the exact form of two-body  correlation functions after a sudden quench [40, 51–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  For simplicity, we consider the Hamiltonian  ˆH = −  �  i  ˆσz  i ˆσz  i+1 − ˜g  �  i  ˆσx  i ,  (A1)  which corresponds to the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (1) with  J = 4 and Γ = ˜gJ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' To get the correlation functions for  the original Hamiltonian, we have to use ˜g = 2Γ/J and  replace time 4t with tJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  After the Jordan-Wigner transformation  ˆσx  i = 2ˆc†  i ˆci − 1,  (A2)  ˆσz  i =  i−1  �  j=1  (1 − 2ˆc†  jˆcj)(ˆci + ˆc†  i)  (A3)  and the Fourier transformation  ˆcj =  1  √  L  �  k  e−ikrj ˆck,  (A4)  k = 2πn  L ,  (A5)  where n = −(L − 1)/2, −(L − 3)/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , −1/2, 1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' ,  (L − 3)/2, (L − 1)/2 for even L or n = −(L − 1)/2,  (L − 3)/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , −2, −1, 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , (L − 3)/2, (L − 1)/2  for odd L, we obtain  ˆH =  �  k  (ˆc†  k ˆc−k)  �  ˜ak −i˜bk  i˜bk −˜ak  � � ˆck  ˆc†  −k  �  ,  (A6)  ˜ak = ˜g + cos k,  (A7)  ˜bk = sin k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A8)  14  Using the Bogoliubov transformation  � ˆck  ˆc†  −k  �  =  �  uk ivk  ivk uk  � � ˆγk  ˆγ†  −k  �  (A9)  uk = cos θk  2 ,  (A10)  vk = sin θk  2 ,  (A11)  tan θk =  sin k  g + cos k  (A12)  satisfying u−k = uk and v−k = −vk, we get  ˆH = 2  �  k  ωk  �  ˆγ†  kˆγk − 1  2  �  ,  (A13)  ωk =  �  ˜g2 + 2˜g cos k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A14)  The coefficients uk and vk can be described by ˜ak, ˜bk,  and ωk as  uk =  ˜ak − ωk  �  2ωk(ωk − ˜ak)  = (˜ak − ωk)√ωk + ˜ak  √2ωk|˜bk|  ,  (A15)  vk =  ˜bk  �  2ωk(ωk − ˜ak)  = sgn(˜bk)√ωk + ˜ak  √2ωk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A16)  In the present paper, we mainly consider the quantum  quench from ˜g = g0 to ˜g = g < ∞ within the disor-  dered phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We write the Hamiltonian before (after)  the quench as ˆH ( ˆH′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' For ˜g → ∞, we have uk → 0 and  vk → sgn(sin k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Longitudinal correlation functions  We evaluate the time-dependent longitudinal correla-  tion functions defined as  ¯Czz(r, t) = ⟨ψ0|ei ˆ  H′tˆσz  i ˆσz  i+re−i ˆ  H′t|ψ0⟩  (A17)  = ⟨ψ0|ei ˆ  H′t(ˆc†  i + ˆci)  �  �  i+r−1  �  j=i  (1 − 2ˆc†  jˆcj)  �  �  (ˆc†  i+r + ˆci+r)e−i ˆ  H′t|ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A18)  Using the equality 1 − 2ˆc†  jˆcj = (ˆc†  j + ˆcj)(ˆc†  j − ˆcj) and  defining the operators  ˆAi = ˆc†  i + ˆci,  ˆBi = ˆc†  i − ˆci,  (A19)  ˆAi(t) = ei ˆ  H′t ˆAie−i ˆ  H′t,  ˆBi(t) = ei ˆ  H′t ˆBie−i ˆ  H′t,  (A20)  we obtain the correlation function  ¯Czz(r, t) = ⟨ψ0| ˆBi(t) ˆAi+1(t) ˆBi+1(t) ˆAi+2(t) ˆBi+2(t) · · ·  ˆAi+r−1(t) ˆBi+r−1(t) ˆBi+r(t)|ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A21)  It can be evaluated by the Pfaffian of a 2r × 2r skew  symmetric matrix A using the Wick’s theorem:  ¯Czz(r, t) = pfA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A22)  The matrix A is given as  A =  � S  G  −GT Q  �  (A23)  with matrices  S =  �  �  �  �  �  �  �  �  �  �  �  �  �  0  S0,1  S0,2  · ·  S0,r−2  S0,r−1  −S0,1  0  S1,2  · ·  S1,r−2  S1,r−1  −S0,2  −S1,2  0  · ·  S2,r−2  S2,r−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  −S0,r−2  −S1,r−2  −S2,r−2  · ·  0  Sr−2,r−1  −S0,r−1  −S1,r−1  −S2,r−1  · ·  −Sr−2,r−1  0  �  �  �  �  �  �  �  �  �  �  �  �  �  ,  (A24)  Q =  �  �  �  �  �  �  �  �  �  �  �  �  �  0  Q0,1  Q0,2  · ·  Q0,r−2  Q0,r−1  −Q0,1  0  Q1,2  · ·  Q1,r−2  Q1,r−1  −Q0,2  −Q1,2  0  · ·  Q2,r−2  Q2,r−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  −Q0,r−2  −Q1,r−2  −Q2,r−2  · ·  0  Qr−2,r−1  −Q0,r−1  −Q1,r−1  −Q2,r−1  · ·  −Qr−2,r−1  0  �  �  �  �  �  �  �  �  �  �  �  �  �  ,  (A25)  G =  �  �  �  �  �  �  �  �  �  �  �  �  �  G0,1  G0,2  G0,3  · ·  G0,r−1  G0,r  G1,1  G1,2  G1,3  · ·  G1,r−1  G1,r  G2,1  G2,2  G2,3  · ·  G2,r−1  G2,r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Gr−2,1  Gr−2,2  Gr−2,3  · ·  Gr−2,r−1  Gr−2,r  Gr−1,1  Gr−1,2  Gr−1,3  · ·  Gr−1,r−1  Gr−1,r  �  �  �  �  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (A26)  Here we define time-dependent correlation functions  Si,j = ⟨ ˆBi(t) ˆBj(t)⟩,  (A27)  Qi,j = ⟨ ˆAi(t) ˆAj(t)⟩,  (A28)  Gi,j = ⟨ ˆBi(t) ˆAj(t)⟩ = −⟨ ˆAj(t) ˆBi(t)⟩  (A29)  and use that they are translational invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We will  obtain the explicit form of evaluating Si,j, Qi,j, and Gi,j  in Appendix A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Transverse correlation functions  We evaluate the time-dependent transverse correlation  functions defined as  ¯Cxx(r, t) = ⟨ψ0|ei ˆ  H′tˆσx  i ˆσx  i+re−i ˆ  H′t|ψ0⟩  (A30)  = ⟨ψ0|ei ˆ  H′t(2ˆc†  i ˆci − 1)(2ˆc†  i+rˆci+r − 1)e−i ˆ  H′t|ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A31)  Using the equality 1 − 2ˆc†  jˆcj = (ˆc†  j + ˆcj)(ˆc†  j − ˆcj) and the  expressions for ˆAi and ˆBi, we obtain  ¯Cxx(r, t) = ⟨ψ0| ˆAi(t) ˆBi(t) ˆAi+r(t) ˆBi+r(t)|ψ0⟩  (A32)  15  = G2  i,i − Qi,i+rSi,i+r + (−Gi+r,i)Gi,i+r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A33)  Subtracting  the  correlation  of  the  local  trans-  verse magnetization,  which is given as ⟨ˆσx  i (t)⟩  =  ⟨ψ0|ei ˆ  H′tˆσx  i e−i ˆ  H′t|ψ0⟩ = −⟨ψ0| ˆAi(t) ˆBi(t)|ψ0⟩ = −Gi,i =  −G0,0, we get the connected correlation function  ¯Cxx  connected(r, t) := ¯Cxx(r, t) − ⟨ˆσx  i (t)⟩⟨ˆσx  i+r(t)⟩  (A34)  = −Qi,i+rSi,i+r − Gi+r,iGi,i+r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A35)  The explicit form of evaluating Si,j, Qi,j, and Gi,j will  be given in Appendix A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Single-particle correlation functions for fermions  Let us focus on t = 0 operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Using the Fourier  transformation ˆcj =  1  √  L  �  k e−ikrj ˆck and the Bogoli-  ubov transformation ˆck = ukˆγk + ivkˆγ−k, and then split-  ting the sum �  k into the positive and negative parts  �  k>0 + �  k<0, we rewrite ˆAi and ˆBi as  ˆAi = ˆa†  i + ˆai,  (A36)  ˆBi = ˆb†  i − ˆbi  (A37)  with  ˆai =  1  √  L  �  k>0  �  eikrj(uk − ivk)ˆγk + e−ikrj(uk + ivk)ˆγ−k  �  ,  (A38)  ˆbi =  1  √  L  �  k>0  �  eikrj(uk + ivk)ˆγk + e−ikrj(uk − ivk)ˆγ−k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A39)  The operators ˆai and ˆbi satisfy  {ˆai, ˆaj} = {ˆbi,ˆbj} = {ˆai,ˆbj} = 0,  (A40)  {ˆai, ˆa†  j} = {ˆbi,ˆb†  j} = δij,  (A41)  {ˆai,ˆb†  j} = {ˆa†  i,ˆbj} =: −Gab  i−j  (A42)  with  Gab  i−j = − 1  √  L  �  k>0  �  eik(ri−rj)(uk − ivk)2  + e−ik(ri−rj)(uk + ivk)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A43)  After the quench, the Heisenberg equation for ˆck(t) is  written as  i d  dtˆck(t) = −2˜a′  kˆck(t) − 2i˜b′  kˆc†  −k(t),  (A44)  ˜a′  k = g + cos k,  (A45)  ˜b′  k = sin k,  (A46)  where the prime symbols indicate the parameters after  the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As in the static case, we can introduce ˜uk(t)  and ˜vk(t) for the Bogoliubov transformation at time t  � ˆck(t)  ˆc†  −k(t)  �  =  �  ˜uk(t) −˜v∗  k(t)  ˜vk(t)  ˜u∗  k(t)  � � ˆγk  ˆγ†  −k  �  (A47)  satisfying  ˜u−k(t) = ˜uk(t),  (A48)  ˜v−k(t) = −˜vk(t),  (A49)  |˜uk(t)|2 + |˜vk(t)|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A50)  Here ˆγk corresponds to the Bogoliubov excitations before  the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (A44) and (A47), ˜uk(t) and ˜vk(t)  should satisfy  i d  dt  �  ˜uk(t)  ˜vk(t)  �  =  �  −2˜a′  k −2i˜b′  k  2i˜b′  k  2˜a′  k  � �  ˜uk(t)  ˜vk(t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A51)  Then, for the sudden quench (g0 → g), ˜uk(t) and ˜vk(t)  are explicitly given as  �  ˜uk(t)  ˜vk(t)  �  =  �  � uk cos 2ω′  kt + i ˜a′  kuk+˜b′  kvk  ω′  k  sin 2ω′  kt  −ivk cos 2ω′  kt +  ˜b′  kuk−˜a′  kvk  ω′  k  sin 2ω′  kt  �  � ,  (A52)  where each variable is represented as  ˜ak = g0 + cos k,  (A53)  ˜bk = sin k,  (A54)  ωk =  �  g2  0 + 2g0 cos k + 1,  (A55)  ω′  k =  �  g2 + 2g cos k + 1,  (A56)  and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (A45) and (A46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The parameters uk and vk are  defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (A15) and (A16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  As in the case of t = 0, the Heisenberg representation  of each operator satisfies  ˆAi(t) = ˆa†  i(t) + ˆai(t),  (A57)  ˆBi(t) = ˆb†  i(t) − ˆbi(t)  (A58)  with  ˆai(t) =  1  √  L  �  k>0  �  eikrj[˜uk(t) + ˜vk(t)]ˆγk  + e−ikrj[˜uk(t) − ˜vk(t)]ˆγ−k  �  ,  (A59)  ˆbi(t) =  1  √  L  �  k>0  �  eikrj[˜uk(t) − ˜vk(t)]ˆγk  + e−ikrj[˜uk(t) + ˜vk(t)]ˆγ−k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A60)  Note that ˆai(t) ̸= ei ˆ  H′tˆaie−i ˆ  H′t and ˆbi(t) ̸= ei ˆ  H′tˆbie−i ˆ  H′t  in our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' the commutation relations for the  operators are  {ˆai(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' ˆaj(t)} = {ˆbi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='ˆbj(t)} = {ˆai(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='ˆbj(t)} = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A61)  16  {ˆai(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' ˆa†  j(t)} =: −Gaa  i−j(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A62)  {ˆbi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='ˆb†  j(t)} =: −Gbb  i−j(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A63)  {ˆai(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='ˆb†  j(t)} = {ˆa†  i(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='ˆbj(t)} =: −Gab  i−j(t)  (A64)  with  Gaa  i−j(t) = − 2  L  �  k>0  �  cos[k(ri − rj)][|˜uk(t)|2 + |˜vk(t)|2]  + i sin[k(ri − rj)][˜uk(t)˜v∗  k(t) + ˜vk(t)˜u∗  k(t)]  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A65)  Gbb  i−j(t) = − 2  L  �  k>0  �  cos[k(ri − rj)][|˜uk(t)|2 + |˜vk(t)|2]  − i sin[k(ri − rj)][˜uk(t)˜v∗  k(t) + ˜vk(t)˜u∗  k(t)]  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A66)  Gab  i−j(t) = − 2  L  �  k>0  �  cos[k(ri − rj)][|˜uk(t)|2 − |˜vk(t)|2]  − i sin[k(ri − rj)][˜uk(t)˜v∗  k(t) − ˜vk(t)˜u∗  k(t)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A67)  Using these results, we obtain  S0,r = ⟨ ˆB0(t) ˆBr(t)⟩ = −⟨{ˆb0(t),ˆb†  r(t)}⟩ = +Gbb  −r(t),  (A68)  Q0,r = ⟨ ˆA0(t) ˆAr(t)⟩ = +⟨{ˆa0(t), ˆa†  r(t)}⟩ = −Gaa  −r(t),  (A69)  G0,r = ⟨ ˆB0(t) ˆAr(t)⟩ = −⟨{ˆb0(t), ˆa†  r(t)}⟩ = +Gab  +r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (A70)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Maximum group velocity  In the 1D transverse-field Ising model, the Lieb-  Robinson bound is obtained as the maximum group ve-  locity determined from the derivative of the band disper-  sion [44–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' It is given as  vLR = max  k  ����  dωk  dk  ���� =  �  2˜g  if ˜g ≤ 1  2  if ˜g ≥ 1  (A71)  for the Hamiltonian defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (A1), and it is ob-  tained as  vLR =  �  Γ  if Γ ≤ J/2  J/2  if Γ ≥ J/2  (A72)  for the original Hamiltonian given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Appendix B: Details of the LSWA  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Bosonic quadratic Hamiltonian  We consider a sudden quench within the disordered  phase for the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We investigate  the effect of small quantum fluctuations around the com-  pletely disordered state at Γ → ∞ using a linear spin-  wave expansion [58–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' As long as we consider a quench  to a strong transverse field so that the transverse mag-  netization is large enough (⟨Sx  i ⟩ ≈ 1/2), this approach  should be a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We specifically study  the parameter region Γ ∈ (Γclassical  c  , ∞), where the clas-  sical transition point obtained by the mean-field approx-  imation [17, 104, 105] is Γclassical  c  = JD with D being  the spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We review the derivation of the  longitudinal correlation functions [59] and then calculate  the transverse correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  We apply the linearized Holstein-Primakoff transfor-  mation, which is given as  ˆSx  i = S − ˆb†  iˆbi,  ˆSz  i =  √  2S  2  (ˆb†  i + ˆbi)  (B1)  before the quench and is represented as  ˆSx  i  ′ = S − ˆa†  iˆai,  ˆSz  i  ′ =  √  2S  2  (ˆa†  i + ˆai)  (B2)  after the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  The prime symbols indicate opera-  tors after the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' After the Fourier transformation  (ˆbi =  1  √  LD  �  k e−ik·riˆbk, ˆai =  1  √  LD  �  k e−ik·riˆak) and  the Bogoliubov transformation, we obtain the Hamilto-  nian for free bosons before the quench (up to constant  terms) as  ˆH =  �  k  Ωk ˆβ†  k ˆβk,  (B3)  ˆbk = sk ˆβk + tk ˆβ†  −k,  (B4)  and that after the quench (up to constant terms) as  ˆH′ =  �  k  Ω′  k ˆα†  k ˆαk,  (B5)  ˆak = s′  k ˆαk + t′  k ˆα†  −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B6)  Here the corresponding dispersions and coefficients are  defined as  Ωk = sgn(Ak)  �  A2  k − B2  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B7)  sk = sgn(Ak)  �  1  2  �|Ak|  |Ωk| + 1  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B8)  tk = −sgn(Bk)  �  1  2  �|Bk|  |Ωk| − 1  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B9)  Ω′  k = sgn(A′  k)  �  A′  k  2 − B′  k  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B10)  s′  k = sgn(A′  k)  �  1  2  �|A′  k|  |Ω′  k| + 1  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B11)  t′  k = −sgn(B′  k)  �  1  2  �|B′  k|  |Ω′  k| − 1  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B12)  where  Ak = −z  2JSγk + Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Bk = −z  2JSγk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B13)  17  A′  k = −z  2J′Sγk + Γ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  B′  k = −z  2J′Sγk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B14)  γk = 1  D  D  �  ν=1  cos kν  (B15)  with z = 2D being the coordination number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' We add  the prime symbols to distinguish parameters after the  quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  At t = 0, the vacuum of both Hamiltonians are the  same, and bosons before the Holstein-Primakoff trans-  formation satisfy ˆbk = ˆak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Then, these operators should  fulfill  � ˆbk  ˆb†  −k  �  =  �  sk tk  tk sk  � � ˆβk  ˆβ†  −k  �  =  �  s′  k t′  k  t′  k s′  k  � � ˆαk  ˆα†  −k  �  =  � ˆak  ˆa†  −k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B16)  This means that Bogoliubov excitations before and after  the quench are connected by  � ˆαk  ˆα†  −k  �  =  �  s′  ksk − t′  ktk s′  ktk − skt′  k  s′  ktk − skt′  k s′  ksk − t′  ktk  � � ˆβk  ˆβ†  −k  �  =:  �  uk vk  vk uk  � � ˆβk  ˆβ†  −k  �  ,  (B17)  where the coefficients satisfy  u2  k − v2  k = s′  k  2 − t′  k  2 = s2  k − t2  k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B18)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Longitudinal correlation functions  We evaluate the time-dependent longitudinal correla-  tion functions defined as  Czz(r, t) = ⟨ψ0|ei ˆ  H′t ˆSz  r ˆSz  0e−i ˆ  H′t|ψ0⟩  (B19)  = S  2 ⟨ψ0|ei ˆ  H′t(ˆb†  r + ˆbr)(ˆb†  0 + ˆb0)e−i ˆ  H′t|ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B20)  Writing  them  in  the  Fourier  space  (ˆbi  =  1  √  LD  �  k e−ik·riˆbk)  and  in  the  Heisenberg  picture,  we obtain  Czz(r, t) =  S  2LD  �  k  eik·r⟨ψ0|[ˆb†  k(t)ˆb†  −k(t) + ˆb†  k(t)ˆbk(t)  + ˆb−k(t)ˆb†  −k(t) + ˆb−k(t)ˆbk(t)]|ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B21)  We then replace all operators ˆbk(t) by ˆβk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Because  ˆbk(t) = s′  k ˆαk(t) + t′  k ˆα†  −k(t), ˆαk(t) = e−iΩ′  ktˆαk, and  ˆαk = uk ˆβk + vk ˆβ†  −k, the following relation holds:  ˆbk(t) = (e−iΩ′  kts′  kuk + e+iΩ′  ktt′  kvk)ˆβk  + (e−iΩ′  kts′  kvk + e+iΩ′  ktt′  kuk)ˆβ†  −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B22)  After straightforward calculations using ˆβk|ψ0⟩ = 0, we  obtain  Czz(r, t) =  S  2LD  �  k  eik·r(s′  k + t′  k)2  (u2  k + v2  k + 2ukvk cos 2Ω′  kt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B23)  Substituting uk and vk with sk, s′  k, tk, and t′  k using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (B17), we get  Czz(r, 0) =  S  2LD  �  k  eik·r(s′  k + t′  k)2(uk + vk)2  (B24)  =  S  2LD  �  k  eik·r(sk + tk)2,  (B25)  ˜Czz(r, t) := Czz(r, t) − Czz(r, 0)  (B26)  =  S  2LD  �  k  eik·r(s′  k + t′  k)22ukvk(cos 2Ω′  kt − 1)  (B27)  =  S  2LD  �  k  eik·r(−2)[(s2  k + t2  k)s′  kt′  k  − (s′  k  2 + t′  k  2)sktk](s′  k + t′  k)2(cos 2Ω′  kt − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B28)  Using the relations defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' (B7-B15), we finally  get [59]  Czz(r, 0) =  S  2LD  �  k  eik·r  Ωk  Ak + Bk  ,  (B29)  ˜Czz(r, t) =  S  2LD  �  k  eik·r AkB′  k − A′  kBk  Ωk(A′  k + B′  k) (cos 2Ω′  kt − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B30)  For Γ → ∞ before the quench, Ωk/(Ak + Bk) = 1 is  satisfied, and hence, Czz(r, 0) = S/2 × δr,Lm (mν: inte-  ger, ν = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , D) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' This means that Czz(r, t) =  ˜Czz(r, t) for 1 ≤ rν ≤ L/2 (ν = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Be-  sides, when J = 0 and J′ ≪ Γ′ < Γ, the intensity  of the correlation would be approximately |Czz(r, t)| =  O[S/LD × �  k B′  k/(A′  k + B′  k)] = O(zS2J′/Γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Transverse correlation functions  We evaluate the time-dependent transverse correlation  functions defined as  Cxx(r, t) = ⟨ψ0|ei ˆ  H′t ˆSx  r ˆSx  0e−i ˆ  H′t|ψ0⟩  (B31)  = ⟨ψ0|ei ˆ  H′t(S − ˆb†  rˆbr)(S − ˆb†  0ˆb0)e−i ˆ  H′t|ψ0⟩  (B32)  and the connected one defined as  Cxx  connected(r, t) = Cxx(r, t) − ⟨ψ0|ei ˆ  H′t ˆSx  re−i ˆ  H′t|ψ0⟩  ⟨ψ0|ei ˆ  H′t ˆSx  0e−i ˆ  H′t|ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B33)  18  Writing  them  in  the  Fourier  space  (ˆbi  =  1  √  LD  �  k e−ik·riˆbk)  and  in  the  Heisenberg  picture,  we obtain  Cxx(r, t) = S2 − 2S  LD  �  k  ⟨ψ0|ˆb†  k(t)ˆbk(t)|ψ0⟩  +  1  L2D  �  k,l,p  ei(k−l)·r⟨ψ0|ˆb†  k(t)ˆbl(t)ˆb†  p(t)ˆbk−l+p(t)|ψ0⟩  (B34)  and  Cxx  connected(r, t) = −  �  S  LD  �  k  ⟨ψ0|ˆb†  k(t)ˆbk(t)|ψ0⟩  �2  +  1  L2D  �  k,l,p  ei(k−l)·r⟨ψ0|ˆb†  k(t)ˆbl(t)ˆb†  p(t)ˆbk−l+p(t)|ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B35)  As in the case of longitudinal correlation functions,  we replace ˆbk(t) by ˆβk and use ˆβk|ψ0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Non-  vanishing terms contain ⟨ψ0|ˆβ−k ˆβ†  −l ˆβ−p ˆβ†  −(k−l+p)|ψ0⟩ =  δk,l, ⟨ψ0|ˆβ−k ˆβl ˆβ†  p ˆβ†  −(k−l+p)|ψ0⟩  =  δk,−p + δl,p, and  ⟨ψ0|ˆβ−k ˆβ†  −k|ψ0⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' After straightforward calculations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  we get  1  LD  �  k  ⟨ψ0|ˆb†  k(t)ˆbk(t)|ψ0⟩  =  1  LD  �  k  (s′  k  2v2  k + t′  k  2u2  k + 2s′  kt′  kukvk cos 2Ω′  kt) (B36)  and  Cxx  connected(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' t) =  �����  1  LD  �  k  eik·r �  [(s′  k  2 + t′  k  2) cos 2Ω′  kt + i sin 2Ω′  kt]ukvk + s′  kt′  k(u2  k + v2  k)  ������  2  +  1  L2D  �  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='l  ei(k−l)·r �  t2  k + 2s′  kt′  kukvk(cos 2Ω′  kt − 1)  � �  s2  l + 2s′  lt′  lulvl(cos 2Ω′  lt − 1)  �  (B37)  =  �����  1  LD  �  k  eik·r �  [(s′  k  2 + t′  k  2) cos 2Ω′  kt + i sin 2Ω′  kt]ukvk + s′  kt′  k(u2  k + v2  k)  ������  2  +  �����  1  LD  �  k  eik·r �  t2  k + 2s′  kt′  kukvk(cos 2Ω′  kt − 1)  �  �����  2  + 1  LD  �  k  �  t2  k + 2s′  kt′  kukvk(cos 2Ω′  kt − 1)  �  × δr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='Lm  (B38)  with mν (ν = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , D) being integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Substituting the  parameters uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' vk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' tk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' s′  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' and t′  k with the parameters  Ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Ωk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' A′  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' B′  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' and Ω′  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' we finally get  Cxx  connected(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' t) =  �����  1  LD  �  k  eik·r  �  −B′  k  2  AkA′  k − BkB′  k  ΩkΩ′  k  2  + AkB′  k − A′  kBk  2ΩkΩ′  k  �A′  k  Ω′  k  cos 2Ω′  kt + i sin 2Ω′  kt  �������  2  +  �����  1  LD  �  k  eik·r  ��  A′  k  2  AkA′  k − BkB′  k  ΩkΩ′  k  2  − 1  2  �  − B′  k  2  AkB′  k − A′  kBk  ΩkΩ′  k  2  cos 2Ω′  kt  ������  2  + 1  LD  �  k  ��  A′  k  2  AkA′  k − BkB′  k  ΩkΩ′  k  2  − 1  2  �  − B′  k  2  AkB′  k − A′  kBk  ΩkΩ′  k  2  cos 2Ω′  kt  �  × δr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='Lm  (B39)  =  �����  1  LD  �  k  eik·r  �  − Bk  2Ωk  + AkB′  k − A′  kBk  2ΩkΩ′  k  �A′  k  Ω′  k  (cos 2Ω′  kt − 1) + i sin 2Ω′  kt  �������  2  +  �����  1  LD  �  k  eik·r  �  1  2  �Ak  Ωk  − 1  �  − B′  k  2  AkB′  k − A′  kBk  ΩkΩ′  k  2  (cos 2Ω′  kt − 1)  ������  2  19  + 1  LD  �  k  �  1  2  �Ak  Ωk  − 1  �  − B′  k  2  AkB′  k − A′  kBk  ΩkΩ′  k  2  (cos 2Ω′  kt − 1)  �  × δr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='Lm  (B40)  with mν (ν = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' , D) being integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  When J = 0 and J′ ≪ Γ′ < Γ, the intensity of the  correlation would be approximately |Cxx  connected(r, t)| =  O[|1/LD × �  k B′  kA′  k/(Ω′  k)2|2] = O(z2S2J′2/Γ′2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Dispersion relation and maximum group velocity  Within the LSWA, the dispersion relation for S = 1/2  is expressed as  Ωk =  �  Γ2 − z  2ΓJγk,  (B41)  = Γ  �  1 − 1  2  zJ  2Γγk − 1  8  �zJ  2Γ  �2  γ2  k + O  ��zJ  2Γ  �3��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B42)  This result is consistent with the dispersion relation  Ωperturb  k  = Γ  �  1 − 1  2  zJ  2Γγk − 1  8  �zJ  2Γ  �2 �  γ2  k − 2  9  �  − 1  16  �zJ  2Γ  �3 �  γ3  k − γk  2 − 3z  z2  �  + O  ��zJ  2Γ  �4��  (B43)  obtained by the perturbation calculation [103] up to  O{[(zJ)/(2Γ)]2} terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  It is widely believed that the Lieb-Robinson bound  should be the maximum group velocity determined from  the derivative of band dispersion [44–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Although the  dispersion obtained by the LSWA does not necessarily  offer the exact Lieb-Robinson bound, we calculate the  reference value using the dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In 1D, the maxi-  mum group velocity of the spin-wave dispersion is given  as  vSW = max  k  ����  dΩk  dk  ���� = J  √  2  �  �1 +  �  1 −  �J  Γ  �2  �  �  −1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  (B44)  For Γ → ∞, the group velocity satisfies vSW → J/2 =  vLR, reproducing the exact maximum group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' On  the other hand, for Γ ∈ (Γclassical  c  , ∞), the LSWA always  gives vSW > J/2 = vLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Its worst (largest) estimate  vSW = J/  √  2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='707J at Γ = Γclassical,1D  c  (= J) is still  tighter than the recent bound 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='51J obtained by the gen-  eral formula for the Lieb-Robinson bound [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  In the same manner, we can extract the group velocity  as vSW = maxk ∇k Ωk from the spin-wave dispersion in  higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' In 2D, the horizontal and diagonal  velocities are given as  vSW,horizontal = J  √  2  �  1 − J  Γ +  �  1 − 2J  Γ  �−1/2  ,  (B45)  vSW,diagonal = J  �  �1 +  �  1 −  �2J  Γ  �2  �  �  −1/2  ,  (B46)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' The maximum velocity along the horizon-  tal (diagonal) axis is estimated to be vSW,horizontal →  J/2 (vSW,diagonal  → J/  √  2) for Γ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  On the  other hand, for both axes, it approaches the value J  (vSW,horizontal, vSW,diagonal → J) for Γ → Γclassical,2D  c  (=  2J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Browaeys and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lahaye, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 16, 132 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Morgado and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Whitlock, AVS Quantum Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 3,  023501 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [3] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Liang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Tian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Tey, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' You, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' B 30, 020305  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Scholl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Schuler, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Williams, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Eberharter,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Barredo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Schymik, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lienhard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Henry,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lahaye, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' L¨auchli, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Browaeys,  Nature 595, 233 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Bluvstein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Omran, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Levine, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Keesling, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Se-  meghini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Ebadi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Michailidis,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Maskara, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Ho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Choi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Serbyn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Greiner,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Vuletic, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lukin, Science 371, 1355 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Ebadi,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Wang,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Levine,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Semeghini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Pichler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Ho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Sachdev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Greiner,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Vuletic, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lukin, Nature 595, 227 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Bernien, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Schwartz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Keesling, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Levine, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Om-  ran, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Pichler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Choi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Zibrov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Endres,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Greiner, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Vuleti´c, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lukin, Nature 551,  579 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Keesling, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Omran, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Levine, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Bernien, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Pich-  ler,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Choi,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Samajdar,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Schwartz,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Silvi,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Sachdev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Zoller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Endres, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Greiner, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Vuleti´c,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lukin, Nature 568, 207 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' de L´es´eleuc, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lienhard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Scholl, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Barredo,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Weber, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' B¨uchler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lahaye, and  20  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Browaeys, Science 365, 775 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Verresen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lukin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Vishwanath, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' X 11, 031005 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Semeghini, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Levine, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Keesling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Ebadi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Bluvstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Verresen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Pichler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Kali-  nowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Samajdar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Omran, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Sachdev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Vish-  wanath, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Greiner, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Vuletic, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lukin, Sci-  ence 374, 1242 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Samajdar, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Ho, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Pichler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lukin, and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Sachdev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content='  [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Samajdar, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Ho, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Christensen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Berg, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Yan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Lahaye,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Browaeys, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Schuler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Goto, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Cheneau, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Asaka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Fukushima, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Schultz, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Mattis, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Fagotti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content='  Cirac, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Phien, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Bengua, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Tuan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Corboz, and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Or´us, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Okunishi, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Nishino, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Hieida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Okunishi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Maeshima,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Hieida, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Akutsu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Okunishi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Xiang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Vidal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Jordan, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Vidal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Vidal, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Troyer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Troyer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Altman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Kollath, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' 2008,  P05018 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Danshita, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Danshita, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Yamashika, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Tsuchiya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Bukov, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content='  [97] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Al-Mohy, Acta Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Rieger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Kunimi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Takasu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Takahashi, and  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Danshita, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Elliott, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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+page_content=' Misawa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfcvxN/content/2301.01407v1.pdf'}
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new file mode 100644
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+arXiv:2301.08388v1  [quant-ph]  20 Jan 2023
+Improving the precision of multiparameter estimation in the teleportation of qutrit
+under amplitude damping noise
+Yan-Ling Li1,∗ Yi-Bo Zeng1, Lin Yao1, and Xing Xiao2†
+1 School of Information Engineering, Jiangxi University of Science and Technology, Ganzhou, China
+2 College of Physics and Electronic Information, Gannan Normal University, Ganzhou, China
+Since the initial discovery of quantum teleportation, it is devoted to transferring unknown quan-
+tum states from one party to another distant partner. However, in the scenarios of remote sensing,
+what people truly care about is the information carried by certain parameters. The problem of
+multiparameter estimation in the framework of qutrit teleportation under amplitude damping (AD)
+noise is studied. Particularly, two schemes are proposed to battle against AD noise and enhance
+the precision of multiparameter estimation by utilizing weak measurement (WM) and environment-
+assisted measurement (EAM). For two-phase parameters encoded in a qutrit state, the analytical
+formulas of the quantum Fisher information matrix (QFIM) can be obtained. The results prove that
+the scheme of EAM outperforms the WM one in the improvements of both independent and simul-
+taneous estimation precision. Remarkably, the EAM scheme can completely ensure the estimation
+precision against the contamination by AD noise. The reason should be attributed to the fact that
+EAM is carried out after the AD noise. Thus, it extracts information from both the system and
+the environment. The findings show that the techniques of WM and EAM are helpful for remote
+quantum sensing and can be generalized to other qutrit-based quantum information tasks under AD
+decoherence.
+Keywords:
+quantum teleportation, multiparameter estimation, weak measurement, environment-assisted
+measurement
+I.
+INTRODUCTION
+Quantum metrology devotes to seeking scenarios where
+quantum resources (entanglement, squeezing and so on)
+can provide enhancements in the parameter estimation
+over the classical strategies.[1–3] The ultimate precision of
+single parameter is determined by the quantum Cram´er-
+Rao bound (CRB).[4] It is well known that single param-
+eter quantum CRB can be always saturated by optimiz-
+ing over all valid quantum measurements. However, most
+high level applications intrinsically involve multiple un-
+known parameters, such as quantum imaging, detection
+of classical electric field, magnetic field and gravitational
+field. All of them fall in the subject of multiparameter es-
+timation. The most important reason for the general in-
+terest of multiparameter estimation is that simultaneous
+quantum estimation of multiple phases can provide bet-
+ter precision than estimating them individually.[5–7] The
+quantum multiparameter CRB characterizes the preci-
+sion of simultaneous estimation. The saturation of mul-
+tiparameter CRB is an open question because different
+parameters usually have different optimal measurements
+which may not commute with each other.
+Except for the saturability problem, it also should be
+noted that the promise of quantum advantages in pa-
+rameter estimation is limited by the presence of noise
+in any realistic experiment, such as dissipation and
+dephasing.[8, 9] Particularly, dissipation plays a crucial
+role in experiments involving photon loss and atomic de-
+cay. The results confirm that it is definitely a reduction
+∗ liyanling0423@gmail.com
+† xiaoxing@gnnu.edu.cn
+of the precision when the noise is considered, but the
+quantum parameter estimation is still possible to surpass
+the classical schemes. In more recent developments, the
+question of how to deal with the noise in quantum param-
+eter estimation was addressed. [10–14] Various approaches
+for protecting parameter precision against noise have
+been proposed, such as quantum error correction,[15–17]
+dynamical decoupling,[18] optimal feedback control,[19]
+correlated effects of the noisy channels[20, 21] and non-
+Markovianity of the environments.[22–24]
+In addition to the strategies mentioned above, weak
+measurement (WM)[25–36] and environment-assited mea-
+surement (EAM)[37–41] are receiving increasing atten-
+tions as new techniques to combat the dissipative noise.
+A WM operation is an extension of the traditional von
+Neumann projective measurement.
+It is reversible be-
+cause the measured system does not totally collapse
+to the eigenstate of the measurement operator. Thus,
+the initial quantum state could be recovered by post
+quantum measurement reversal (QMR) operation with
+a certain probability.
+This merit enables WM to be
+one of the potential candidates for further suppression
+of the noise in quantum information processing. Simi-
+larly, the combination of EAM and QMR also devotes
+to recovering the initial quantum states.
+So far, the
+power of WM and EAM has been demonstrated in pro-
+tecting entanglement,[25, 26, 30] quantum discord[27, 33, 42]
+and other quantum correlations.[43] Particularly, the pro-
+posals for improving the precision of single-parameter
+quantum estimation by the WM and EAM have been
+proposed in recent studies.[31, 41, 44] For multiparameter
+quantum estimation, systematic study of WM and EAM
+to enhance the precision under decoherence is still lack-
+ing.
+
+2
+Although the problem of multiparameter quantum es-
+timation could be illustrated in the single-qubit sys-
+tem in proof of principle (e.g., the simultaneous estima-
+tion of weight and phase parameters), it is more nat-
+urally to estimate multiple phase parameters in higher
+dimensional systems.
+Qutrit-based (and, more gen-
+erally, qudit-based) systems have been confirmed to
+provide significant advantages in the context of quan-
+tum technology, such as quantum communications,[45–47]
+quantum error correction,[48] quantum simulation,[49]
+quantum computation[50] and high-fidelity magic state
+distillation.[51] Moreover, it is of particular interest to
+note that the qutrit-based system can offer an enhanced
+precision of magnetic-field measurement.[52]
+Motivated by the above considerations, we investigate
+the problem of multiple phases estimation in the frame-
+work of qutrit teleportation under the influence of ampli-
+tude damping (AD) noise. We focus on how to improve
+the estimation precision in the qutrit teleportation with
+the assistance of WM and EAM. The explicit expres-
+sion of the quantum Fisher information matrix (QFIM)
+of phase parameters φ1 and φ2 encoded in the teleported
+qutrit is obtained. We shall show that both WM and
+EAM can improve the precisions of individual and si-
+multaneous estimations in a probatilistic way. The per-
+formance of simultaneous estimation is better than in-
+dividual estimation under either WM scheme or EAM
+scheme. Finally, the efficiency analysis proves that the
+EAM scheme outperforms the WM one. Our results pro-
+vide two possible strategies for suppressing the AD deco-
+herence and enhancing the precision of multiple phases
+estimation using qutrit-based quantum states, in which
+the multiparameter nature of the problem leads to an
+intrinsic benefit when exploiting the techniques of WM
+and EAM.
+This paper is organized as follows: In Section II, we
+briefly review the theories of quantum multiparameter
+estimation, AD noise, WM, QMR, and quantum gates
+for qutrit. In Section III, we present the teleportation
+of an unknown qutrit in the presence of AD noise and
+discuss the estimation precision of phase parameters en-
+coded in the teleported qutrit. In order to suppress the
+influence of AD noise, we propose a scheme to improve
+the precision of phase parameters estimation by WM and
+QMR in Section IV. Moreover, in Section V, we exploit
+another improved scheme drawing support from EAM
+and QMR. Finally, a comprehensively comparison of the
+two schemes from the perspective of multiparameter es-
+timation and a brief summary are given in Section VI.
+II.
+PRELIMINARIES
+A.
+Quantum multiparameter estimation
+Assuming that the general case of simultaneously es-
+timating parameters θ = (θ1, . . . , θa, . . . , θb, . . . , θn)T are
+encoded in a d-dimensional density matrix ρ = ρ(θ). Fur-
+ther, let ˆθ = (ˆθ1, . . . , ˆθa, . . . , ˆθb, . . . , ˆθn)T be an estimator
+of θ, and ˆθa is the estimator of θa. For unbiased estima-
+tors, the quantum CRB establishes a lower bound to the
+covariance matrix in terms of the QFIM.[4, 53]
+Cov(ˆθ) ≥ F−1(θ)
+(1)
+where F(θ) is the QFIM. The entries of QFIM for the
+parameters θa and θb with the spectral decomposition
+ρ = �d−1
+i=0 λi |ψi⟩ ⟨ψi| can be written as[7]
+Fab =
+d−1
+�
+i=0
+(∂aλi) (∂bλi)
+λi
++
+�
+i̸=j,λi+λj̸=0
+2 (λi − λj)2
+λi + λj
+Re (⟨ψi | ∂aψj⟩ ⟨∂bψj | ψi⟩)
+(2)
+Notice that the QFI of parameter θa is just the diago-
+nal element of the QFIM. In the independent estimation
+scenario, lower bound on the total variance of all param-
+eters to be estimated in the system can be calculated by
+summing the inverse of diagonal elements of the QFIM[4]
+δind =
+n
+�
+k=1
+F−1
+kk
+(3)
+However, if we want to estimate the parameters si-
+multaneously, a lower bound on the total variance of all
+parameters is determined by the trace of the inverse of
+the QFIM[54]
+δsim = Tr F−1(θ)
+(4)
+To compare the performance of separate and simulta-
+neous estimates, a ratio is defined as[55]
+R =
+δind
+1
+nδsim
+(5)
+in which n denotes the number of parameters to be es-
+timated. The value of R satisfies 0 < R ≤ n and R > 1
+indicates that the performance of simultaneous estima-
+tion is better than that of individual estimation.
+B.
+Amplitude damping noise for qutrit
+The amplitude damping noise is the standard one for
+a dissipative interaction between a quantum system and
+its environment.[56] For example, the model of AD noise
+can be applied to describe the photon loss of an opti-
+cal field, or spontaneous emission of an atomic system
+weakly coupled to a zero temperature environment in
+the Born-Markov approximation. For the qutrit-based
+system, the situation is more complicated since there are
+three different configurations to be considered, i.e., V , Λ
+and Ξ configurations.[57] Here, we will focus on the V -
+configuration. If the environment is in a vacuum state,
+
+3
+the amplitude damping noise of the V -type qutrit can be
+represented by the following Kraus operators[58]
+E0 =
+
+
+1
+0
+0
+0 √1 − d1
+0
+0
+0
+√1 − d2
+
+ , E1 =
+
+
+0 √d1 0
+0
+0
+0
+0
+0
+0
+
+
+E2 =
+
+
+0 0 √d2
+0 0 0
+0 0 0
+
+
+(6)
+where d1 = 1 − e−γ1t and d2 = 1 − e−γ2t. γ1 and γ2
+are the spontaneous emission rates of the upper levels |1⟩
+and |2⟩, respectively.
+C.
+Weak measurement and Quantum measurement
+reversal for qutrit
+The WM considered in this paper is the positive
+operator-valued measure (POVM) or partial-collapse
+measurement originally discussed by Korotkov.[59, 60] In
+the qutrit case, the measurement operators can be ex-
+pressed as[61]
+M0 =
+
+
+1
+0
+0
+0 √1 − p
+0
+0
+0
+√1 − q
+
+ , M1 =
+
+
+0
+0
+0
+0 √p 0
+0
+0
+0
+
+
+M2 =
+
+
+0 0
+0
+0 0
+0
+0 0 √q
+
+
+(7)
+where 0 ⩽ p, q ⩽ 1 describe the strengths of WM. Notice
+that the measurement operators M1 and M2 are equiva-
+lent to the von Neumann projective measurements which
+are irrevocable. Only the measurement operator M0 is a
+WM for the qutrit system. Such a WM could be reversed
+by the operation of QMR which can be written as[61]
+Mr =
+
+
+�
+(1 − pr) (1 − qr)
+0
+0
+0
+√1 − qr
+0
+0
+0
+√1 − pr
+
+ (8)
+where 0 ⩽ pr, qr ⩽ 1 are the strengths of QMR.
+D.
+Quantum Gates for qutrit
+The quantum circuit of qutrit teleportation involves
+some qutrit-based logic gates and operations. Here, we
+introduce the elementary quantum gates or operators for
+3-dimension (3D) systems. The Not gate and phase gate
+for single qutrit are[62, 63]
+Xi|m⟩ = |m ⊕ i⟩
+Zk|m⟩ = ωkm|m⟩
+(9)
+where
+⊕
+denoting
+addition
+module
+3
+and
+ω
+≡
+exp(2πi/3).
+The generalized Hadamard gate of qutrit is
+H =
+1
+√
+3
+2
+�
+m,n=0
+e2πimn/3|m⟩⟨n| =
+1
+√
+3
+
+
+1
+1
+1
+1 e2πi/3 e4πi/3
+1 e4πi/3 e2πi/3
+
+
+(10)
+RC and LC denotes the generalized CNOT Right-Shift
+gate and CNOT Left-Shift gate for two qutrits. They are
+as follows
+RC|m⟩ ⊗ |n⟩ = |m⟩ ⊗ |n ⊕ m⟩
+LC|m⟩ ⊗ |n⟩ = |m⟩ ⊗ |n ⊖ m⟩
+(11)
+with ⊖ denoting subtraction module 3.
+III.
+PRECISION OF PHASE ESTIMATION OF
+THE TELEPORTATED QUTRIT
+The two phase parameters φ1 and φ2 that we try to
+estimate are encoded in a qutrit state |ψ⟩in
+|ψ⟩in = α|0⟩ + βeiφ1|1⟩ + δeiφ2|2⟩
+(12)
+where α, β, δ are all real numbers and α2 + β2 + δ2 = 1.
+Alice wishes to transfer the parameterized phase informa-
+tion to Bob through the procedure of teleportation. For
+this, they should establish the shared entanglement. Sup-
+pose Charlie has prepared a maximally entangled state
+in the form of
+|ψ⟩23 =
+1
+√
+3(|00⟩ + |11⟩ + |22⟩)
+(13)
+Then he distributes qutrits 2 and 3 to Alice and Bob
+through the independent AD channels, as shown in Fig-
+ure 1.
+This represents a typical situation, where the
+shared entangled state is generated and distributed by
+the third party through the AD channel. Then the ini-
+tially entangled state shown in Equation (13) evolves into
+a mixed state in the presence of AD noise.
+ρ1 =
+�
+i,j
+Eijρ0E†
+ij
+(14)
+where ρ0 = |ψ⟩23 ⟨ψ|, Eij = E(2)
+i
+⊗ E(3)
+j
+, i, j = 0, 1, 2.
+For the sake of simplicity, we will present the analytic
+expressions only for d1 = d2 = d = 1 − e−γt because
+the general analytic expressions are too complicated to
+present. After experiencing the AD noise, the non-zero
+elements of ρ1 the shared state in the basis of {|3j + k +
+1⟩ = |j, k⟩} can be written as the following
+ρ11 = 1
+3 + 2d2
+3
+ρ22 = ρ33 = ρ44 = ρ77 = 1
+3d(1 − d)
+ρ55 = ρ99 = ρ59 = ρ∗
+95 = 1
+3(1 − d)2
+ρ15 = ρ∗
+51 = ρ19 = ρ∗
+91 = 1
+3(1 − d)
+(15)
+
+4
+��tr����
+��������
+��������
+����������������
+�����������������
+���������������������
+maximal
+entangled
+state
+��������
+��������
+�����
+�����
+���
+�
+ !"#
+$%&2
+()
+Z*%&1
+FIG. 1. (color online) Schematic illustrations of qutrit-based teleportation under the AD noise.
+According to the circuit of quantum teleportation
+shown in Figure 1, Bob finally obtains the output state
+ρout which can be given by
+ρout =
+�
+�
+A
+3 + (1 − A)α2
+Bαβe−iφ1
+Bαδe−iφ2
+Bαβeiφ1
+A
+3 + (1 − A)β2
+Bβδe−i(φ2−φ1)
+Bαδeiφ2
+Bβδei(φ2−φ1)
+A
+3 + (1 − A)δ2
+�
+�
+(16)
+where A = 2d − 2d2 and B = 1 + 1
+3d2 − 4
+3d. In order to
+calculate the QFIM of the φ1 and φ2, one can diagonalize
+Equation (16) and obtain the non-zero eigenvalues and
+eigenvectors. Choosing α = β = δ =
+1
+√
+3 and substituting
+them into Equation (2), the QFIM for the parameters φ1
+and φ2 can be calculated as
+FAD =
+�
+4
+√
+2
+3
+1
+ζ1
+4
+9
+1
+ζ1
+4
+9
+1
+ζ1
+4
+√
+2
+3
+1
+ζ1
+�
+(17)
+where ζ1 =
+d2−4d+9
+(d2−4d+3)2 .
+Then the total variances of phase parameters φ1 and
+φ2 for independent and simultaneous estimations yield to
+δind
+AD = 3
+√
+2
+4 ζ1
+δsim
+AD = 27
+√
+2
+34 ζ1
+(18)
+It can be found that they are only determined by ζ1 but
+with different scaling factors. Figure 2 shows ζ1 as a
+function of γt.
+It is obvious that the value of ζ1 in-
+creases sharply with the increasing of γt. The increasing
+of variance means that the estimation precision of the
+phase parameters φ1 and φ2 decrease rapidly as the AD
+noise increases. In the following, based on the techniques
+of WM and EAM, we propose two schemes to improve
+the precision of the two-parameter estimation under AD
+noise.
+IV.
+IMPROVING THE PRECISION OF
+PARAMETER ESTIMATION BY WM AND QMR
+In this section, we turn to show how the precision of
+parameter estimation can be improved by the synergistic
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+1
+5
+10
+50
+100
+500
+FIG. 2. (color online) ζ1 as a function of γt.
+action of WM and QMR. In the procedure of entangle-
+ment establishment, two WM operations are performed
+on the entangled qutrits before they pass through the
+AD noise, and finally two QMRs are carried out, re-
+spectively. After these operations, the entangled state
+of Equation (13) becomes
+ρ2 =
+�
+Mr
+�
+ij
+Eij(M0ρ0M†
+0)E†
+ijM†
+r
+�
+/W
+(19)
+where M0 = M (2)
+0
+⊗ M (3)
+0
+and Mr = M (2)
+r
+⊗ M (3)
+r
+.
+W = 1
+3 ¯p2
+r ¯q2
+r
+�
+d2¯p2 + d2¯q2 + 1
+�
++ 2
+3d ¯d¯pr¯qr(¯p2¯qr + ¯pr¯q2) +
+1
+3d
+2(¯p2
+r ¯q2 + ¯q2
+r ¯p2) is the normalization parameter, and
+¯o = 1 − o.
+Assuming that qutrits 2 and 3 are identi-
+cal and both subject to the same measurement strengths
+of WM and QMR. Then the non-zero elements of the
+
+5
+shared entangled state ρ2 are as follows
+ρ11 =
+1
+3W ¯p2
+r¯q2
+r[1 + d2¯p2 + d2¯q2]
+ρ22 = ρ44 =
+1
+3W ¯pr¯q2
+rd ¯d¯p2
+ρ33 = ρ77 =
+1
+3W ¯p2
+r¯qrd ¯d¯q2
+ρ55 =
+1
+3W ¯q2
+rd
+2¯p2
+ρ99 =
+1
+3W ¯p2
+rd
+2¯q2
+ρ15 = ρ∗
+51 =
+1
+3W ¯pr¯q2
+r ¯d¯p
+ρ19 = ρ∗
+91 =
+1
+3W ¯p2
+r¯qr ¯d¯q
+ρ59 = ρ∗
+95 =
+1
+3W ¯pr¯qrd
+2¯p¯q
+(20)
+Through the teleportation procedure shown in Fig-
+ure 1, the output state will be given as
+ρWM
+out =
+�
+�
+C
+3 + (1 − C)α2
+Dαβe−iφ1
+Dαδe−iφ2
+Dαβeiφ1
+C
+3 + (1 − C)β2
+Dβδe−i(φ2−φ1)
+Dαδeiφ2
+Dβδei(φ2−φ1)
+C
+3 + (1 − C)δ2
+�
+�
+(21)
+where C
+=
+d ¯d¯pr¯qr(¯p2¯qr + ¯q2¯pr)/W
+and
+D
+=
+1
+3W ¯d¯pr¯qr(¯pr¯q + ¯d¯p¯q + ¯p¯qr).
+Since we have previously assumed that d1 = d2 = d, it
+is consequently to assume p = q and pr = qr. The QFIM
+for the parameters φ1 and φ2 can be obtained
+FWM =
+�
+4
+√
+2
+3
+1
+ζ2
+4
+9
+1
+ζ2
+4
+9
+1
+ζ2
+4
+√
+2
+3
+1
+ζ2
+�
+(22)
+where ζ2 =
+fh+2h2
+3f 2
+, f =
+1
+3 ¯d¯p¯p2
+r(2¯pr + ¯d¯p) and h =
+1
+3 ¯p2
+r[¯p2
+r(1 + 2d2¯p2) + 4d ¯d¯p2¯pr + 2d
+2¯p2].
+From Equations (3), (4) and (22), the lower bound
+for the independent and simultaneous estimations on the
+total variance of phase parameters φ1 and φ2 can be cal-
+culated as
+δind
+WM = 3
+√
+2
+4 ζ2
+δsim
+WM = 27
+√
+2
+34 ζ2
+(23)
+As we all known, the smaller the variance, the higher
+the precision of multiparameter estimation. Therefore we
+want to minimize ζ2. The minimum ζ2 can be derived
+from the conditions ∂ζ2
+∂pr = 0 and ∂2ζ2
+∂p2r > 0. The result
+turns out to be
+popt
+r,WM =
+1
+2(1 + 2d2¯p2)
+�
+2 + ¯d + d¯p + 2d2¯p2(2 + ¯p ¯d)
+− ¯d¯p
+�
+9 − 4d¯p(1 − d¯p)2(2 − d¯p)
+�
+(24)
+Substituting the optimal QMR strength into Equation
+(23), the optimal ζopt
+2
+can be easily obtained. Here, we
+only show the numerical results, because the general an-
+alytic expression is too complicated to present. In Fig-
+ure 3a we plot the parameter ζopt
+2
+as a function of γt
+for different WM strengths p under the optimal QMR
+strength. Unlike the results in Figure 2 where the to-
+tal variances increases rapidly due to the AD noise (i.e.,
+diverging in the no-operation case), Figure 3a shows
+that WM and QMR can be indeed used for suppress-
+ing the total variances of both independent and simul-
+taneous estimations and hence improving the estimation
+performance. In particular, in the case of γt > 0.5, the
+combined action of WM and QMR operations keeps the
+growth of ζ2 within a small range (i.e, converging in the
+WM cases). This means that this scheme works much
+better in the severe decoherence regime. Meanwhile, ac-
+cording to the results of different WM strengths in Fig-
+ure 3a, it is obvious that the larger the WM strength,
+the higher the estimation precision of the phase param-
+eters φ1 and φ2. When the WM strength p → 1, the
+estimation precision will not be affected by the AD noise.
+The physical mechanism can be understood by two
+steps: Firstly, the pre-WM is performed to reduce the
+ratio of states |1⟩ and |2⟩ and increase the ratio of |0⟩ of
+the qutrit. Since |0⟩ is immune to AD noise, the state
+after WM becomes more robust to AD noise. Secondly,
+WM is a non-completely destructive measurement, the
+state after WM can be reversed by QMR in a probatilis-
+tic way. Therefore, the larger the WM strength p is, the
+better the improvement of estimation precision under AD
+noise, as shown in Figure 3a.
+Due to the fact that both WM and QMR are non-
+unitary operations, such a method is not determinis-
+tic.
+The successful probability could be calculated by
+following the standard postulate of quantum measure-
+ment with two steps.
+Firstly, the probability of ob-
+taining the measurement outcome of M0 = M (2)
+0
+⊗
+M (3)
+0
+is P1 = tr
+�
+M†
+0M0ρ
+�
+.
+Then, the reduced state
+will pass through the AD channel.
+Since the AD
+channel is trace-preserving, it doesn’t change the suc-
+cessful probability.
+Secondly, the probability of ob-
+taining the measurement outcome of QMR is P2 =
+tr
+�
+W†
+totWtot
+��
+ij Eij(M0ρ0M†
+0)E†
+ij
+� �
+. The final prob-
+ability of this scheme depends on WM and QMR per-
+formed successfully in sequence, i.e., PWM = P1P2, which
+exactly equals to the normalization factor W.
+Under
+the condition of Equation (24), the successful probabil-
+ity yields to
+P opt
+WM =(1 − popt
+r,WM)4
+�2
+3d2¯p2 + 1
+3
+�
++ 4
+3(1 − popt
+r,WM)3d ¯d¯p2
++2
+3(1 − popt
+r,WM)2d
+2¯p2
+(25)
+It can be seen that the successful probability decreases
+with the increasing strength of WM, as shown in Fig-
+ure 3b.
+This result implies that the improvement of
+the estimation precision is at the price of reducing the
+probability of success.
+
+6
+no operation
+p=0
+p=0.5
+p=0.9
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+1
+5
+10
+50
+100
+500
+(a)
+p=0
+p=0.5
+p=0.9
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.00000
+0.00005
+0.00010
+0.00015
+0.00020
+FIG. 3. (color online) (a) ζopt
+2
+as a function of γt. (b) P opt
+WM as a function of γt for different values of p.
+V.
+IMPROVING THE PRECISION OF
+PARAMETER ESTIMATION BY EAM AND
+QMR
+In this section, we present another scheme to improve
+the precision of parameter estimation under the AD noise
+by EAM and QMR. The idea is based on the fact that
+some of the Kraus operators Eij of AD noise maybe re-
+versible. From Equation (6), we will find that only E00
+is reversible. Thus, if we can choose the reversible Kraus
+operator E00 during the evolution, then we can reverse
+the impact of AD noise by appropriate QMR. The selec-
+tion of E00 is implemented by monitoring the environ-
+ment coupled with the system. Only when the detection
+of environment is null, then the following QMR opera-
+tion is carried out. The other measurement results with
+clicks are discarded.
+Initially, two qutrits are prepared in the entangled
+state shown in Equation (13) and both the environments
+are in vacuum state. Suppose the outcome of EAM is no
+click, then the entangled state will be mapped to
+ρ3 =
+�
+MrE00ρ0E†
+00M†
+r
+�
+/U
+(26)
+where U = 1
+3(¯p2
+r¯q2
+r + d
+2¯q2
+r + d
+2¯p2
+r) is the normalization
+parameter. The non-zero elements of ρ3 are
+ρ11 = 1
+3U ¯p2
+r ¯q2
+r
+ρ55 = 1
+3U d
+2¯q2
+r
+ρ99 = 1
+3U d
+2¯p2
+r
+ρ15 = ρ∗
+51 =
+1
+3U ¯pr ¯d¯q2
+r
+ρ19 = ρ∗
+91 =
+1
+3U ¯p2
+r ¯d¯qr
+ρ59 = ρ∗
+95 =
+1
+3U ¯prd
+2¯qr
+(27)
+Following the quantum teleportation protocol shown
+in Figure 1, the output state can be obtained
+ρEAM
+out
+=
+
+
+α2
+Gαβe−iφ1
+Gαδe−iφ2
+Gαβeiφ1
+β2
+Gβδe−i(φ2−φ1)
+Gαδeiφ2
+Gβδei(φ2−φ1)
+δ2
+
+(28)
+where G =
+¯d¯pr ¯qr( ¯d+¯pr+¯qr)
+3U
+.
+We still assume that pr = qr, then the QFIM for the
+parameters φ1 and φ2 can be obtained by diagonalizing
+ρEAM
+out , which gives
+FEAM =
+�
+4
+√
+2
+3
+1
+ζ3
+4
+9
+1
+ζ3
+4
+9
+1
+ζ3
+4
+√
+2
+3
+1
+ζ3
+�
+(29)
+where ζ3 = uv+2v2
+3u2
+, u = ¯d( ¯d + 2¯q2
+r) and v = ¯q2
+r + 2d
+2.
+To demonstrate the power of EAM and QMR opera-
+tions on the multiparameter estimation, we calculate the
+lower bound of the total variances of the phase param-
+eters φ1 and φ2 for the independent and simultaneous
+estimations
+δind
+EAM = 3
+√
+2
+4 ζ3
+δsim
+EAM = 27
+√
+2
+34 ζ3
+(30)
+Similarly, the optimal strength qr corresponding to the
+minimal ζ3 can be derived from the conditions ∂ζ3
+∂qr = 0
+and ∂2ζ3
+∂q2r > 0. The result is given by
+qopt
+r,EAM = d
+(31)
+Figure 4a shows the dynamics of parameter ζ3 as a
+function of γt for different QMR strengths qr. It is ob-
+vious that the choose of QMR plays a significant role
+in this scheme. When the QMR is not performed (i.e.,
+qr = 0) or not selected to the optimal situation, ζ3 still di-
+verges after a short time period. In the case of qr = 0.5,
+
+7
+no operation
+qr=0
+qr=0.5
+qr=0.9
+qr=d
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+1
+5
+10
+50
+100
+500
+(a)
+qr=0
+qr=d
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b)
+FIG. 4. (color online) (a) ζ3 as a function of γt. (b) PEAM as a function γt for different values of qr
+.
+we find the lower bound of the total variance will first
+reach the minimal value and then begin to increase after
+passing through the minimum.
+As the QMR strength
+qr increases, the minimum of total variance will appear
+later but the trend of ζ3 seems to be diverged in the long
+time limit. However, by choosing the optimal strength
+of QMR, i.e., qopt
+r,EAM = d, it is intriguing to note that
+the lower bound of the total variance will always be kept
+at the optimal value, thus the improvement of parameter
+estimation precision will be the best at this time. The
+underlying reason is that the introducing of EAM post-
+selects the result of quantum system suffered from the
+decoherence process E00. As we mentioned before, such
+process can be reversed by a proper QMR.
+Considering that both EAM and QMR are non-unitary
+operations, the scheme is also probatilistic. The success-
+ful probability is given by
+PEAM = 1
+3(1 − qr)4 + 2
+3(1 − d)2(1 − qr)2
+(32)
+which reduces to P opt
+EAM = (1 − d)4 when qopt
+r,EAM = d.
+Figure 4b sketches the success probability of the EAM
+scheme as a function of γt. Here, we only choose two
+cases: without QMR qr = 0 and with the optimal QMR
+qr = d, because the other cases are unhelpful. The dot-
+dashed line denotes the success probability P opt
+EAM with
+which the total variance achieves the minimal value. A
+simple comparison between Figure 3b and Figure 4b
+shows that the success probability of EAM scheme is
+higher than that of the WM scheme.
+Particularly, in
+the case of γt = 0.5, the success probability PWM tends
+to be close to 0 while P opt
+EAM is larger than 0.1.
+VI.
+DISCUSSIONS AND CONCLUSIONS
+We have shown that both WM scheme and EAM
+scheme enable to improve the multiparameter estima-
+tion precision by selecting the appropriate measurement
+strength of QMR. However, the probabilistic nature of
+them inspires us to consider the efficiency of them in im-
+proving the precision of multiparameter estimation. To
+ensure an equitable comparison of these two schemes,
+we introduce the difference in an average improvement
+of the precision with both schemes work at the optimal
+measurement strength of QMR.
+∆ =
+1
+ζopt
+3
+× P opt
+EAM −
+1
+ζopt
+2
+× P opt
+WM
+(33)
+The quantity ∆ measures the superiority of EAM scheme
+compared to WM scheme in terms of improving the pre-
+cision of multiparameter estimation.
+The behavior of ∆ as a function of γt and p is shown
+in Figure 5. An interesting finding is that ∆ is always
+larger than 0. Especially in the regime of γt < 0.5, the
+efficiency of EAM scheme is significantly better than that
+of WM scheme. Such an advantage stems from the differ-
+ent order of action between the two types of measurement
+and AD noise. The WM is performed before the qutrit
+suffered AD noise, while the EAM is carried out after the
+AD noise. This means that the EAM not only collects
+the information from the system but also gathers the in-
+formation from the environment. As a consequence, the
+EAM scheme more effectively improves the estimation
+precision in the qutrit teleportation.
+Another interesting problem needs to be clarified is
+whether the superiority of simultaneous estimation still
+holds for these two schemes. For two-parameter estima-
+tion, we have n = 2. According to the Equations (18),
+(23), and (30), the ratio of Equation (5) turns out to
+be R = RWM = REAM =
+17
+9 . The ratio R =
+17
+9 > 1
+indicates that the performance of simultaneous estima-
+tion is always better than that of individual estimation
+regardless of whether WM and EAM are involved or not.
+In summary, we revisited the multiparameter estima-
+tion problem from the perspective of quantum teleporta-
+
+8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. 5. (color online) The difference ∆ versus γt and the WM
+strength p.
+tion, focusing on two phase parameters φ1 and φ2 en-
+coded in a teleported qutrit state.
+We proposed two
+schemes for combating the AD noise and improving the
+estimation precision. The first one is based on WM and
+QMR. We showed that the precisions of independent and
+simultaneous estimations can be improved with the same
+scaling factor. However, the efficiency is highly depen-
+dent on the strength of WM. Only when p → 1, the best
+estimation precision can be achieved but with a very low
+success probability. The second scheme is based on EAM
+and QMR. Crucially, we found that the best estimation
+precision can be maintained by selecting the appropri-
+ate strength of QMR. We adopted a quantity ∆ which
+takes into account both estimation precision and prob-
+ability of success, as figure of merit to account for the
+performance of WM and EAM schemes. As one might
+expect, the EAM scheme works more precisely and ef-
+ficiently because the post-measurement of EAM collects
+the information of system and environment at the same
+time. We also highlighted that simultaneous estimation
+is always advantageous regardless of whether WM and
+EAM are involved or not. Our model realizes the tele-
+portation of multiparameter information for a qutrit sys-
+tem and improves the estimation precision through two
+different schemes. Although both schemes are probabilis-
+tic, we believe that they still provide a new perspective
+of actively battling against decoherence in the scenarios
+remote sensing.
+ACKNOWLEDGMENTS
+This work is supported by Jiangxi Provincial Natural
+Science Foundation under Grant Nos. 20212ACB211004
+and 20212BAB201014. And it is supported by the Funds
+of the National Natural Science Foundation of China un-
+der Grant Nos. 12265004 and 61765007.
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+page_content=' Jiangxi University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Ganzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' China  2 College of Physics and Electronic Information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Gannan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Ganzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' China  Since the initial discovery of quantum teleportation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' it is devoted to transferring unknown quan-  tum states from one party to another distant partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' However, in the scenarios of remote sensing,  what people truly care about is the information carried by certain parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The problem of  multiparameter estimation in the framework of qutrit teleportation under amplitude damping (AD)  noise is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Particularly, two schemes are proposed to battle against AD noise and enhance  the precision of multiparameter estimation by utilizing weak measurement (WM) and environment-  assisted measurement (EAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' For two-phase parameters encoded in a qutrit state, the analytical  formulas of the quantum Fisher information matrix (QFIM) can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The results prove that  the scheme of EAM outperforms the WM one in the improvements of both independent and simul-  taneous estimation precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Remarkably, the EAM scheme can completely ensure the estimation  precision against the contamination by AD noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The reason should be attributed to the fact that  EAM is carried out after the AD noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Thus, it extracts information from both the system and  the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The findings show that the techniques of WM and EAM are helpful for remote  quantum sensing and can be generalized to other qutrit-based quantum information tasks under AD  decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Keywords:  quantum teleportation, multiparameter estimation, weak measurement, environment-assisted  measurement  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  INTRODUCTION  Quantum metrology devotes to seeking scenarios where  quantum resources (entanglement, squeezing and so on)  can provide enhancements in the parameter estimation  over the classical strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [1–3] The ultimate precision of  single parameter is determined by the quantum Cram´er-  Rao bound (CRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [4] It is well known that single param-  eter quantum CRB can be always saturated by optimiz-  ing over all valid quantum measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' However, most  high level applications intrinsically involve multiple un-  known parameters, such as quantum imaging, detection  of classical electric field, magnetic field and gravitational  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' All of them fall in the subject of multiparameter es-  timation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The most important reason for the general in-  terest of multiparameter estimation is that simultaneous  quantum estimation of multiple phases can provide bet-  ter precision than estimating them individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [5–7] The  quantum multiparameter CRB characterizes the preci-  sion of simultaneous estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The saturation of mul-  tiparameter CRB is an open question because different  parameters usually have different optimal measurements  which may not commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Except for the saturability problem, it also should be  noted that the promise of quantum advantages in pa-  rameter estimation is limited by the presence of noise  in any realistic experiment, such as dissipation and  dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [8, 9] Particularly, dissipation plays a crucial  role in experiments involving photon loss and atomic de-  cay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The results confirm that it is definitely a reduction  ∗ liyanling0423@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='com  † xiaoxing@gnnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='cn  of the precision when the noise is considered, but the  quantum parameter estimation is still possible to surpass  the classical schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In more recent developments, the  question of how to deal with the noise in quantum param-  eter estimation was addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [10–14] Various approaches  for protecting parameter precision against noise have  been proposed, such as quantum error correction,[15–17]  dynamical decoupling,[18] optimal feedback control,[19]  correlated effects of the noisy channels[20, 21] and non-  Markovianity of the environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [22–24]  In addition to the strategies mentioned above, weak  measurement (WM)[25–36] and environment-assited mea-  surement (EAM)[37–41] are receiving increasing atten-  tions as new techniques to combat the dissipative noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  A WM operation is an extension of the traditional von  Neumann projective measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  It is reversible be-  cause the measured system does not totally collapse  to the eigenstate of the measurement operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Thus,  the initial quantum state could be recovered by post  quantum measurement reversal (QMR) operation with  a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  This merit enables WM to be  one of the potential candidates for further suppression  of the noise in quantum information processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Simi-  larly, the combination of EAM and QMR also devotes  to recovering the initial quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  So far, the  power of WM and EAM has been demonstrated in pro-  tecting entanglement,[25, 26, 30] quantum discord[27, 33, 42]  and other quantum correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [43] Particularly, the pro-  posals for improving the precision of single-parameter  quantum estimation by the WM and EAM have been  proposed in recent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [31, 41, 44] For multiparameter  quantum estimation, systematic study of WM and EAM  to enhance the precision under decoherence is still lack-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  2  Although the problem of multiparameter quantum es-  timation could be illustrated in the single-qubit sys-  tem in proof of principle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=', the simultaneous estima-  tion of weight and phase parameters), it is more nat-  urally to estimate multiple phase parameters in higher  dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Qutrit-based (and, more gen-  erally, qudit-based) systems have been confirmed to  provide significant advantages in the context of quan-  tum technology, such as quantum communications,[45–47]  quantum error correction,[48] quantum simulation,[49]  quantum computation[50] and high-fidelity magic state  distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [51] Moreover, it is of particular interest to  note that the qutrit-based system can offer an enhanced  precision of magnetic-field measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [52]  Motivated by the above considerations, we investigate  the problem of multiple phases estimation in the frame-  work of qutrit teleportation under the influence of ampli-  tude damping (AD) noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' We focus on how to improve  the estimation precision in the qutrit teleportation with  the assistance of WM and EAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The explicit expres-  sion of the quantum Fisher information matrix (QFIM)  of phase parameters φ1 and φ2 encoded in the teleported  qutrit is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' We shall show that both WM and  EAM can improve the precisions of individual and si-  multaneous estimations in a probatilistic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The per-  formance of simultaneous estimation is better than in-  dividual estimation under either WM scheme or EAM  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Finally, the efficiency analysis proves that the  EAM scheme outperforms the WM one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Our results pro-  vide two possible strategies for suppressing the AD deco-  herence and enhancing the precision of multiple phases  estimation using qutrit-based quantum states, in which  the multiparameter nature of the problem leads to an  intrinsic benefit when exploiting the techniques of WM  and EAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  This paper is organized as follows: In Section II, we  briefly review the theories of quantum multiparameter  estimation, AD noise, WM, QMR, and quantum gates  for qutrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In Section III, we present the teleportation  of an unknown qutrit in the presence of AD noise and  discuss the estimation precision of phase parameters en-  coded in the teleported qutrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In order to suppress the  influence of AD noise, we propose a scheme to improve  the precision of phase parameters estimation by WM and  QMR in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Moreover, in Section V, we exploit  another improved scheme drawing support from EAM  and QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Finally, a comprehensively comparison of the  two schemes from the perspective of multiparameter es-  timation and a brief summary are given in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  PRELIMINARIES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Quantum multiparameter estimation  Assuming that the general case of simultaneously es-  timating parameters θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' , θa, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' , θb, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' , θn)T are  encoded in a d-dimensional density matrix ρ = ρ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Fur-  ther, let ˆθ = (ˆθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' , ˆθa, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' , ˆθb, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' , ˆθn)T be an estimator  of θ, and ˆθa is the estimator of θa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' For unbiased estima-  tors, the quantum CRB establishes a lower bound to the  covariance matrix in terms of the QFIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [4, 53]  Cov(ˆθ) ≥ F−1(θ)  (1)  where F(θ) is the QFIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The entries of QFIM for the  parameters θa and θb with the spectral decomposition  ρ = �d−1  i=0 λi |ψi⟩ ⟨ψi| can be written as[7]  Fab =  d−1  �  i=0  (∂aλi) (∂bλi)  λi  +  �  i̸=j,λi+λj̸=0  2 (λi − λj)2  λi + λj  Re (⟨ψi | ∂aψj⟩ ⟨∂bψj | ψi⟩)  (2)  Notice that the QFI of parameter θa is just the diago-  nal element of the QFIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In the independent estimation  scenario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' lower bound on the total variance of all param-  eters to be estimated in the system can be calculated by  summing the inverse of diagonal elements of the QFIM[4]  δind =  n  �  k=1  F−1  kk  (3)  However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' if we want to estimate the parameters si-  multaneously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' a lower bound on the total variance of all  parameters is determined by the trace of the inverse of  the QFIM[54]  δsim = Tr F−1(θ)  (4)  To compare the performance of separate and simulta-  neous estimates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' a ratio is defined as[55]  R =  δind  1  nδsim  (5)  in which n denotes the number of parameters to be es-  timated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The value of R satisfies 0 < R ≤ n and R > 1  indicates that the performance of simultaneous estima-  tion is better than that of individual estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Amplitude damping noise for qutrit  The amplitude damping noise is the standard one for  a dissipative interaction between a quantum system and  its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [56] For example, the model of AD noise  can be applied to describe the photon loss of an opti-  cal field, or spontaneous emission of an atomic system  weakly coupled to a zero temperature environment in  the Born-Markov approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' For the qutrit-based  system, the situation is more complicated since there are  three different configurations to be considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=', V , Λ  and Ξ configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [57] Here, we will focus on the V -  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' If the environment is in a vacuum state,  3  the amplitude damping noise of the V -type qutrit can be  represented by the following Kraus operators[58]  E0 =  \uf8eb  \uf8ed  1  0  0  0 √1 − d1  0  0  0  √1 − d2  \uf8f6  \uf8f8 , E1 =  \uf8eb  \uf8ed  0 √d1 0  0  0  0  0  0  0  \uf8f6  \uf8f8  E2 =  \uf8eb  \uf8ed  0 0 √d2  0 0 0  0 0 0  \uf8f6  \uf8f8  (6)  where d1 = 1 − e−γ1t and d2 = 1 − e−γ2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' γ1 and γ2  are the spontaneous emission rates of the upper levels |1⟩  and |2⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Weak measurement and Quantum measurement  reversal for qutrit  The WM considered in this paper is the positive  operator-valued measure (POVM) or partial-collapse  measurement originally discussed by Korotkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [59, 60] In  the qutrit case, the measurement operators can be ex-  pressed as[61]  M0 =  \uf8eb  \uf8ed  1  0  0  0 √1 − p  0  0  0  √1 − q  \uf8f6  \uf8f8 , M1 =  \uf8eb  \uf8ed  0  0  0  0 √p 0  0  0  0  \uf8f6  \uf8f8  M2 =  \uf8eb  \uf8ed  0 0  0  0 0  0  0 0 √q  \uf8f6  \uf8f8  (7)  where 0 ⩽ p, q ⩽ 1 describe the strengths of WM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Notice  that the measurement operators M1 and M2 are equiva-  lent to the von Neumann projective measurements which  are irrevocable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Only the measurement operator M0 is a  WM for the qutrit system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Such a WM could be reversed  by the operation of QMR which can be written as[61]  Mr =  \uf8eb  \uf8ed  �  (1 − pr) (1 − qr)  0  0  0  √1 − qr  0  0  0  √1 − pr  \uf8f6  \uf8f8 (8)  where 0 ⩽ pr, qr ⩽ 1 are the strengths of QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Quantum Gates for qutrit  The quantum circuit of qutrit teleportation involves  some qutrit-based logic gates and operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Here, we  introduce the elementary quantum gates or operators for  3-dimension (3D) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The Not gate and phase gate  for single qutrit are[62, 63]  Xi|m⟩ = |m ⊕ i⟩  Zk|m⟩ = ωkm|m⟩  (9)  where  ⊕  denoting  addition  module  3  and  ω  ≡  exp(2πi/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  The generalized Hadamard gate of qutrit is  H =  1  √  3  2  �  m,n=0  e2πimn/3|m⟩⟨n| =  1  √  3  \uf8ee  \uf8f0  1  1  1  1 e2πi/3 e4πi/3  1 e4πi/3 e2πi/3  \uf8f9  \uf8fb  (10)  RC and LC denotes the generalized CNOT Right-Shift  gate and CNOT Left-Shift gate for two qutrits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' They are  as follows  RC|m⟩ ⊗ |n⟩ = |m⟩ ⊗ |n ⊕ m⟩  LC|m⟩ ⊗ |n⟩ = |m⟩ ⊗ |n ⊖ m⟩  (11)  with ⊖ denoting subtraction module 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  PRECISION OF PHASE ESTIMATION OF  THE TELEPORTATED QUTRIT  The two phase parameters φ1 and φ2 that we try to  estimate are encoded in a qutrit state |ψ⟩in  |ψ⟩in = α|0⟩ + βeiφ1|1⟩ + δeiφ2|2⟩  (12)  where α, β, δ are all real numbers and α2 + β2 + δ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Alice wishes to transfer the parameterized phase informa-  tion to Bob through the procedure of teleportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' For  this, they should establish the shared entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Sup-  pose Charlie has prepared a maximally entangled state  in the form of  |ψ⟩23 =  1  √  3(|00⟩ + |11⟩ + |22⟩)  (13)  Then he distributes qutrits 2 and 3 to Alice and Bob  through the independent AD channels, as shown in Fig-  ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  This represents a typical situation, where the  shared entangled state is generated and distributed by  the third party through the AD channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Then the ini-  tially entangled state shown in Equation (13) evolves into  a mixed state in the presence of AD noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  ρ1 =  �  i,j  Eijρ0E†  ij  (14)  where ρ0 = |ψ⟩23 ⟨ψ|, Eij = E(2)  i  ⊗ E(3)  j  , i, j = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  For the sake of simplicity, we will present the analytic  expressions only for d1 = d2 = d = 1 − e−γt because  the general analytic expressions are too complicated to  present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' After experiencing the AD noise, the non-zero  elements of ρ1 the shared state in the basis of {|3j + k +  1⟩ = |j, k⟩} can be written as the following  ρ11 = 1  3 + 2d2  3  ρ22 = ρ33 = ρ44 = ρ77 = 1  3d(1 − d)  ρ55 = ρ99 = ρ59 = ρ∗  95 = 1  3(1 − d)2  ρ15 = ρ∗  51 = ρ19 = ρ∗  91 = 1  3(1 − d)  (15)  4  ��tr����  ��������  ��������  ����������������  �����������������  ���������������������  maximal  entangled  state  ��������  ��������  �����  �����  ���  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' "#  $%&2  ()  Z*%&1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' (color online) Schematic illustrations of qutrit-based teleportation under the AD noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  According to the circuit of quantum teleportation  shown in Figure 1, Bob finally obtains the output state  ρout which can be given by  ρout =  �  �  A  3 + (1 − A)α2  Bαβe−iφ1  Bαδe−iφ2  Bαβeiφ1  A  3 + (1 − A)β2  Bβδe−i(φ2−φ1)  Bαδeiφ2  Bβδei(φ2−φ1)  A  3 + (1 − A)δ2  �  �  (16)  where A = 2d − 2d2 and B = 1 + 1  3d2 − 4  3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In order to  calculate the QFIM of the φ1 and φ2, one can diagonalize  Equation (16) and obtain the non-zero eigenvalues and  eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Choosing α = β = δ =  1  √  3 and substituting  them into Equation (2), the QFIM for the parameters φ1  and φ2 can be calculated as  FAD =  �  4  √  2  3  1  ζ1  4  9  1  ζ1  4  9  1  ζ1  4  √  2  3  1  ζ1  �  (17)  where ζ1 =  d2−4d+9  (d2−4d+3)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Then the total variances of phase parameters φ1 and  φ2 for independent and simultaneous estimations yield to  δind  AD = 3  √  2  4 ζ1  δsim  AD = 27  √  2  34 ζ1  (18)  It can be found that they are only determined by ζ1 but  with different scaling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Figure 2 shows ζ1 as a  function of γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  It is obvious that the value of ζ1 in-  creases sharply with the increasing of γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The increasing  of variance means that the estimation precision of the  phase parameters φ1 and φ2 decrease rapidly as the AD  noise increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In the following, based on the techniques  of WM and EAM, we propose two schemes to improve  the precision of the two-parameter estimation under AD  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  IMPROVING THE PRECISION OF  PARAMETER ESTIMATION BY WM AND QMR  In this section, we turn to show how the precision of  parameter estimation can be improved by the synergistic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  1  5  10  50  100  500  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' (color online) ζ1 as a function of γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  action of WM and QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In the procedure of entangle-  ment establishment, two WM operations are performed  on the entangled qutrits before they pass through the  AD noise, and finally two QMRs are carried out, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' After these operations, the entangled state  of Equation (13) becomes  ρ2 =  �  Mr  �  ij  Eij(M0ρ0M†  0)E†  ijM†  r  �  /W  (19)  where M0 = M (2)  0  ⊗ M (3)  0  and Mr = M (2)  r  ⊗ M (3)  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  W = 1  3 ¯p2  r ¯q2  r  �  d2¯p2 + d2¯q2 + 1  �  + 2  3d ¯d¯pr¯qr(¯p2¯qr + ¯pr¯q2) +  1  3d  2(¯p2  r ¯q2 + ¯q2  r ¯p2) is the normalization parameter, and  ¯o = 1 − o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Assuming that qutrits 2 and 3 are identi-  cal and both subject to the same measurement strengths  of WM and QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Then the non-zero elements of the  5  shared entangled state ρ2 are as follows  ρ11 =  1  3W ¯p2  r¯q2  r[1 + d2¯p2 + d2¯q2]  ρ22 = ρ44 =  1  3W ¯pr¯q2  rd ¯d¯p2  ρ33 = ρ77 =  1  3W ¯p2  r¯qrd ¯d¯q2  ρ55 =  1  3W ¯q2  rd  2¯p2  ρ99 =  1  3W ¯p2  rd  2¯q2  ρ15 = ρ∗  51 =  1  3W ¯pr¯q2  r ¯d¯p  ρ19 = ρ∗  91 =  1  3W ¯p2  r¯qr ¯d¯q  ρ59 = ρ∗  95 =  1  3W ¯pr¯qrd  2¯p¯q  (20)  Through the teleportation procedure shown in Fig-  ure 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' the output state will be given as  ρWM  out =  �  �  C  3 + (1 − C)α2  Dαβe−iφ1  Dαδe−iφ2  Dαβeiφ1  C  3 + (1 − C)β2  Dβδe−i(φ2−φ1)  Dαδeiφ2  Dβδei(φ2−φ1)  C  3 + (1 − C)δ2  �  �  (21)  where C  =  d ¯d¯pr¯qr(¯p2¯qr + ¯q2¯pr)/W  and  D  =  1  3W ¯d¯pr¯qr(¯pr¯q + ¯d¯p¯q + ¯p¯qr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Since we have previously assumed that d1 = d2 = d, it  is consequently to assume p = q and pr = qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The QFIM  for the parameters φ1 and φ2 can be obtained  FWM =  �  4  √  2  3  1  ζ2  4  9  1  ζ2  4  9  1  ζ2  4  √  2  3  1  ζ2  �  (22)  where ζ2 =  fh+2h2  3f 2  , f =  1  3 ¯d¯p¯p2  r(2¯pr + ¯d¯p) and h =  1  3 ¯p2  r[¯p2  r(1 + 2d2¯p2) + 4d ¯d¯p2¯pr + 2d  2¯p2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  From Equations (3), (4) and (22), the lower bound  for the independent and simultaneous estimations on the  total variance of phase parameters φ1 and φ2 can be cal-  culated as  δind  WM = 3  √  2  4 ζ2  δsim  WM = 27  √  2  34 ζ2  (23)  As we all known, the smaller the variance, the higher  the precision of multiparameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Therefore we  want to minimize ζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The minimum ζ2 can be derived  from the conditions ∂ζ2  ∂pr = 0 and ∂2ζ2  ∂p2r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The result  turns out to be  popt  r,WM =  1  2(1 + 2d2¯p2)  �  2 + ¯d + d¯p + 2d2¯p2(2 + ¯p ¯d)  − ¯d¯p  �  9 − 4d¯p(1 − d¯p)2(2 − d¯p)  �  (24)  Substituting the optimal QMR strength into Equation  (23), the optimal ζopt  2  can be easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Here, we  only show the numerical results, because the general an-  alytic expression is too complicated to present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In Fig-  ure 3a we plot the parameter ζopt  2  as a function of γt  for different WM strengths p under the optimal QMR  strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Unlike the results in Figure 2 where the to-  tal variances increases rapidly due to the AD noise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=',  diverging in the no-operation case), Figure 3a shows  that WM and QMR can be indeed used for suppress-  ing the total variances of both independent and simul-  taneous estimations and hence improving the estimation  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In particular, in the case of γt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='5, the  combined action of WM and QMR operations keeps the  growth of ζ2 within a small range (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='e, converging in the  WM cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' This means that this scheme works much  better in the severe decoherence regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Meanwhile, ac-  cording to the results of different WM strengths in Fig-  ure 3a, it is obvious that the larger the WM strength,  the higher the estimation precision of the phase param-  eters φ1 and φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' When the WM strength p → 1, the  estimation precision will not be affected by the AD noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  The physical mechanism can be understood by two  steps: Firstly, the pre-WM is performed to reduce the  ratio of states |1⟩ and |2⟩ and increase the ratio of |0⟩ of  the qutrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Since |0⟩ is immune to AD noise, the state  after WM becomes more robust to AD noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Secondly,  WM is a non-completely destructive measurement, the  state after WM can be reversed by QMR in a probatilis-  tic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Therefore, the larger the WM strength p is, the  better the improvement of estimation precision under AD  noise, as shown in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Due to the fact that both WM and QMR are non-  unitary operations, such a method is not determinis-  tic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  The successful probability could be calculated by  following the standard postulate of quantum measure-  ment with two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Firstly, the probability of ob-  taining the measurement outcome of M0 = M (2)  0  ⊗  M (3)  0  is P1 = tr  �  M†  0M0ρ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Then, the reduced state  will pass through the AD channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Since the AD  channel is trace-preserving, it doesn’t change the suc-  cessful probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Secondly, the probability of ob-  taining the measurement outcome of QMR is P2 =  tr  �  W†  totWtot  ��  ij Eij(M0ρ0M†  0)E†  ij  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The final prob-  ability of this scheme depends on WM and QMR per-  formed successfully in sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=', PWM = P1P2, which  exactly equals to the normalization factor W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Under  the condition of Equation (24), the successful probabil-  ity yields to  P opt  WM =(1 − popt  r,WM)4  �2  3d2¯p2 + 1  3  �  + 4  3(1 − popt  r,WM)3d ¯d¯p2  +2  3(1 − popt  r,WM)2d  2¯p2  (25)  It can be seen that the successful probability decreases  with the increasing strength of WM, as shown in Fig-  ure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  This result implies that the improvement of  the estimation precision is at the price of reducing the  probability of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  6  no operation  p=0  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='5  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='0  1  5  10  50  100  500  (a)  p=0  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='00005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='00020  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' (color online) (a) ζopt  2  as a function of γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' (b) P opt  WM as a function of γt for different values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  IMPROVING THE PRECISION OF  PARAMETER ESTIMATION BY EAM AND  QMR  In this section, we present another scheme to improve  the precision of parameter estimation under the AD noise  by EAM and QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The idea is based on the fact that  some of the Kraus operators Eij of AD noise maybe re-  versible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' From Equation (6), we will find that only E00  is reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Thus, if we can choose the reversible Kraus  operator E00 during the evolution, then we can reverse  the impact of AD noise by appropriate QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The selec-  tion of E00 is implemented by monitoring the environ-  ment coupled with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Only when the detection  of environment is null, then the following QMR opera-  tion is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The other measurement results with  clicks are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Initially, two qutrits are prepared in the entangled  state shown in Equation (13) and both the environments  are in vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Suppose the outcome of EAM is no  click, then the entangled state will be mapped to  ρ3 =  �  MrE00ρ0E†  00M†  r  �  /U  (26)  where U = 1  3(¯p2  r¯q2  r + d  2¯q2  r + d  2¯p2  r) is the normalization  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The non-zero elements of ρ3 are  ρ11 = 1  3U ¯p2  r ¯q2  r  ρ55 = 1  3U d  2¯q2  r  ρ99 = 1  3U d  2¯p2  r  ρ15 = ρ∗  51 =  1  3U ¯pr ¯d¯q2  r  ρ19 = ρ∗  91 =  1  3U ¯p2  r ¯d¯qr  ρ59 = ρ∗  95 =  1  3U ¯prd  2¯qr  (27)  Following the quantum teleportation protocol shown  in Figure 1, the output state can be obtained  ρEAM  out  =  \uf8eb  \uf8ed  α2  Gαβe−iφ1  Gαδe−iφ2  Gαβeiφ1  β2  Gβδe−i(φ2−φ1)  Gαδeiφ2  Gβδei(φ2−φ1)  δ2  \uf8f6  \uf8f8(28)  where G =  ¯d¯pr ¯qr( ¯d+¯pr+¯qr)  3U  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  We still assume that pr = qr, then the QFIM for the  parameters φ1 and φ2 can be obtained by diagonalizing  ρEAM  out , which gives  FEAM =  �  4  √  2  3  1  ζ3  4  9  1  ζ3  4  9  1  ζ3  4  √  2  3  1  ζ3  �  (29)  where ζ3 = uv+2v2  3u2  , u = ¯d( ¯d + 2¯q2  r) and v = ¯q2  r + 2d  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  To demonstrate the power of EAM and QMR opera-  tions on the multiparameter estimation, we calculate the  lower bound of the total variances of the phase param-  eters φ1 and φ2 for the independent and simultaneous  estimations  δind  EAM = 3  √  2  4 ζ3  δsim  EAM = 27  √  2  34 ζ3  (30)  Similarly, the optimal strength qr corresponding to the  minimal ζ3 can be derived from the conditions ∂ζ3  ∂qr = 0  and ∂2ζ3  ∂q2r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The result is given by  qopt  r,EAM = d  (31)  Figure 4a shows the dynamics of parameter ζ3 as a  function of γt for different QMR strengths qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' It is ob-  vious that the choose of QMR plays a significant role  in this scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' When the QMR is not performed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=',  qr = 0) or not selected to the optimal situation, ζ3 still di-  verges after a short time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' In the case of qr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='5,  7  no operation  qr=0  qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='9  qr=d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='0  1  5  10  50  100  500  (a)  qr=0  qr=d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' (color online) (a) ζ3 as a function of γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' (b) PEAM as a function γt for different values of qr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  we find the lower bound of the total variance will first  reach the minimal value and then begin to increase after  passing through the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  As the QMR strength  qr increases, the minimum of total variance will appear  later but the trend of ζ3 seems to be diverged in the long  time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' However, by choosing the optimal strength  of QMR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=', qopt  r,EAM = d, it is intriguing to note that  the lower bound of the total variance will always be kept  at the optimal value, thus the improvement of parameter  estimation precision will be the best at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The  underlying reason is that the introducing of EAM post-  selects the result of quantum system suffered from the  decoherence process E00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' As we mentioned before, such  process can be reversed by a proper QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Considering that both EAM and QMR are non-unitary  operations, the scheme is also probatilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The success-  ful probability is given by  PEAM = 1  3(1 − qr)4 + 2  3(1 − d)2(1 − qr)2  (32)  which reduces to P opt  EAM = (1 − d)4 when qopt  r,EAM = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Figure 4b sketches the success probability of the EAM  scheme as a function of γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Here, we only choose two  cases: without QMR qr = 0 and with the optimal QMR  qr = d, because the other cases are unhelpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The dot-  dashed line denotes the success probability P opt  EAM with  which the total variance achieves the minimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' A  simple comparison between Figure 3b and Figure 4b  shows that the success probability of EAM scheme is  higher than that of the WM scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Particularly, in  the case of γt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='5, the success probability PWM tends  to be close to 0 while P opt  EAM is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  DISCUSSIONS AND CONCLUSIONS  We have shown that both WM scheme and EAM  scheme enable to improve the multiparameter estima-  tion precision by selecting the appropriate measurement  strength of QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' However, the probabilistic nature of  them inspires us to consider the efficiency of them in im-  proving the precision of multiparameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' To  ensure an equitable comparison of these two schemes,  we introduce the difference in an average improvement  of the precision with both schemes work at the optimal  measurement strength of QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  ∆ =  1  ζopt  3  × P opt  EAM −  1  ζopt  2  × P opt  WM  (33)  The quantity ∆ measures the superiority of EAM scheme  compared to WM scheme in terms of improving the pre-  cision of multiparameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  The behavior of ∆ as a function of γt and p is shown  in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' An interesting finding is that ∆ is always  larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Especially in the regime of γt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='5, the  efficiency of EAM scheme is significantly better than that  of WM scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Such an advantage stems from the differ-  ent order of action between the two types of measurement  and AD noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The WM is performed before the qutrit  suffered AD noise, while the EAM is carried out after the  AD noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' This means that the EAM not only collects  the information from the system but also gathers the in-  formation from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' As a consequence, the  EAM scheme more effectively improves the estimation  precision in the qutrit teleportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Another interesting problem needs to be clarified is  whether the superiority of simultaneous estimation still  holds for these two schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' For two-parameter estima-  tion, we have n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' According to the Equations (18),  (23), and (30), the ratio of Equation (5) turns out to  be R = RWM = REAM =  17  9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The ratio R =  17  9 > 1  indicates that the performance of simultaneous estima-  tion is always better than that of individual estimation  regardless of whether WM and EAM are involved or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  In summary, we revisited the multiparameter estima-  tion problem from the perspective of quantum teleporta-  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' (color online) The difference ∆ versus γt and the WM  strength p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  tion, focusing on two phase parameters φ1 and φ2 en-  coded in a teleported qutrit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  We proposed two  schemes for combating the AD noise and improving the  estimation precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The first one is based on WM and  QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' We showed that the precisions of independent and  simultaneous estimations can be improved with the same  scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' However, the efficiency is highly depen-  dent on the strength of WM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Only when p → 1, the best  estimation precision can be achieved but with a very low  success probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The second scheme is based on EAM  and QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Crucially, we found that the best estimation  precision can be maintained by selecting the appropri-  ate strength of QMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' We adopted a quantity ∆ which  takes into account both estimation precision and prob-  ability of success, as figure of merit to account for the  performance of WM and EAM schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' As one might  expect, the EAM scheme works more precisely and ef-  ficiently because the post-measurement of EAM collects  the information of system and environment at the same  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' We also highlighted that simultaneous estimation  is always advantageous regardless of whether WM and  EAM are involved or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Our model realizes the tele-  portation of multiparameter information for a qutrit sys-  tem and improves the estimation precision through two  different schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Although both schemes are probabilis-  tic, we believe that they still provide a new perspective  of actively battling against decoherence in the scenarios  remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work is supported by Jiangxi Provincial Natural  Science Foundation under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 20212ACB211004  and 20212BAB201014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' And it is supported by the Funds  of the National Natural Science Foundation of China un-  der Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 12265004 and 61765007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Datta, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Phys: X  2016, 1, 621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Yuan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='  2020, 63, 230322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 2007, 99, 160502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Yao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' A 2015, 91, 052105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' (Berlin) 2022, 534,  2100412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Lee, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Kim, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='  2012, 8, 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  [26] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Man, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xia, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' An, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' A 2012, 86,  012325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xiao, Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 2013, 12, 3067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Yu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' A 2014, 89, 022318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Yao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xiao, Laser Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' 2016, 15, 3881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  [31] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Yao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' A 2016, 93, 012307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  [32] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xiao, Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  2017, 16, 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' Wang, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' (Amsterdam, Neth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=') 2018, 390, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  [34] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Xie, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Plus  2020, 135, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  [35] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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+page_content=' 2013, 67, 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Note  that the Equation (6) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' [60] is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' The au-  thors have corrected this typo in arxiv: 1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='4692v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  [62] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Fujii, arxiv: 0106018, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content='  [63] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Karimipour, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
+page_content=' Bahraminasab, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE_T4oBgHgl3EQf-xwz/content/2301.08388v1.pdf'}
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